shapes and geometries: metrics, analysis, differential...

Shapes andGeometries

Advances in Design and ControlSIAM’s Advances in Design and Control series consists of texts and monographs dealing with all areas of design and control and their applications. Topics of interest include shape optimization, multidisciplinary design, trajectory optimization, feedback, and optimal control. The series focuses on the mathematical and computational aspects of engineering design and control that are usable in a wide variety of scientific and engineering disciplines.

Editor-in-ChiefRalph C. Smith, North Carolina State University

Editorial BoardAthanasios C. Antoulas, Rice UniversitySiva Banda, Air Force Research LaboratoryBelinda A. Batten, Oregon State UniversityJohn Betts, The Boeing Company (retired)Stephen L. Campbell, North Carolina State University Michel C. Delfour, University of Montreal Max D. Gunzburger, Florida State University J. William Helton, University of California, San Diego Arthur J. Krener, University of California, DavisKirsten Morris, University of WaterlooRichard Murray, California Institute of TechnologyEkkehard Sachs, University of Trier

Series VolumesDelfour, M. C. and Zolésio, J.-P., Shapes and Geometries: Metrics, Analysis, Differential Calculus, and Optimization, Second EditionHovakimyan, Naira and Cao, Chengyu, L1 Adaptive Control Theory: Guaranteed Robustness with Fast AdaptationSpeyer, Jason L. and Jacobson, David H., Primer on Optimal Control TheoryBetts, John T., Practical Methods for Optimal Control and Estimation Using Nonlinear Programming, Second EditionShima, Tal and Rasmussen, Steven, eds., UAV Cooperative Decision and Control: Challenges and Practical ApproachesSpeyer, Jason L. and Chung, Walter H., Stochastic Processes, Estimation, and ControlKrstic, Miroslav and Smyshlyaev, Andrey, Boundary Control of PDEs: A Course on Backstepping DesignsIto, Kazufumi and Kunisch, Karl, Lagrange Multiplier Approach to Variational Problems and ApplicationsXue, Dingyü, Chen, YangQuan, and Atherton, Derek P., Linear Feedback Control: Analysis and Design with MATLABHanson, Floyd B., Applied Stochastic Processes and Control for Jump-Diffusions: Modeling, Analysis, and Computation Michiels, Wim and Niculescu, Silviu-Iulian, Stability and Stabilization of Time-Delay Systems: An Eigenvalue- Based ApproachIoannou, Petros and Fidan, Barıs, Adaptive Control TutorialBhaya, Amit and Kaszkurewicz, Eugenius, Control Perspectives on Numerical Algorithms and Matrix ProblemsRobinett III, Rush D., Wilson, David G., Eisler, G. Richard, and Hurtado, John E., Applied Dynamic Programming for Optimization of Dynamical SystemsHuang, J., Nonlinear Output Regulation: Theory and ApplicationsHaslinger, J. and Mäkinen, R. A. E., Introduction to Shape Optimization: Theory, Approximation, and ComputationAntoulas, Athanasios C., Approximation of Large-Scale Dynamical SystemsGunzburger, Max D., Perspectives in Flow Control and OptimizationDelfour, M. C. and Zolésio, J.-P., Shapes and Geometries: Analysis, Differential Calculus, and OptimizationBetts, John T., Practical Methods for Optimal Control Using Nonlinear ProgrammingEl Ghaoui, Laurent and Niculescu, Silviu-Iulian, eds., Advances in Linear Matrix Inequality Methods in ControlHelton, J. William and James, Matthew R., Extending H∞ Control to Nonlinear Systems: Control of Nonlinear Systems to Achieve Performance Objectives

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Society for Industrial and Applied MathematicsPhiladelphia

Shapes andGeometriesMetrics, Analysis, Differential Calculus, and OptimizationSECOND EDITION

M. C. DelfourUniversité de MontréalMontréal, QuébecCanada

J.-P. ZolésioNational Center for Scientific Research (CNRS) and National Institute for Research in Computer Science and Control (INRIA)Sophia Antipolis France

is a registered trademark.

Copyright © 2011 by the Society for Industrial and Applied Mathematics

10 9 8 7 6 5 4 3 2 1

All rights reserved. Printed in the United States of America. No part of this book may be reproduced, stored, or transmitted in any manner without the written permission of the publisher. For information, write to the Society for Industrial and Applied Mathematics, 3600 Market Street, 6th Floor, Philadelphia, PA 19104-2688 USA.

Trademarked names may be used in this book without the inclusion of a trademark symbol. These names are used in an editorial context only; no infringement of trademark is intended.

The research of the first author was supported by the Canada Council, which initiated the work presented in this book through a Killam Fellowship; the National Sciences and Engineering Research Council of Canada; and the FQRNT program of the Ministère de l’Éducation du Québec.

Library of Congress Cataloging-in-Publication Data Delfour, Michel C., 1943- Shapes and geometries : metrics, analysis, differential calculus, and optimization / M. C. Delfour, J.-P. Zolésio. -- 2nd ed. p. cm. Includes bibliographical references and index. ISBN 978-0-898719-36-9 (hardcover : alk. paper) 1. Shape theory (Topology) I. Zolésio, J.-P. II. Title. QA612.7.D45 2011 514’.24--dc22 2010028846

This book is dedicated to Alice, Jeanne, Jean, and Roger

!

Contents

List of Figures xvii

Preface xix1 Objectives and Scope of the Book . . . . . . . . . . . . . . . . . . . . xix2 Overview of the Second Edition . . . . . . . . . . . . . . . . . . . . . xx3 Intended Audience . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxii4 Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxiii

1 Introduction: Examples, Background, and Perspectives 11 Orientation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1 Geometry as a Variable . . . . . . . . . . . . . . . . . . . . . 11.2 Outline of the Introductory Chapter . . . . . . . . . . . . . . 3

2 A Simple One-Dimensional Example . . . . . . . . . . . . . . . . . . 33 Buckling of Columns . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 Eigenvalue Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 Optimal Triangular Meshing . . . . . . . . . . . . . . . . . . . . . . . 76 Modeling Free Boundary Problems . . . . . . . . . . . . . . . . . . . 10

6.1 Free Interface between Two Materials . . . . . . . . . . . . . 116.2 Minimal Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . 12

7 Design of a Thermal Diffuser . . . . . . . . . . . . . . . . . . . . . . 137.1 Description of the Physical Problem . . . . . . . . . . . . . . 137.2 Statement of the Problem . . . . . . . . . . . . . . . . . . . . 147.3 Reformulation of the Problem . . . . . . . . . . . . . . . . . . 167.4 Scaling of the Problem . . . . . . . . . . . . . . . . . . . . . . 167.5 Design Problem . . . . . . . . . . . . . . . . . . . . . . . . . . 17

8 Design of a Thermal Radiator . . . . . . . . . . . . . . . . . . . . . . 188.1 Statement of the Problem . . . . . . . . . . . . . . . . . . . . 188.2 Scaling of the Problem . . . . . . . . . . . . . . . . . . . . . . 20

9 A Glimpse into Segmentation of Images . . . . . . . . . . . . . . . . 219.1 Automatic Image Processing . . . . . . . . . . . . . . . . . . 219.2 Image Smoothing/Filtering by Convolution and Edge Detectors 22

9.2.1 Construction of the Convolution of I . . . . . . . . 239.2.2 Space-Frequency Uncertainty Relationship . . . . . 239.2.3 Laplacian Detector . . . . . . . . . . . . . . . . . . . 25

vii

viii Contents

9.3 Objective Functions Defined on the Whole Edge . . . . . . . 269.3.1 Eulerian Shape Semiderivative . . . . . . . . . . . . 269.3.2 From Local to Global Conditions on the Edge . . . 27

9.4 Snakes, Geodesic Active Contours, and Level Sets . . . . . . 289.4.1 Objective Functions Defined on the Contours . . . . 289.4.2 Snakes and Geodesic Active Contours . . . . . . . . 289.4.3 Level Set Method . . . . . . . . . . . . . . . . . . . 299.4.4 Velocity Carried by the Normal . . . . . . . . . . . 309.4.5 Extension of the Level Set Equations . . . . . . . . 31

9.5 Objective Function Defined on the Whole Image . . . . . . . 329.5.1 Tikhonov Regularization/Smoothing . . . . . . . . . 329.5.2 Objective Function of Mumford and Shah . . . . . . 329.5.3 Relaxation of the (N − 1)-Hausdorff Measure . . . . 339.5.4 Relaxation to BV-, Hs-, and SBV-Functions . . . . 339.5.5 Cracked Sets and Density Perimeter . . . . . . . . . 35

10 Shapes and Geometries: Background and Perspectives . . . . . . . . 3610.1 Parametrize Geometries by Functions or Functions by

Geometries? . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3610.2 Shape Analysis in Mechanics and Mathematics . . . . . . . . 3910.3 Characteristic Functions: Surface Measure and Geometric

Measure Theory . . . . . . . . . . . . . . . . . . . . . . . . . 4110.4 Distance Functions: Smoothness, Normal, and Curvatures . . 4110.5 Shape Optimization: Compliance Analysis and Sensitivity

Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4310.6 Shape Derivatives . . . . . . . . . . . . . . . . . . . . . . . . 4410.7 Shape Calculus and Tangential Differential Calculus . . . . . 4610.8 Shape Analysis in This Book . . . . . . . . . . . . . . . . . . 46

11 Shapes and Geometries: Second Edition . . . . . . . . . . . . . . . . 4711.1 Geometries Parametrized by Functions . . . . . . . . . . . . . 4811.2 Functions Parametrized by Geometries . . . . . . . . . . . . . 5011.3 Shape Continuity and Optimization . . . . . . . . . . . . . . 5211.4 Derivatives, Shape and Tangential Differential Calculuses, and

Derivatives under State Constraints . . . . . . . . . . . . . . 53

2 Classical Descriptions of Geometries and Their Properties 551 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 552 Notation and Definitions . . . . . . . . . . . . . . . . . . . . . . . . . 56

2.1 Basic Notation . . . . . . . . . . . . . . . . . . . . . . . . . . 562.2 Abelian Group Structures on Subsets of a Fixed Holdall D . 56

2.2.1 First Abelian Group Structure on (P(D),) . . . . 572.2.2 Second Abelian Group Structure on (P(D),) . . . 58

2.3 Connected Space, Path-Connected Space, and GeodesicDistance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

2.4 Bouligand’s Contingent Cone, Dual Cone, and Normal Cone 592.5 Sobolev Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . 60

2.5.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . 60

Contents ix

2.5.2 The Space Wm,p0 (Ω) . . . . . . . . . . . . . . . . . . 61

2.5.3 Embedding of H10 (Ω) into H1

0 (D) . . . . . . . . . . 622.5.4 Projection Operator . . . . . . . . . . . . . . . . . . 63

2.6 Spaces of Continuous and Differentiable Functions . . . . . . 632.6.1 Continuous and Ck Functions . . . . . . . . . . . . 632.6.2 Holder (C0,ℓ) and Lipschitz (C0,1) Continuous

Functions . . . . . . . . . . . . . . . . . . . . . . . . 652.6.3 Embedding Theorem . . . . . . . . . . . . . . . . . 652.6.4 Identity Ck,1(Ω) = W k+1,∞(Ω): From Convex to

Path-Connected Domains via the Geodesic Distance 663 Sets Locally Described by an Homeomorphism or a Diffeomorphism 67

3.1 Sets of Classes Ck and Ck,ℓ . . . . . . . . . . . . . . . . . . . 673.2 Boundary Integral, Canonical Density, and Hausdorff Measures 70

3.2.1 Boundary Integral for Sets of Class C1 . . . . . . . 703.2.2 Integral on Submanifolds . . . . . . . . . . . . . . . 713.2.3 Hausdorff Measures . . . . . . . . . . . . . . . . . . 72

3.3 Fundamental Forms and Principal Curvatures . . . . . . . . . 734 Sets Globally Described by the Level Sets of a Function . . . . . . . 755 Sets Locally Described by the Epigraph of a Function . . . . . . . . 78

5.1 Local C0 Epigraphs, C0 Epigraphs, and Equi-C0 Epigraphsand the Space H of Dominating Functions . . . . . . . . . . . 79

5.2 Local Ck,ℓ-Epigraphs and Holderian/Lipschitzian Sets . . . . 875.3 Local Ck,ℓ-Epigraphs and Sets of Class Ck,ℓ . . . . . . . . . . 895.4 Locally Lipschitzian Sets: Some Examples and Properties . . 92

5.4.1 Examples and Continuous Linear Extensions . . . . 925.4.2 Convex Sets . . . . . . . . . . . . . . . . . . . . . . 935.4.3 Boundary Measure and Integral for Lipschitzian Sets 945.4.4 Geodesic Distance in a Domain and in Its Boundary 975.4.5 Nonhomogeneous Neumann and Dirichlet Problems 100

6 Sets Locally Described by a Geometric Property . . . . . . . . . . . 1016.1 Definitions and Main Results . . . . . . . . . . . . . . . . . . 1026.2 Equivalence of Geometric Segment and C0 Epigraph

Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1046.3 Equivalence of the Uniform Fat Segment and the Equi-C0

Epigraph Properties . . . . . . . . . . . . . . . . . . . . . . . 1096.4 Uniform Cone/Cusp Properties and Holderian/Lipschitzian

Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1136.4.1 Uniform Cone Property and Lipschitzian Sets . . . 1146.4.2 Uniform Cusp Property and Holderian Sets . . . . . 115

6.5 Hausdorff Measure and Dimension of the Boundary . . . . . 116

3 Courant Metrics on Images of a Set 1231 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1232 Generic Constructions of Micheletti . . . . . . . . . . . . . . . . . . . 124

2.1 Space F(Θ) of Transformations of RN . . . . . . . . . . . . . 1242.2 Diffeomorphisms for B(RN,RN) and C∞

0 (RN,RN) . . . . . . 136

x Contents

2.3 Closed Subgroups G . . . . . . . . . . . . . . . . . . . . . . . 1382.4 Courant Metric on the Quotient Group F(Θ)/G . . . . . . . 1402.5 Assumptions for Bk(RN,RN), Ck(RN,RN), and Ck

0 (RN,RN) 1432.5.1 Checking the Assumptions . . . . . . . . . . . . . . 1432.5.2 Perturbations of the Identity and Tangent Space . . 147

2.6 Assumptions for Ck,1(RN,RN) and Ck,10 (RN,RN) . . . . . . 149

2.6.1 Checking the Assumptions . . . . . . . . . . . . . . 1492.6.2 Perturbations of the Identity and Tangent Space . . 151

3 Generalization to All Homeomorphisms and Ck-Diffeomorphisms . . 153

4 Transformations Generated by Velocities 1591 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1592 Metrics on Transformations Generated by Velocities . . . . . . . . . 161

2.1 Subgroup GΘ of Transformations Generated by Velocities . . 1612.2 Complete Metrics on GΘ and Geodesics . . . . . . . . . . . . 1662.3 Constructions of Azencott and Trouve . . . . . . . . . . . . . 169

3 Semiderivatives via Transformations Generated by Velocities . . . . 1703.1 Shape Function . . . . . . . . . . . . . . . . . . . . . . . . . . 1703.2 Gateaux and Hadamard Semiderivatives . . . . . . . . . . . . 1703.3 Examples of Families of Transformations of Domains . . . . . 173

3.3.1 C∞-Domains . . . . . . . . . . . . . . . . . . . . . . 1733.3.2 Ck-Domains . . . . . . . . . . . . . . . . . . . . . . 1753.3.3 Cartesian Graphs . . . . . . . . . . . . . . . . . . . 1763.3.4 Polar Coordinates and Star-Shaped Domains . . . . 1773.3.5 Level Sets . . . . . . . . . . . . . . . . . . . . . . . . 178

4 Unconstrained Families of Domains . . . . . . . . . . . . . . . . . . . 1804.1 Equivalence between Velocities and Transformations . . . . . 1804.2 Perturbations of the Identity . . . . . . . . . . . . . . . . . . 1834.3 Equivalence for Special Families of Velocities . . . . . . . . . 185

5 Constrained Families of Domains . . . . . . . . . . . . . . . . . . . . 1935.1 Equivalence between Velocities and Transformations . . . . . 1935.2 Transformation of Condition (V2D) into a Linear

Constraint . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2006 Continuity of Shape Functions along Velocity Flows . . . . . . . . . 202

5 Metrics via Characteristic Functions 2091 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2092 Abelian Group Structure on Measurable Characteristic Functions . . 210

2.1 Group Structure on Xµ(RN) . . . . . . . . . . . . . . . . . . 2102.2 Measure Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . 2112.3 Complete Metric for Characteristic Functions in

Lp-Topologies . . . . . . . . . . . . . . . . . . . . . . . . . . . 2123 Lebesgue Measurable Characteristic Functions . . . . . . . . . . . . 214

3.1 Strong Topologies and C∞-Approximations . . . . . . . . . . 2143.2 Weak Topologies and Microstructures . . . . . . . . . . . . . 2153.3 Nice or Measure Theoretic Representative . . . . . . . . . . . 220

Contents xi

3.4 The Family of Convex Sets . . . . . . . . . . . . . . . . . . . 2233.5 Sobolev Spaces for Measurable Domains . . . . . . . . . . . . 224

4 Some Compliance Problems with Two Materials . . . . . . . . . . . 2284.1 Transmission Problem and Compliance . . . . . . . . . . . . 2284.2 The Original Problem of Cea and Malanowski . . . . . . . . . 2354.3 Relaxation and Homogenization . . . . . . . . . . . . . . . . 239

5 Buckling of Columns . . . . . . . . . . . . . . . . . . . . . . . . . . . 2406 Caccioppoli or Finite Perimeter Sets . . . . . . . . . . . . . . . . . . 244

6.1 Finite Perimeter Sets . . . . . . . . . . . . . . . . . . . . . . . 2456.2 Decomposition of the Integral along Level Sets . . . . . . . . 2516.3 Domains of Class W ε,p(D), 0 ≤ ε < 1/p, p ≥ 1, and a Cascade

of Complete Metric Spaces . . . . . . . . . . . . . . . . . . . 2526.4 Compactness and Uniform Cone Property . . . . . . . . . . . 254

7 Existence for the Bernoulli Free Boundary Problem . . . . . . . . . . 2587.1 An Example: Elementary Modeling of the Water Wave . . . 2587.2 Existence for a Class of Free Boundary Problems . . . . . . . 2607.3 Weak Solutions of Some Generic Free Boundary Problems . . 262

7.3.1 Problem without Constraint . . . . . . . . . . . . . 2627.3.2 Constraint on the Measure of the Domain Ω . . . . 264

7.4 Weak Existence with Surface Tension . . . . . . . . . . . . . 265

6 Metrics via Distance Functions 2671 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2672 Uniform Metric Topologies . . . . . . . . . . . . . . . . . . . . . . . 268

2.1 Family of Distance Functions Cd(D) . . . . . . . . . . . . . . 2682.2 Pompeiu–Hausdorff Metric on Cd(D) . . . . . . . . . . . . . . 2692.3 Uniform Complementary Metric Topology and Cc

d(D) . . . . 2752.4 Families Cc

d(E;D) and Ccd,loc(E;D) . . . . . . . . . . . . . . . 278

3 Projection, Skeleton, Crack, and Differentiability . . . . . . . . . . . 2794 W 1,p-Metric Topology and Characteristic Functions . . . . . . . . . 292

4.1 Motivations and Main Properties . . . . . . . . . . . . . . . . 2924.2 Weak W 1,p-Topology . . . . . . . . . . . . . . . . . . . . . . . 296

5 Sets of Bounded and Locally Bounded Curvature . . . . . . . . . . . 2995.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 301

6 Reach and Federer’s Sets of Positive Reach . . . . . . . . . . . . . . 3036.1 Definitions and Main Properties . . . . . . . . . . . . . . . . 3036.2 Ck-Submanifolds . . . . . . . . . . . . . . . . . . . . . . . . . 3106.3 A Compact Family of Sets with Uniform Positive Reach . . . 315

7 Approximation by Dilated Sets/Tubular Neighborhoods and CriticalPoints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 316

8 Characterization of Convex Sets . . . . . . . . . . . . . . . . . . . . 3188.1 Convex Sets and Properties of dA . . . . . . . . . . . . . . . . 3188.2 Semiconvexity and BV Character of dA . . . . . . . . . . . . 3208.3 Closed Convex Hull of A and Fenchel Transform of dA . . . . 3228.4 Families of Convex Sets Cd(D), Cc

d(D), Ccd(E;D), and

Ccd,loc(E;D) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323

xii Contents

9 Compactness Theorems for Sets of Bounded Curvature . . . . . . . . 3249.1 Global Conditions in D . . . . . . . . . . . . . . . . . . . . . 3259.2 Local Conditions in Tubular Neighborhoods . . . . . . . . . . 327

7 Metrics via Oriented Distance Functions 3351 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3352 Uniform Metric Topology . . . . . . . . . . . . . . . . . . . . . . . . 337

2.1 The Family of Oriented Distance Functions Cb(D) . . . . . . 3372.2 Uniform Metric Topology . . . . . . . . . . . . . . . . . . . . 339

3 Projection, Skeleton, Crack, and Differentiability . . . . . . . . . . . 3444 W 1,p(D)-Metric Topology and the Family C0

b (D) . . . . . . . . . . . 3494.1 Motivations and Main Properties . . . . . . . . . . . . . . . . 3494.2 Weak W 1,p-Topology . . . . . . . . . . . . . . . . . . . . . . . 352

5 Boundary of Bounded and Locally Bounded Curvature . . . . . . . . 3545.1 Examples and Limit of Tubular Norms as h Goes to Zero . . 355

6 Approximation by Dilated Sets/Tubular Neighborhoods . . . . . . . 3587 Federer’s Sets of Positive Reach . . . . . . . . . . . . . . . . . . . . . 361

7.1 Approximation by Dilated Sets/Tubular Neighborhoods . . . 3617.2 Boundaries with Positive Reach . . . . . . . . . . . . . . . . . 363

8 Boundary Smoothness and Smoothness of bA . . . . . . . . . . . . . 3659 Sobolev or Wm,p Domains . . . . . . . . . . . . . . . . . . . . . . . . 37310 Characterization of Convex and Semiconvex Sets . . . . . . . . . . . 375

10.1 Convex Sets and Convexity of bA . . . . . . . . . . . . . . . . 37510.2 Families of Convex Sets Cb(D), Cb(E;D), and

Cb,loc(E;D) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37910.3 BV Character of bA and Semiconvex Sets . . . . . . . . . . . 380

11 Compactness and Sets of Bounded Curvature . . . . . . . . . . . . . 38111.1 Global Conditions on D . . . . . . . . . . . . . . . . . . . . . 38211.2 Local Conditions in Tubular Neighborhoods . . . . . . . . . . 382

12 Finite Density Perimeter and Compactness . . . . . . . . . . . . . . 38513 Compactness and Uniform Fat Segment Property . . . . . . . . . . . 387

13.1 Main Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 38713.2 Equivalent Conditions on the Local Graph Functions . . . . . 391

14 Compactness under the Uniform Fat Segment Property and a Boundon a Perimeter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39314.1 De Giorgi Perimeter of Caccioppoli Sets . . . . . . . . . . . . 39314.2 Finite Density Perimeter . . . . . . . . . . . . . . . . . . . . . 394

15 The Families of Cracked Sets . . . . . . . . . . . . . . . . . . . . . . 39416 A Variation of the Image Segmentation Problem of Mumford

and Shah . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40016.1 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . 40016.2 Cracked Sets without the Perimeter . . . . . . . . . . . . . . 401

16.2.1 Technical Lemmas . . . . . . . . . . . . . . . . . . . 40116.2.2 Another Compactness Theorem . . . . . . . . . . . 40216.2.3 Proof of Theorem 16.1 . . . . . . . . . . . . . . . . . 402

16.3 Existence of a Cracked Set with Minimum Density Perimeter 405

Contents xiii

16.4 Uniform Bound or Penalization Term in the ObjectiveFunction on the Density Perimeter . . . . . . . . . . . . . . . 407

8 Shape Continuity and Optimization 4091 Introduction and Generic Examples . . . . . . . . . . . . . . . . . . . 409

1.1 First Generic Example . . . . . . . . . . . . . . . . . . . . . . 4111.2 Second Generic Example . . . . . . . . . . . . . . . . . . . . . 4111.3 Third Generic Example . . . . . . . . . . . . . . . . . . . . . 4111.4 Fourth Generic Example . . . . . . . . . . . . . . . . . . . . . 412

2 Upper Semicontinuity and Maximization of the First Eigenvalue . . 4123 Continuity of the Transmission Problem . . . . . . . . . . . . . . . . 4174 Continuity of the Homogeneous Dirichlet Boundary Value Problem . 418

4.1 Classical, Relaxed, and Overrelaxed Problems . . . . . . . . . 4184.2 Classical Dirichlet Boundary Value Problem . . . . . . . . . . 4214.3 Overrelaxed Dirichlet Boundary Value Problem . . . . . . . . 423

4.3.1 Approximation by Transmission Problems . . . . . . 4234.3.2 Continuity with Respect to X(D) in the

Lp(D)-Topology . . . . . . . . . . . . . . . . . . . . 4244.4 Relaxed Dirichlet Boundary Value Problem . . . . . . . . . . 425

5 Continuity of the Homogeneous Neumann Boundary Value Problem 4266 Elements of Capacity Theory . . . . . . . . . . . . . . . . . . . . . . 429

6.1 Definition and Basic Properties . . . . . . . . . . . . . . . . . 4296.2 Quasi-continuous Representative and H1-Functions . . . . . . 4316.3 Transport of Sets of Zero Capacity . . . . . . . . . . . . . . . 432

7 Crack-Free Sets and Some Applications . . . . . . . . . . . . . . . . 4347.1 Definitions and Properties . . . . . . . . . . . . . . . . . . . . 4347.2 Continuity and Optimization over L(D, r,O,λ) . . . . . . . . 437

7.2.1 Continuity of the Classical Homogeneous DirichletBoundary Condition . . . . . . . . . . . . . . . . . . 437

7.2.2 Minimization/Maximization of the FirstEigenvalue . . . . . . . . . . . . . . . . . . . . . . . 438

8 Continuity under Capacity Constraints . . . . . . . . . . . . . . . . . 4409 Compact Families Oc,r(D) and Lc,r(O, D) . . . . . . . . . . . . . . . 447

9.1 Compact Family Oc,r(D) . . . . . . . . . . . . . . . . . . . . 4479.2 Compact Family Lc,r(O, D) and Thick Set Property . . . . . 4509.3 Maximizing the Eigenvalue λA(Ω) . . . . . . . . . . . . . . . 4529.4 State Constrained Minimization Problems . . . . . . . . . . . 4539.5 Examples with a Constraint on the Gradient . . . . . . . . . 454

9 Shape and Tangential Differential Calculuses 4571 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4572 Review of Differentiation in Topological Vector Spaces . . . . . . . . 458

2.1 Definitions of Semiderivatives and Derivatives . . . . . . . . . 4582.2 Derivatives in Normed Vector Spaces . . . . . . . . . . . . . . 4612.3 Locally Lipschitz Functions . . . . . . . . . . . . . . . . . . . 4652.4 Chain Rule for Semiderivatives . . . . . . . . . . . . . . . . . 465

xiv Contents

2.5 Semiderivatives of Convex Functions . . . . . . . . . . . . . . 4672.6 Hadamard Semiderivative and Velocity Method . . . . . . . . 469

3 First-Order Shape Semiderivatives and Derivatives . . . . . . . . . . 4713.1 Eulerian and Hadamard Semiderivatives . . . . . . . . . . . . 4713.2 Hadamard Semidifferentiability and Courant Metric

Continuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4763.3 Perturbations of the Identity and Gateaux and Frechet

Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4763.4 Shape Gradient and Structure Theorem . . . . . . . . . . . . 479

4 Elements of Shape Calculus . . . . . . . . . . . . . . . . . . . . . . . 4824.1 Basic Formula for Domain Integrals . . . . . . . . . . . . . . 4824.2 Basic Formula for Boundary Integrals . . . . . . . . . . . . . 4844.3 Examples of Shape Derivatives . . . . . . . . . . . . . . . . . 486

4.3.1 Volume of Ω and Surface Area of Γ . . . . . . . . . 4864.3.2 H1(Ω)-Norm . . . . . . . . . . . . . . . . . . . . . . 4874.3.3 Normal Derivative . . . . . . . . . . . . . . . . . . . 488

5 Elements of Tangential Calculus . . . . . . . . . . . . . . . . . . . . 4915.1 Intrinsic Definition of the Tangential Gradient . . . . . . . . 4925.2 First-Order Derivatives . . . . . . . . . . . . . . . . . . . . . 4955.3 Second-Order Derivatives . . . . . . . . . . . . . . . . . . . . 4965.4 A Few Useful Formulae and the Chain Rule . . . . . . . . . . 4975.5 The Stokes and Green Formulae . . . . . . . . . . . . . . . . 4985.6 Relation between Tangential and Covariant Derivatives . . . 4985.7 Back to the Example of Section 4.3.3 . . . . . . . . . . . . . . 501

6 Second-Order Semiderivative and Shape Hessian . . . . . . . . . . . 5016.1 Second-Order Derivative of the Domain Integral . . . . . . . 5026.2 Basic Formula for Domain Integrals . . . . . . . . . . . . . . 5046.3 Nonautonomous Case . . . . . . . . . . . . . . . . . . . . . . 5056.4 Autonomous Case . . . . . . . . . . . . . . . . . . . . . . . . 5106.5 Decomposition of d2J(Ω; V (0), W (0)) . . . . . . . . . . . . . 515

10 Shape Gradients under a State Equation Constraint 5191 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5192 Min Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 521

2.1 An Illustrative Example and a Shape Variational Principle . 5212.2 Function Space Parametrization . . . . . . . . . . . . . . . . 5222.3 Differentiability of a Minimum with Respect to a

Parameter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5232.4 Application of the Theorem . . . . . . . . . . . . . . . . . . . 5262.5 Domain and Boundary Integral Expressions of the Shape

Gradient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5303 Buckling of Columns . . . . . . . . . . . . . . . . . . . . . . . . . . . 5324 Eigenvalue Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . 535

4.1 Transport of Hk0 (Ω) by W k,∞-Transformations of RN . . . . 536

4.2 Laplacian and Bi-Laplacian . . . . . . . . . . . . . . . . . . . 5374.3 Linear Elasticity . . . . . . . . . . . . . . . . . . . . . . . . . 546

Contents xv

5 Saddle Point Formulation and Function Space Parametrization . . . 5515.1 An Illustrative Example . . . . . . . . . . . . . . . . . . . . . 5515.2 Saddle Point Formulation . . . . . . . . . . . . . . . . . . . . 5525.3 Function Space Parametrization . . . . . . . . . . . . . . . . 5535.4 Differentiability of a Saddle Point with Respect to a

Parameter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5555.5 Application of the Theorem . . . . . . . . . . . . . . . . . . . 5595.6 Domain and Boundary Expressions for the Shape Gradient . 561

6 Multipliers and Function Space Embedding . . . . . . . . . . . . . . 5626.1 The Nonhomogeneous Dirichlet Problem . . . . . . . . . . . . 5626.2 A Saddle Point Formulation of the State Equation . . . . . . 5636.3 Saddle Point Expression of the Objective Function . . . . . . 5646.4 Verification of the Assumptions of Theorem 5.1 . . . . . . . . 566

Elements of Bibliography 571

Index of Notation 615

Index 619

List of Figures

1.1 Graph of J(a). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.2 Column of height one and cross section area A under the load ℓ. . . 51.3 Triangulation and basis function associated with node Mi. . . . . . . 81.4 Fixed domain D and its partition into Ω1 and Ω2. . . . . . . . . . . 111.5 Heat spreading scheme for high-power solid-state devices. . . . . . . 141.6 (A) Volume Ω and its boundary Σ; (B) Surface A generating Ω;

(C) Surface D generating !Ω. . . . . . . . . . . . . . . . . . . . . . . . 151.7 Volume Ω and its cross section. . . . . . . . . . . . . . . . . . . . . . 191.8 Volume !Ω and its generating surface A. . . . . . . . . . . . . . . . . 201.9 Image I of objects and their segmentation in the frame D. . . . . . . 221.10 Image I containing black curves or cracks in the frame D. . . . . . . 221.11 Example of a two-dimensional strongly cracked set. . . . . . . . . . . 351.12 Example of a surface with facets associated with a ball. . . . . . . . 37

2.1 Diffeomorphism gx from U(x) to B. . . . . . . . . . . . . . . . . . . 682.2 Local epigraph representation (N = 2). . . . . . . . . . . . . . . . . . 792.3 Domain Ω0 and its image T (Ω0) spiraling around the origin. . . . . . 912.4 Domain Ω0 and its image T (Ω0) zigzagging towards the origin. . . . 922.5 Examples of arbitrary and axially symmetrical O around the

direction d = Ax(0, eN ). . . . . . . . . . . . . . . . . . . . . . . . . . 1092.6 The cone x + AxC(λ,ω) in the direction AxeN . . . . . . . . . . . . . 1142.7 Domain Ω for N = 2, 0 < α < 1, e2 = (0, 1), ρ = 1/6, λ = (1/6)α,

h(θ) = θα. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1182.8 f(x) = dC(x)1/2 constructed on the Cantor set C for 2k + 1 = 3. . . 119

4.1 Transport of Ω by the velocity field V . . . . . . . . . . . . . . . . . . 171

5.1 Smiling sun Ω and expressionless sun !Ω. . . . . . . . . . . . . . . . . 2205.2 Disconnected domain Ω = Ω0 ∪ Ω1 ∪ Ω2. . . . . . . . . . . . . . . . . 2275.3 Fixed domain D and its partition into Ω1 and Ω2. . . . . . . . . . . 2285.4 The function f(x, y) = 56 (1− |x|− |y|)6. . . . . . . . . . . . . . . . 2345.5 Optimal distribution and isotherms with k1 = 2 (black) and k2 = 1

(white) for the problem of section 4.1. . . . . . . . . . . . . . . . . . 2355.6 Optimal distribution and isotherms with k1 = 2 (black) and k2 = 1

(white) for the problem of Cea and Malanowski. . . . . . . . . . . . 239

xvii

xviii List of Figures

5.7 The staircase. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 248

6.1 Skeletons Sk (A), Sk (#A), and Sk (∂A) = Sk (A) ∪ Sk (#A). . . . . . 2806.2 Nonuniqueness of the exterior normal. . . . . . . . . . . . . . . . . . 2866.3 Vertical stripes of Example 4.1. . . . . . . . . . . . . . . . . . . . . . 2936.4 ∇dA for Examples 5.1, 5.2, and 5.3. . . . . . . . . . . . . . . . . . . 3016.5 Set of critical points of A. . . . . . . . . . . . . . . . . . . . . . . . . 318

7.1 ∇bA for Examples 5.1, 5.2, and 5.3. . . . . . . . . . . . . . . . . . . . 3567.2 W 1,p-convergence of a sequence of open subsets An : n ≥ 1 of R2

with uniformly bounded density perimeter to a set with emptyinterior. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 387

7.3 Example of a two-dimensional strongly cracked set. . . . . . . . . . . 3967.4 The two-dimensional strongly cracked set of Figure 7.3 in an open

frame D. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4007.5 The two open components Ω1 and Ω2 of the open domain Ω for

N = 2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 406

Preface

1 Objectives and Scope of the BookThe objective of this book is to give a comprehensive presentation of mathematicalconstructions and tools that can be used to study problems where the modeling,optimization, or control variable is no longer a set of parameters or functions butthe shape or the structure of a geometric object. In that context, a good analyticalframework and good modeling techniques must be able to handle the occurrence ofsingular behaviors whenever they are compatible with the mechanics or the physicsof the problems at hand. In some optimization problems, the natural intuitivenotion of a geometric domain undergoes mutations into relaxed entities such asmicrostructures. So the objects under consideration need not be smooth open do-mains, or even sets, as long as they still makes sense mathematically.

This book covers the basic mathematical ideas, constructions, and methodsthat come from different fields of mathematical activities and areas of applicationsthat have often evolved in parallel directions. The scope of research is frighteninglybroad because it touches on areas that include classical geometry, modern partial dif-ferential equations, geometric measure theory, topological groups, and constrainedoptimization, with applications to classical mechanics of continuous media such asfluid mechanics, elasticity theory, fracture theory, modern theories of optimal de-sign, optimal location and shape of geometric objects, free and moving boundaryproblems, and image processing. Innovative modeling or new issues raised in someapplications force a new look at the fundamentals of well-established mathematicalareas such as geometry, to relax basic notions of volume, perimeter, and curvatureor boundary value problems, and to find suitable relaxations of solutions. In thatspirit, Henri Lebesgue was probably a pioneer when he relaxed the intuitive notionof volume to the one of measure on an equivalence class of measurable sets in 1907.He was followed in that endeavor in the early 1950s by the celebrated work of E. DeGiorgi, who used the relaxed notion of perimeter defined on the class of Caccioppolisets to solve Plateau’s problem of minimal surfaces.

The material that is pertinent to the study of geometric objects and the en-tities and functions that are defined on them would necessitate an encyclopedicinvestment to bring together the basic theories and their fields of applications. Thisobjective is obviously beyond the scope of a single book and two authors. The

xix

xx Preface

coverage of this book is more modest. Yet, it contains most of the important fun-damentals at this stage of evolution of this expanding field.

Even if shape analysis and optimization have undergone considerable and im-portant developments on the theoretical and numerical fronts, there are still culturalbarriers between areas of applications and between theories. The whole field is ex-tremely active, and the best is yet to come with fundamental structures and toolsbeginning to emerge. It is hoped that this book will help to build new bridges andstimulate cross-fertilization of ideas and methods.

2 Overview of the Second EditionThe second edition is almost a new book. All chapters from the first edition havebeen updated and, in most cases, considerably enriched with new material. Manychapters or parts of chapters have been completely rewritten following the devel-opments in the field over the past 10 years. The book went from 9 to 10 chapterswith a more elaborate sectioning of each chapter in order to produce a much moredetailed table of contents. This makes it easier to find specific material.

A series of illustrative generic examples has been added right at the begin-ning of the introductory Chapter 1 to motivate the reader and illustrate the basicdilemma: parametrize geometries by functions or functions by geometries? This isfollowed by the big picture: a section on background and perspectives and a moredetailed presentation of the second edition.

The former Chapter 2 has been split into Chapter 2 on the classical descrip-tions and properties of domains and sets and a new Chapter 3, where the importantmaterial on Courant metrics and the generic constructions of A. M. Micheletti havebeen reorganized and expanded. Basic definitions and material have been addedand regrouped at the beginning of Chapter 2: Abelian group structure on subsetsof a set, connected and path-connected spaces, function spaces, tangent and dualcones, and geodesic distance. The coverage of domains that verify some segmentproperty and have a local epigraph representation has been considerably expanded,and Lipschitzian (graph) domains are now dealt with as a special case.

The new Chapter 3 on domains and submanifolds that are the image of afixed set considerably expands the material of the first edition by bringing up thegeneral assumptions behind the generic constructions of A. M. Micheletti that leadto the Courant metrics on the quotient space of families of transformations bysubgroups of isometries such as identities, rotations, translations, or flips. Thegeneral results apply to a broad range of groups of transformations of the Euclideanspace and to arbitrary closed subgroups. New complete metrics on the whole spacesof homeomorphisms and Ck-diffeomorphisms are also introduced to extend classicalresults for transformations of compact manifolds to general unbounded closed setsand open sets that are crack-free. This material is central in classical mechanicsand physics and in modern applications such as imaging and detection.

The former Chapter 7 on transformations versus flows of velocities has beenmoved right after the Courant metrics as Chapter 4 and considerably expanded.It now specializes the results of Chapter 3 to spaces of transformations that are

2. Overview of the Second Edition xxi

generated by the flow of a velocity field over a generic time interval. One importantmotivation is to introduce a notion of semiderivatives as well as a tractable criterionfor continuity with respect to Courant metrics. Another motivation for the velocitypoint of view is the general framework of R. Azencott and A. Trouve starting in1994 with applications in imaging. They construct complete metrics in relationwith geodesic paths in spaces of diffeomorphisms generated by a velocity field.

The former Chapter 3 on the relaxation to measurable sets and Chapters 4 and5 on distance and oriented distance functions have become Chapters 5, 6, and 7.Those chapters have been renamed Metrics Generated by . . . in order to emphasizeone of the main thrusts of the book: the construction of complete metrics on shapesand geometries.1 Those chapters emphasize the function analytic description of setsand domains: construction of metric topologies and characterization of compactfamilies of sets or submanifolds in the Euclidean space. In that context, we are nowdealing with equivalence classes of sets that may or may not have an invariant openor closed representative in the class. For instance, they include Lebesgue measurablesets and Federer’s sets of positive reach. Many of the classical properties of sets canbe recovered from the smoothness or function analytic properties of those functions.

The former Chapter 6 on optimization of shape functions has been completelyrewritten and expanded as Chapter 8 on shape continuity and optimization. Withmeaningful metric topologies, we can now speak of continuity of a geometric objec-tive functional such as the volume, the perimeter, the mean curvature, etc., compactfamilies of sets, and existence of optimal geometries. The chapter concentrates oncontinuity issues related to shape optimization problems under state equation con-straints. A special family of state constrained problems are the ones for which theobjective function is defined as an infimum over a family of functions over a fixeddomain or set such as the eigenvalue problems. We first characterize the continuityof the transmission problem and the upper semicontinuity of the first eigenvalue ofthe generalized Laplacian with respect to the domain. We then study the conti-nuity of the solution of the homogeneous Dirichlet and Neumann boundary valueproblems with respect to their underlying domain of definition since they requiredifferent constructions and topologies that are generic of the two types of boundaryconditions even for more complex nonlinear partial differential equations. An intro-duction is also given to the concepts and results from capacity theory from whichvery general families of sets stable with respect to boundary conditions can be con-structed. Note that some material has been moved from one chapter to another.For instance, section 7 on the continuity of the Dirichlet boundary problem in theformer Chapter 3 has been merged with the content of the former Chapter 4 in thenew Chapter 8.

The former Chapters 8 and 9 have become Chapters 9 and 10. They aredevoted to a modern version of the shape calculus, an introduction to the tangentialdifferential calculus, and the shape derivatives under a state equation constraint. InChapters 3, 5, 6, and 7, we have constructed complete metric spaces of geometries.Those spaces are nonlinear and nonconvex. However, several of them have a group

1This is in line with current trends in the literature such as in the work of the 2009 Abel Prizewinner M. Gromov [1] and its applications in imaging by G. Sapiro [1] and F. Memoli and G.Sapiro [1] to identify objects up to an isometry.

xxii Preface

structure and, in some cases, it is possible to construct C1-paths in the groupfrom velocity fields. This leads to the notion of Eulerian semiderivative that issomehow the analogue of a derivative on a smooth manifold. In fact, two typesof semiderivatives are of interest: the weaker Gateaux style semiderivative and thestronger Hadamard style semiderivative. In the latter case, the classical chain ruleis still available even for nondifferentiable functions. In order to prepare the groundfor shape derivatives, an enriched self-contained review of the pertinent material onsemiderivatives and derivatives in topological vector spaces is provided.

The important Chapter 10 concentrates on two generic examples often encoun-tered in shape optimization. The first one is associated with the so-called complianceproblems, where the shape functional is itself the minimum of a domain-dependentenergy functional. The special feature of such functionals is that the adjoint statecoincides with the state. This obviously leads to considerable simplifications in theanalysis. In that case, it will be shown that theorems on the differentiability ofthe minimum of a functional with respect to a real parameter readily give explicitexpressions of the Eulerian semiderivative even when the minimizer is not unique.The second one will deal with shape functionals that can be expressed as the saddlepoint of some appropriate Lagrangian. As in the first example, theorems on thedifferentiability of the saddle point of a functional with respect to a real parameterreadily give explicit expressions of the Eulerian semiderivative even when the so-lution of the saddle point equations is not unique. Avoiding the differentiation ofthe state equation with respect to the domain is particularly advantageous in shapeproblems.

3 Intended AudienceThe targeted audience is applied mathematicians and advanced engineers and sci-entists, but the book is also suitable for a broader audience of mathematicians as arelatively well-structured initiation to shape analysis and calculus techniques. Someof the chapters are fairly self-contained and of independent interest. They can beused as lecture notes for a mini-course. The material at the beginning of eachchapter is accessible to a broad audience, while the latter sections may sometimesrequire more mathematical maturity. Thus the book can be used as a graduate textas well as a reference book. It complements existing books that emphasize specificmechanical or engineering applications or numerical methods. It can be considereda companion to the book of J. Soko!lowski and J.-P. Zolesio [9], Introductionto Shape Optimization, published in 1992.

Earlier versions of parts of this book have been used as lecture notes in grad-uate courses at the Universite de Montreal in 1986–1987, 1993–1994, 1995–1996,and 1997–1998 and at international meetings, workshops, or schools: Seminaire deMathematiques Superieures on Shape Optimization and Free Boundaries (Montreal,Canada, June 25 to July 13, 1990), short course on Shape Sensitivity Analysis(Kenitra, Morocco, December 1993), course of the COMETT MATARI EuropeanProgram on Shape Optimization and Mutational Equations (Sophia-Antipolis,France, September 27 to October 1, 1993), CRM Summer School on Boundaries,

4. Acknowledgments xxiii

Interfaces and Transitions (Banff, Canada, August 6–18, 1995), and CIME courseon Optimal Design (Troia, Portugal, June 1998).

4 AcknowledgmentsThe first author is pleased to acknowledge the support of the Canada Council, whichinitiated the work presented in this book through a Killam Fellowship; the constantsupport of the National Sciences and Engineering Research Council of Canada;and the FQRNT program of the Ministere de l’Education du Quebec. Manythanks also to Louise Letendre and Andre Montpetit of the Centre de RecherchesMathematiques, who provided their technical support, experience, and talent overthe extended period of gestation of this book.

Michel DelfourJean-Paul Zolesio

August 13, 2009

Chapter 1

Introduction: Examples,Background, andPerspectives

1 Orientation1.1 Geometry as a VariableThe central object of this book1 is the geometry as a variable. As in the theoryof functions of real variables, we need a differential calculus, spaces of geometries,evolution equations, and other familiar concepts in analysis when the variable isno longer a scalar, a vector, or a function, but is a geometric domain. This ismotivated by many important problems in science and engineering that involve thegeometry as a modeling, design, or control variable. In general the geometric objectswe shall consider will not be parametrized or structured. Yet we are not startingfrom scratch, and several building blocks are already available from many fields:geometric measure theory, physics of continuous media, free boundary problems,the parametrization of geometries by functions, the set derivative as the inverse ofthe integral, the parametrization of functions by geometries, the Pompeiu–Hausdorffmetric, and so on.

As is often the case in mathematics, spaces of geometries and notions of de-rivatives with respect to the geometry are built from well-established elements offunctional analysis and differential calculus. There are many ways to structurefamilies of geometries. For instance, a domain can be made variable by considering

1The numbering of equations, theorems, lemmas, corollaries, definitions, examples, and remarksis by chapter. When a reference to another chapter is necessary it is always followed by the wordsin Chapter and the number of the chapter. For instance, “equation (2.18) in Chapter 9.” Thetext of theorems, lemmas, and corollaries is slanted; the text of definitions, examples, and remarksis normal shape and ended by a square . This makes it possible to aesthetically emphasizecertain words especially in definitions. The bibliography is by author in alphabetical order. Foreach author or group of coauthors, there is a numbering in square brackets starting with [1]. Areference to an item by a single author is of the form J. Dieudonne [3] and a reference to anitem with several coauthors S. Agmon, A. Douglis, and L. Nirenberg [2]. Boxed formulae orstatements are used in some chapters for two distinct purposes. First, they emphasize certainimportant definitions, results, or identities; second, in long proofs of some theorems, lemmas, orcorollaries, they isolate key intermediary results which will be necessary to more easily follow thesubsequent steps of the proof.

1

2 Chapter 1. Introduction: Examples, Background, and Perspectives

the images of a fixed domain by a family of diffeomorphisms that belong to somefunction space over a fixed domain. This naturally occurs in physics and mechan-ics, where the deformations of a continuous body or medium are smooth, or in thenumerical analysis of optimal design problems when working on a fixed grid. Thisconstruction naturally leads to a group structure induced by the composition ofthe diffeomorphisms. The underlying spaces are no longer topological vector spacesbut groups that can be endowed with a nice complete metric space structure byintroducing the Courant metric. The practitioner might or might not want to usethe underlying mathematical structure associated with his or her constructions, butit is there and it contains information that might guide the theory and influencethe choice of the numerical methods used in the solution of the problem at hand.

The parametrization of a fixed domain by a fixed family of diffeomorphismsobviously limits the family of variable domains. The topology of the images is simi-lar to the topology of the fixed domain. Singularities that were not already presentthere cannot be created in the images. Other constructions make it possible to con-siderably enlarge the family of variable geometries and possibly open the doors topathological geometries that are no longer open sets with a nice boundary. Insteadof parametrizing the domains by functions or diffeomorphisms, certain families offunctions can be parametrized by sets. A single function completely specifies a setor at least an equivalence class of sets. This includes the distance functions and thecharacteristic function, but also the support function from convex analysis. Per-haps the best known example of that construction is the Pompeiu–Hausdorff metrictopology. This is a very weak topology that does not preserve the volume of a set.When the volume, the perimeter, or the curvatures are important, such functionsmust be able to yield relaxed definitions of volume, perimeter, or curvatures. Thecharacteristic function that preserves the volume has many applications. It playeda fundamental role in the integration theory of Henri Lebesgue at the beginning ofthe 20th century. It was also used in the 1950s by E. De Giorgi to define a relaxednotion of perimeter in the theory of minimal surfaces.

Another technique that has been used successfully in free or moving boundaryproblems, such as motion by mean curvature, shock waves, or detonation theory,is the use of level sets of a function to describe a free or moving boundary. Suchfunctions are often the solution of a system of partial differential equations. This isanother way to build new tools from functional analysis. The choice of familiesof function parametrized sets or of families of set parametrized functions, or otherappropriate constructions, is obviously problem dependent, much like the choiceof function spaces of solutions in the theory of partial differential equations oroptimization problems. This is one aspect of the geometry as a variable. Anotheraspect is to build the equivalent of a differential calculus and the computationaland analytical tools that are essential in the characterization and computation ofgeometries. Again, we are not starting from scratch and many building blocks arealready available, but many questions and issues remain open.

This book aims at covering a small but fundamental part of that program. Wehad to make difficult choices and refer the reader to appropriate books and referencesfor background material such as geometric measure theory and specialized topicssuch as homogenization theory and microstructures which are available in excellent

2. A Simple One-Dimensional Example 3

books in English. It was unfortunately not possible to include references to theconsiderable literature on numerical methods, free and moving boundary problems,and optimization.

1.2 Outline of the Introductory ChapterWe first give a series of generic examples where the shape or the geometry is themodeling, control, or optimization variable. They will be used in the subsequentchapters to illustrate the many ways such problems can be formulated. The firstexample is the celebrated problem of the optimal shape of a column formulated byLagrange in 1770 to prevent buckling. The extremization of the eigenvalues hasalso received considerable attention in the engineering literature. The free inter-face between two regions with different physical or mechanical properties is anothergeneric problem that can lead in some cases to a mixing or a microstructure. Twotypical problems arising from applications to condition the thermal environment ofsatellites are described in sections 7 and 8. The first one is the design of a thermaldiffuser of minimal weight subject to an inequality constraint on the output thermalpower flux. The second one is the design of a thermal radiator to effectively radiatelarge amounts of thermal power to space. The geometry is a volume of revolutionaround an axis that is completely specified by its height and the function which spec-ifies its lateral boundary. Finally, we give a glimpse at image segmentation, whichis an example of shape/geometric identification problems. Many chapters of thisbook are of direct interest to imaging sciences.

Section 10 presents some background and perspectives. A fundamental issue isto find tractable and preferably analytical representations of a geometry as a variablethat are compatible with the problems at hand. The generic examples suggest twotypes of representations: the ones where the geometry is parametrized by functionsand the ones where a family of functions is parametrized by the geometry. As isalways the case, the choice is very much problem dependent. In the first case, thetopology of the variable sets is fixed; in the second case the families of sets are muchlarger and topological changes are included. The book presents the two points ofview. Finally, section 11 sketches the material in the second edition of the book.

2 A Simple One-Dimensional ExampleA general feature of minimization problems with respect to a shape or a geometrysubject to a state equation constraint is that they are generally not convex andthat, when they have a solution, it is generally not unique. This is illustrated inthe following simple example from J. Cea [2]: minimize the objective function

J(a) def=∫ a

0|ya(x)− 1|2 dx,

where a ≥ 0 and ya is the solution of the boundary value problem (state equation)

d2ya

dx2 (x) = −2 in Ωadef= (0, a),

dya

dx(0) = 0, ya(a) = 0. (2.1)


Here the one-dimensional geometric domain Ωa = ]0, a[ is the minimizing variable.We recognize the classical structure of a control problem, except that the minimizingvariable is no longer under the integral sign but in the limits of the integral sign.One consequence of this difference is that even the simplest problems will usuallynot be convex or convexifiables. They will require a special analysis.

In this example it is easy to check that the solution of the state equation is

ya(x) = a2 − x2 and J(a) =815

a5 − 43a3 + a.

The graph of J , shown in Figure 1.1, is not the graph of a convex function. Itsglobal minimum in a0 = 0, local maximum in a1, and local minimum in a2,

a1def=

√34

(1− 1√

3

), a2

def=

√34

(1 +

1√3

),

are all different.

a1 a2

Figure 1.1. Graph of J(a).

To avoid a trivial solution, a strictly positive lower bound must be put on a.A unique minimizing solution is obtained for a ≥ a1 where the gradient of J is zero.For 0 < a < a2, the minimum will occur at the preset lower or upper bound on a.

3 Buckling of ColumnsThe next example illustrates the fact that even simple problems can be nondiffer-entiable with respect to the geometry. This is generic of all eigenvalue problemswhen the eigenvalue is not simple.

One of the early optimal design problems was formulated by J. L. Lagrange[1] in 1770 (cf. I. Todhunter and K. Pearson [1]) and later studied by theDanish mathematician and astronomer T. Clausen [1] in 1849. It consists infinding the best profile of a vertical column of fixed volume to prevent buckling.

3. Buckling of Columns 5

It turns out that this problem is in fact a hidden maximization of an eigenvalue.Many incorrect solutions had been published until 1992. This problem and otherproblems related to columns have been revisited in a series of papers by S. J.Cox [1], S. J. Cox and M. L. Overton [1], S. J. Cox [2], and S. J. Cox andC. M. McCarthy [1]. Since Lagrange many authors have proposed solutions, buta complete theoretical and numerical solution for the buckling of a column was givenonly in 1992 by S. J. Cox and M. L. Overton [1]. The difficulty was that theeigenvalue is not simple and hence not differentiable with respect to the geometry.

Consider a normalized column of unit height and unit volume (see Figure 1.2).Denote by ℓ the magnitude of the normalized axial load and by u the resultingtransverse displacement. Assume that the potential energy is the sum of the bendingand elongation energies

∫ 1

0EI |u′′|2 dx− ℓ

∫ 1

0|u′|2 dx,

where I is the second moment of area of the column’s cross section and E is itsYoung’s modulus. For sufficiently small load ℓ the minimum of this potential energywith respect to all admissible u is zero. Euler’s buckling load λ of the column is thelargest ℓ for which this minimum is zero. This is equivalent to finding the followingminimum:

λdef= inf

0 =u∈V

∫ 10 EI |u′′|2 dx∫ 10 |u′|2 dx

, (3.1)

where V = H20 (0, 1) corresponds to the clamped case, but other types of bound-

ary conditions can be contemplated. This is an eigenvalue problem with a specialRayleigh quotient.

Assume that E is constant and that the second moment of area I(x) of thecolumn’s cross section at the height x, 0 ≤ x ≤ 1, is equal to a constant c times its

0

x

1

normalized loadℓ

cross section areaA(x)

Figure 1.2. Column of height one and cross section area A under the load ℓ.


cross-sectional area A(x),

I(x) = c A(x) and∫ 1

0A(x) dx = 1.

Normalizing λ by cE and taking into account the engineering constraints

∃0 < A0 < A1, ∀x ∈ [0, 1], 0 < A0 ≤ A(x) ≤ A1,

we finally get

supA∈A

λ(A), λ(A) def= inf0 =u∈V

∫ 10 A |u′′|2 dx∫ 10 |u′|2 dx

, (3.2)

A def=

A ∈ L2(0, 1) : A0 ≤ A ≤ A1 and∫ 1

0A(x) dx = 1

. (3.3)

4 Eigenvalue ProblemsLet D be a bounded open Lipschitzian domain in RN and A ∈ L∞(D;L(RN,RN))be a matrix function defined on D such that

∗A = A and αI ≤ A ≤ βI (4.1)

for some coercivity and continuity constants 0 < α ≤ β and ∗A is the transpose ofA. Consider the minimization or the maximization of the first eigenvalue

supΩ∈A(D)

λA(Ω)

infΩ∈A(D)

λA(Ω)

∣∣∣∣∣∣∣λA(Ω) def= inf

0 =ϕ∈H10 (Ω)

∫Ω A∇ϕ ·∇ϕ dx∫

Ω |ϕ|2 dx, (4.2)

where A(D) is a family of admissible open subsets of D (cf., for instance, sections 2,7, and 9 of Chapter 8).

In the vectorial case, consider the following linear elasticity problem: findU ∈ H1

0 (Ω)3 such that

∀W ∈ H10 (Ω)3,

∫

ΩCε(U)·· ε(W ) dx =

∫

ΩF · W dx (4.3)

for some distributed loading F ∈ L2(Ω)3 and a constitutive law C which is a bilinearsymmetric transformation of

Sym3def=τ ∈ L(R3;R3) : ∗τ = τ

, σ·· τ def=

∑

1≤i,j≤3

σij τij

(L(R3;R3) is the space of all linear transformations of R3 or 3× 3-matrices) underthe following assumption.

Assumption 4.1.The constitutive law is a transformation C ∈ Sym3 for which there exists a constantα > 0 such that Cτ ·· τ ≥ α τ ·· τ for all τ ∈ Sym3.

5. Optimal Triangular Meshing 7

For instance, for the Lame constants µ > 0 and λ ≥ 0, the special constitutivelaw Cτ = 2µ τ + λ tr τ I satisfies Assumption 4.1 with α = 2µ.

The associated bilinear form is

aΩ(U, W ) def=∫

ΩCε(U)·· ε(W ) dx,

where U is a vector function, D(U) is the Jacobian matrix of U , and

ε(U) def=12

(D(U) + ∗D(U))

is the strain tensor. The first eigenvalue is the minimum of the Rayleigh quotient

λ(Ω) = inf

aΩ(U, U)∫Ω |U |2 dx

: ∀U ∈ H10 (Ω)3, U = 0

.

A typical problem is to find the sensitivity of the first eigenvalue with respect tothe shape of the domain Ω. In 1907, J. Hadamard [1] used displacements alongthe normal to the boundary Γ of a C∞-domain to compute the derivative of thefirst eigenvalue of the clamped plate. As in the case of the column, this problem isnot differentiable with respect to the geometry when the eigenvalue is not simple.

5 Optimal Triangular MeshingThe shape calculus that will be developed in Chapters 9 and 10 for problems gov-erned by partial differential equations (the continuous model) will be readily ap-plicable to their discrete model as in the finite element discretization of ellipticboundary value problems. However, some care has to be exerted in the choice ofthe formula for the gradient, since the solution of a finite element discretizationproblem is usually less smooth than the solution of its continuous counterpart.

Most shape objective functionals will have two basic formulas for their shapegradient: a boundary expression and a volume expression. The boundary expressionis always nicer and more compact but can be applied only when the solution of theunderlying partial differential equation is smooth and in most cases smoother thanthe finite element solution. This leads to serious computational errors. The rightformula to use is the less attractive volume expression that requires only the samesmoothness as the finite element solution. Numerous computational experimentsconfirm that fact (cf., for instance, E. J. Haug and J. S. Arora [1] or E. J. Haug,K. K. Choi, and V. Komkov [1]). With the volume expression, the gradient ofthe objective function with respect to internal and boundary nodes can be readilyobtained by plugging in the right velocity field.

A large class of linear elliptic boundary value problems can be expressed asthe minimum of a quadratic function over some Hilbert space. For instance, let Ωbe a bounded open domain in RN with a smooth boundary Γ. The solution u ofthe boundary value problem

−∆u = f in Ω, u = 0 on Γ


is the minimizing element in the Sobolev space H10 (Ω) of the energy functional

E(v, Ω) def=∫

Ω|∇v|2 − 2f v dx,

J(Ω) def= infv∈H1

0 (Ω)E(v, Ω) = E(u, Ω) = −

∫

Ω|∇u|2 dx.

The elements of this problem are a Hilbert space V , a continuous symmetrical co-ercive bilinear form on V , and a continuous linear form ℓ on V . With this notation

∃u ∈ V, E(u) = infv∈V

E(v), E(v) def= a(v, v)− 2 ℓ(v)

and u is the unique solution of the variational equation

∃u ∈ U, ∀v ∈ V, a(u, v) = ℓ(v).

In the finite element approximation of the solution u, a finite-dimensionalsubspace Vh of V is used for some small mesh parameter h. The solution of theapproximate problem is given by

∃uh ∈ Vh, E(uh) = infvh∈Vh

E(vh), ⇒ ∃uh ∈ Uh, ∀vh ∈ Vh, a(uh, vh) = ℓ(vh).

It is easy to show that the error can be expressed as follows:

a(u− uh, u− uh) = ∥u− uh∥2V = 2 [E(uh)− E(u)] .

Assume that Ω is a polygonal domain in RN. In the finite element method, thedomain is partitioned into a set τh of small triangles by introducing nodes in Ω

Mdef= Mi ∈ Ω : 1 ≤ i ≤ p

∂Mdef= Mi ∈ ∂Ω : p + 1 ≤ i ≤ p + q

and Mdef= M ∪ ∂M

for some integers p ≥ N + 1 and q ≥ 1 (see Figure 1.3). Therefore the triangular-ization τh = τh(M), the solution space Vh = Vh(M), and the solution uh = uh(M)are functions of the positions of the nodes of the set M . Assuming that the total

Mi 10

0

0

0 0

0

Figure 1.3. Triangulation and basis function associated with node Mi.

5. Optimal Triangular Meshing 9

number of nodes is fixed, consider the following optimal triangularization problem:

infM

j(M), j(M) def= E(uh(τh(M)), Ω) = infvh∈Vh(M)

E(vh, Ω),

∥u− uh∥2V =∫

Ω|∇(u− uh)|2 dx = 2

[E(uh, Ω(τh(M)))− E(u, Ω)

]

= 2[J(Ω(τh(M)))− J(Ω)

],

J(Ω(τh(M))) def= infvh∈Vh

E(vh, Ω(τh(M))) = E(uh, Ω(τh(M))) = −∫

Ω|∇uh|2 dx.

The objective is to compute the partial derivative of j(M) with respect to theℓth component (Mi)ℓ of the node Mi:

∂j

∂(Mi)ℓ(M).

This partial derivative can be computed by using the velocity method for the specialvelocity field (cf. M. C. Delfour, G. Payre, and J.-P. Zolesio [3])

Viℓ(x) = bMi(x) eℓ,

where bMi ∈ Vh is the (piecewise P 1) basis function associated with the node Mi:bMi(Mj) = δij for all i, j. In that method each point X of the plane is movedaccording to the solution of the vector differential equation

dx

dt(t) = V (x(t)), x(0) = X.

This yields a transformation X .→ Tt(X) def= x(t;X) : R2 → R2 of the plane, and itis natural to introduce the following notion of semiderivative:

dJ(Ω; V ) def= limt0

J(Tt(Ω))− J(Ω)t

.

For t ≥ 0 small, the velocity field must be chosen in such a way that trianglesare moved onto triangles and the point Mi is moved in the direction eℓ:

Mi →Mit = Mi + t eℓ.

This is achieved by choosing the following velocity field:

Viℓ(t, x) = bMit(x) eℓ,

where bMit is the piecewise P 1 basis function associated with node Mit: bMit(Mj) =δij for all i, j. This yields the family of transformations

Tt(x) = x + t bMi(x) eℓ

which moves the node Mi to Mi + t eℓ and hence

∂j

∂(Mi)ℓ(M) = dJ(Ω; Viℓ).


Going back to our original example, introduce the shape functional

J(Ω) def= infv∈H1

0 (Ω)E(Ω, v) = −

∫

Ω|∇u|2 dx, E(Ω, v) =

∫

Ω|∇v|2 − 2 f v dx.

In Chapter 9, we shall show that we have the following boundary and volume ex-pressions for the derivative of J(Ω):

dJ(Ω; V ) = −∫

Γ

∣∣∣∣∂u

∂n

∣∣∣∣2

V · n dΓ,

dJ(Ω; V ) =∫

ΩA′(0)∇u ·∇u− 2 [div V (0)f +∇f · V (0)]u dx,

A′(0) = div V (0) I − ∗DV (0)−DV (0).

For a P 1-approximation

Vhdef=v ∈ C0(Ω) : v|K ∈ P 1(K), ∀K ∈ τh

and the trace of the normal derivative on Γ is not defined. Thus, it is necessary touse the volume expression. For the velocity field Viℓ

DViℓ = eℓ∗∇bMi , div DViℓ = eℓ ·∇bMi ,

A′(0) = eℓ ·∇bMi I − eℓ∗∇bMi −∇bMi

∗eℓ.

Since

∂j

∂(Mi)ℓ(M) = dJ(Ω; Viℓ),

we finally obtain the formula for the derivative of the function j(M) with respectto node Mi in the direction eℓ:

∂j

∂(Mi)ℓ(M) =

∫

Ω[eℓ ·∇bMi I − eℓ

∗∇bMi −∇bMi∗eℓ]∇; uh ·∇uh

− 2 [eℓ ·∇bMi f +∇f · eℓ bMi ]uh dx.

Since the support of bMi consists of the triangles having Mi as a vertex, the gradientwith respect to the nodes can be constructed piece by piece by visiting each node.

6 Modeling Free Boundary ProblemsThe first step towards the solution of a shape optimization is the mathematicalmodeling of the problem. Physical phenomena are often modeled on relativelysmooth or nice geometries. Adding an objective functional to the model will usuallypush the system towards rougher geometries or even microstructures. For instance,in the optimal design of plates the optimization of the profile of a plate led to highlyoscillating profiles that looked like a comb with abrupt variations ranging from zero

6. Modeling Free Boundary Problems 11

to maximum thickness. The phenomenon began to be understood in 1975 with thepaper of N. Olhoff [1] for circular plates with the introduction of the mechanicalnotion of stiffeners. The optimal plate was a virtual plate, a microstructure, thatis a homogenized geometry. Another example is the Plateau problem of minimalsurfaces that experimentally exhibits surfaces with singularities. In both cases, itis mathematically natural to replace the geometry by a characteristic function, afunction that is equal to 1 on the set and 0 outside the set. Instead of optimizingover a restricted family of geometries, the problem is relaxed to the optimizationover a set of measurable characteristic functions that contains a much larger familyof geometries, including the ones with boundary singularities and/or an arbitrarynumber of holes.

6.1 Free Interface between Two Materials

Consider the optimal design problem studied by J. Cea and K. Malanowski [1]in 1970, where the optimization variable is the distribution of two materials withdifferent physical characteristics within a fixed domain D. It cannot a priori beassumed that the two regions are separated by a smooth interface and that eachregion is connected. This problem will be covered in more details in section 4 ofChapter 5.

Let D ⊂ RN be a bounded open domain with Lipschitzian boundary ∂D.Assume for the moment that the domain D is partitioned into two subdomains Ω1and Ω2 separated by a smooth interface ∂Ω1 ∩ ∂Ω2 as illustrated in Figure 1.4.Domain Ω1 (resp., Ω2) is made up of a material characterized by a constant k1 > 0(resp., k2 > 0). Let y be the solution of the transmission problem

⎧⎨

⎩

− k1y = f in Ω1 and − k2y = f in Ω2,

y = 0 on ∂D and k1∂y

∂n1+ k2

∂y

∂n2= 0 on Ω1 ∩ Ω2,

(6.1)

where n1 (resp., n2) is the unit outward normal to Ω1 (resp., Ω2) and f is a givenfunction in L2(D). Assume that k1 > k2. The objective is to maximize the equiva-lent of the compliance

J(Ω1) = −∫

Dfy dx (6.2)

Ω2Ω1

Figure 1.4. Fixed domain D and its partition into Ω1 and Ω2.


over all domains Ω1 in D subject to the following constraint on the volume ofmaterial k1 which occupies the part Ω1 of D:

m(Ω1) ≤ α, 0 < α < m(D) (6.3)

for some constant α.If χ denotes the characteristic function of the domain Ω1,

χ(x) = 1 if x ∈ Ω1 and 0 if x /∈ Ω1,

the compliance J(χ) = J(Ω1) can be expressed as the infimum over the Sobolevspace H1

0 (D) of an energy functional defined on the fixed set D:

J(χ) = minϕ∈H1

0 (D)E(χ,ϕ), (6.4)

E(χ,ϕ) def=∫

D(k1 χ+ k2 (1− χ)) |∇ϕ|2 − 2χfϕ dx. (6.5)

J(χ) can be minimized or maximized over some appropriate family of characteristicfunctions or with respect to their relaxation to functions between 0 and 1 that wouldcorrespond to microstructures. As in the eigenvalue problem, the objective functionis an infimum, but here the infimum is over a space that does not depend on thefunction χ that specifies the geometric domain. This will be handled by the specialtechniques of Chapter 10 for the differentiation of the minimum of a functional.

6.2 Minimal SurfacesThe celebrated Plateau’s problem, named after the Belgian physicist and profes-sor J. A. F. Plateau [1] (1801–1883), who did experimental observations on thegeometry of soap films around 1873, also provides a nice example where the geome-try is a variable. It consists in finding the surface of least area among those boundedby a given curve. One of the difficulties in studying the minimal surface problem isthe description of such surfaces in the usual language of differential geometry. Forinstance, the set of possible singularities is not known.

Measure theoretic methods such as k-currents (k-dim surfaces) were used byE. R. Reifenberg [1, 2, 3, 4] around 1960, H. Federer and W. H. Fleming [1]in 1960 (normals and integral currents), F. J. Almgren, Jr. [1] in 1965 (varifolds),and H. Federer [5] in 1969.

In the early 1950s, E. De Giorgi [1, 2, 3] and R. Caccioppoli [1] considereda hypersurface in the N -dimensional Euclidean space RN as the boundary of aset. In order to obtain a boundary measure, they restricted their attention to setswhose characteristic function is of bounded variation. Their key property is anassociated natural notion of perimeter that extends the classical surface measure ofthe boundary of a smooth set to the larger family of Caccioppoli sets named afterthe celebrated Neapolitan mathematician Renato Caccioppoli.2

2In 1992 his tormented personality was remembered in a film directed by Mario Martone, TheDeath of a Neapolitan Mathematician (Morte di un matematico napoletano).

7. Design of a Thermal Diffuser 13

Caccioppoli sets occur in many shape optimization problems (or free boundaryproblems), where a surface tension is present on the (free) boundary, such as in thefree interface water/soil in a dam (C. Baiocchi, V. Comincioli, E. Magenes,and G. A. Pozzi [1]) in 1973 and in the free boundary of a water wave (M. Souliand J.-P. Zolesio [1, 2, 3, 4, 5]) in 1988. More details will be given in Chapter 5.

7 Design of a Thermal DiffuserShape optimization problems are everywhere in engineering, physics, and medicine.We choose two illustrative examples that were proposed by the Canadian SpaceProgram in the 1980s. The first one is the design of a thermal diffuser to condi-tion the thermal environment of electronic devices in communication satellites; thesecond one is the design of a thermal radiator that will be described in the next sec-tion. There are more and more design and control problems coming from medicine.For instance, the design of endoprotheses such as valves, stents, and coils in bloodvessels or left ventricular assistance devices (cardiac pumps) in interventional car-diology helps to improve the health of patients and minimize the consequences andcosts of therapeutical interventions by going to mini-invasive procedures.

7.1 Description of the Physical ProblemThis problem arises in connection with the use of high-power solid-state devices(HPSSD) in communication satellites (cf. M. C. Delfour, G. Payre, and J.-P. Zolesio [1]). An HPSSD dissipates a large amount of thermal power (typ.> 50 W) over a relatively small mounting surface (typ. 1.25 cm2). Yet, its junctiontemperature is required to be kept moderately low (typ. 110C). The thermalresistance from the junction to the mounting surface is known for any particularHPSSD (typ. 1C/W), so that the mounting surface is required to be kept ata lower temperature than the junction (typ. 60C). In a space application thethermal power must ultimately be dissipated to the environment by the mechanismof radiation. However, to radiate large amounts of thermal power at moderately lowtemperatures, correspondingly large radiating areas are required. Thus we have therequirement to efficiently spread the high thermal power flux (TPF) at the HPSSDsource (typ. 40 W/cm2) to a low TPF at the radiator (typ. 0.04 W/cm2) so thatthe source temperature is maintained at an acceptably low level (typ. < 60C)at the mounting surface. The efficient spreading task is best accomplished usingheatpipes, but the snag in the scheme is that heatpipes can accept only a limitedmaximum TPF from a source (typ. max 4 W/cm2).

Hence we are led to the requirement for a thermal diffuser. This device isinserted between the HPSSD and the heatpipes and reduces the TPF at the source(typ. > 40 W/cm2) to a level acceptable to the heatpipes (typ. > 4 W/cm2). Theheatpipes then sufficiently spread the heat over large space radiators, reducing theTPF from a level at the diffuser (typ. 4 W/cm2) to that at the radiator (typ. 0.04W/cm2). This scheme of heat spreading is depicted in Figure 1.5.

It is the design of the thermal diffuser which is the problem at hand. Wemay assume that the HPSSD presents a uniform thermal power flux to the diffuser


heatpipesaddle

heatpipes

radiator to spaceschematic drawingnot to scale

thermaldiffuser high-power

solid-state diffuser

Figure 1.5. Heat spreading scheme for high-power solid-state devices.

at the HPSSD/diffuser interface. Heatpipes are essentially isothermalizing devices,and we may assume that the diffuser/heatpipe saddle interface is indeed isothermal.Any other surfaces of the diffuser may be treated as adiabatic.

7.2 Statement of the Problem

Assume that the thermal diffuser is a volume Ω symmetrical about the z-axis(cf. Figure 1.6 (A)) whose boundary surface is made up of three regular pieces:the mounting surface Σ1 (a disk perpendicular to the z-axis with center in (r, z) =(0, 0)), the lateral adiabatic surface Σ2, and the interface Σ3 between the dif-fuser and the heatpipe saddle (a disk perpendicular to the z-axis with center in(r, z) = (0, L)).

The temperature distribution over this volume Ω is the solution of the station-ary heat equation k∆T = 0 (∆T , the Laplacian of T ) with the following boundaryconditions on the surface Σ = Σ1 ∪ Σ2 ∪ Σ3 (the boundary of Ω):

k∂T

∂n= qin on Σ1, k

∂T

∂n= 0 on Σ2, T = T3 (constant) on Σ3, (7.1)

where n always denotes the outward unit normal to the boundary surface Σ and∂T/∂n is the normal derivative to the boundary surface Σ,

∂T

∂n= ∇T · n (∇T = the gradient of T ). (7.2)

The parameters appearing in (7.1) are

k = thermal conductivity (typ. 1.8W/cm×C),

qin = uniform inward thermal power flux at the source (positive constant).


The radius R0 of the mounting surface Σ1 is fixed so that the boundary surface Σ1is already given in the design problem.

For practical considerations, we assume that the diffuser is solid without in-terior hollows or cutouts. The class of shapes for the diffuser is characterized bythe design parameter L > 0 and the positive function R(z), 0 < z ≤ L, withR(0) = R0 > 0. They are volumes of revolution Ω about the z-axis generated bythe surface A between the z-axis and the function R(z) (cf. Figure 1.6 (B)), that is,

Ω def=(x, y, z) : 0 < z < L, x2 + y2 < R(z)2

. (7.3)

So the shape of Ω is completely specified by the length L > 0 and the functionR(z) > 0 on the interval [0, L].

(A) (B) (C)

z z

L Σ3

Σ2

Σ1

L

0 0

ζ

1

0y y ρ0 0 0R0 1

AR(z) S4

S3

S2

S1

D

Figure 1.6. (A) Volume Ω and its boundary Σ; (B) Surface A generatingΩ; (C) Surface D generating Ω.

Assuming that the diffuser is made up of a homogeneous material of uniformdensity (no hollow) the design objective is to minimize the volume

J(Ω) def=∫

Ωdx = π

∫ L

0R(z)2 dz (7.4)

subject to a uniform constraint on the outward thermal power flux at the interfaceΣ3 between the diffuser and the heatpipe saddle:

supp∈Σ3

−k∂T

∂z(p) ≤ qout or k

∂T

∂n+ qout ≥ 0 on Σ3, (7.5)

where qout is a specified positive constant.It is readily seen that the minimization problem (7.4) subject to the constraint

(7.5) (where T is the solution of the heat equation with the boundary conditions(7.1)) is independent of the fixed temperature T3 on the boundary Σ3. In otherwords the optimal shape Ω, if it exists, is independent of T3. As a result, from nowon we set T3 equal to 0.


7.3 Reformulation of the ProblemIn a shape optimization problem the formulation is important from both the the-oretical and the numerical viewpoints. In particular condition (7.5) is difficult tonumerically handle since it involves the pointwise evaluation of the normal derivativeon the piece of boundary Σ3. This problem can be reformulated as the minimizationof T on Σ3, where T is now the solution of a variational inequality. Consider thefollowing minimization problem over the subspace of functions that are positive orzero on Σ3:

V +(Ω) def=v ∈ H1(Ω) : v|Σ3 ≥ 0

, (7.6)

infv∈V +(Ω)

∫

Ω

12|∇v|2 dx−

∫

Σ1

qin v dΣ +∫

Σ3

qout v dΣ. (7.7)

H1(Ω) is the usual Sobolev space on the domain Ω, and the inequality on Σ3has to be interpreted quasi-everywhere in the capacity sense. Leaving aside thosetechnicalities, the minimizing solution of (7.7) is characterized by

−k ∆T = 0 in Ω, k∂T

∂n= qin on Σ1, k

∂T

∂n= 0 on Σ2,

T ≥ 0,

(k∂T

∂n+ qout

)≥ 0, T

(k∂T

∂n+ qout

)= 0 on Σ3.

(7.8)

The former constraint (7.5) is verified and replaced by the new constraint

T = 0 on Σ3. (7.9)

If there exists a nonempty domain Ω of the form (7.3) such that T = 0 on Σ3, theproblem is feasible.

In this formulation the pointwise constraint on the normal derivative of thetemperature on Σ3 has been replaced by a pointwise constraint on the less de-manding trace of the temperature on Σ3. Yet, we now have to solve a variationalinequality instead of a variational equation for the temperature T .

7.4 Scaling of the ProblemIn the above formulations the shape parameter L and the shape function R are notindependent of each other since the function R is defined on the interval [0, L]. Thismotivates the following changes of variables and the introduction of the dimension-less temperature y:

x .→ ξ1 =x

R0, y .→ ξ2 =

y

R0, z .→ ζ =

z

L, 0 ≤ ζ ≤ 1,

L =L

R0, R(ζ) =

R(Lζ)R0

,

y(ξ1, ξ2, ζ) =k

LqinT (R0ξ1, R0ξ2, Lζ),

Ddef=

(ξ1, ξ2, ζ) : 0 < ζ < 1, ξ21 + ξ22 < R(z)2

.


The parameter L now appears as a coefficient in the partial differential equation

L2(∂2y

∂ξ21+∂2y

∂ξ22

)+∂2y

∂ζ2 = 0 in D

with the following boundary conditions on the boundary S = S1 ∪ S2 ∪ S3 of D:

∂y

∂νA= 1 on S1,

∂y

∂νA= 0 on S2, y = 0 on S3, (7.10)

where ν denotes the outward normal to the boundary surface S and ∂y/∂νA is theconormal derivative to the boundary surface S,

∂y

∂νA= L2

(ν1∂y

∂ξ1+ ν2

∂y

∂ξ2

)+ ν3

∂y

∂ζ.

Finally, the optimal design problem depends only on the ratio q = qout/qin throughthe constraint

∂y

∂νA+

qout

qin≥ 0 on S3.

The design variables are the parameter L > 0 and the function R > 0 now definedon the fixed interval [0, 1].

7.5 Design ProblemThe fact that this specific design problem can be reduced to finding a parameterand a function gives the false or unfounded impression that it can now be solvedby standard mathematical programming and numerical methods. Early work onsuch problems revealed a different reality, such as oscillating boundaries and con-vergence towards nonphysical designs. Clearly, the geometry refused to be handledby standard methods without a better understanding of the underlying physics andits inception in the modeling of the geometric variable.

At the theoretical level, the existence of solution requires a concept of conti-nuity with respect to the geometry of the solution of either the heat equation withan inequality constraint on the TPF or the variational inequality with an equalityconstraint on the temperature. The other element is the lower semicontinuity of theobjective functional that is not too problematic for the volume functional as long asthe chosen topology on the geometry preserves the continuity of the volume func-tional. For instance, the classical Hausdorff metric topology does not preserve thevolume. In the context of fluid mechanics (cf., for instance, O. Pironneau [1]), itmeans that a drag minimizing sequence of sets with constant volume may convergeto a set with twice the volume (cf. Example 4.1 in Chapter 6). A wine makingindustry exploiting the convergence in the Hausdorff metric topology could yieldmiraculous profits.

Other serious issues are, for instance, the lack of differentiability of the solutionof a variational inequality at the continuous level that will inadvertently affect thedifferentiability or the evolution of a gradient method at the discrete level. We shall


see that there is not only one topology for shapes but a whole range that selectivelypreserve some but not all of the geometrical features. Again the right choice isproblem dependent, much like the choice of the right Sobolev space in the theoryof partial differential equations.

8 Design of a Thermal RadiatorCurrent trends indicate that future communications satellites and spacecrafts willgrow ever larger, consume ever more electrical power, and dissipate larger amountsof thermal energy. Various techniques and devices can be deployed to conditionthe thermal environment for payload boxes within a spacecraft, but it is desirableto employ those which offer good performance for low cost, low weight, and highreliability. A thermal radiator (or radiating fin) which accepts a given TPF from apayload box and radiates it directly to space can offer good performance and highreliability at low cost. However, without careful design, such a radiator can be un-necessarily bulky and heavy. It is the mass-optimized design of the thermal radiatorwhich is the problem at hand (cf. M. C. Delfour, G. Payre, and J.-P. Zole-sio [2]). We may assume that the payload box presents a uniform TPF (typ. 0.1 to1.0 W/cm2) into the radiator at the box/radiator interface. The radiating surfaceis a second surface mirror which consists of a sheet of glass whose inner surface hassilver coating. We may assume that the TPF out of the radiator/space interfaceis governed by the T 4 radiation law, although we must account also for a constantTPF (typ. 0.01 W/cm2) into this interface from the sun. Any other surfaces ofthe radiator may be treated as adiabatic. Two constraints restrict freedom in thedesign of the thermal radiator:

(i) the maximum temperature at the box/radiator interface is not to exceed someconstant (typ. 50); and

(ii) no part of the radiator is to be thinner than some constant (typ. 1 mm).

8.1 Statement of the ProblemAssume that the radiator is a volume Ω symmetrical about the z-axis (cf. Figure 1.7)whose boundary surface is made up of three regular pieces: the contact surface Σ1(a disk perpendicular to the z-axis with center at the point (r, z) = (0, 0)), thelateral adiabatic surface Σ2, and the radiating surface Σ3 (a disk perpendicular tothe z-axis with center at (r, z) = (0, L)). More precisely

Σ1 =(x, y, z) : z = 0 and x2 + y2 ≤ R2

0

,

Σ2 =(x, y, z) : x2 + y2 = R(z)2, 0 ≤ z ≤ L

,

Σ3 =(x, y, z) : z = L and x2 + y2 ≤ R(L)2

,

(8.1)

where the radius R0 > 0 (typ. 10 cm), the length L > 0, and the function

R : [0, L]→ R, R(0) = R0, R(z) > 0, 0 ≤ z ≤ L, (8.2)

are given (R, the field of real numbers).

8. Design of a Thermal Radiator 19

z z

L

0

Σ3

Σ2

Σ1

R00ry

Figure 1.7. Volume Ω and its cross section.

The temperature distribution (in Kelvin degrees) over the volume Ω is thesolution of the stationary heat equation

∆T = 0 (the Laplacian of T ) (8.3)

with the following boundary conditions on the surface Σ = Σ1 ∪ Σ2 ∪ Σ3 (theboundary of Ω):

k∂T

∂n= qin on Σ1, k

∂T

∂n= 0 on Σ2, k

∂T

∂n= −σεT |T |3 + qs on Σ3, (8.4)

where n denotes the outward normal to the boundary surface Σ, ∂T/∂n is thenormal derivative on the boundary surface Σ, and

∂T

∂n= ∇T · n (∇T = the gradient of T ).

The parameters appearing in (8.1)–(8.4) are

k = thermal conductivity (typ. 1.8 W/cm×C),

qin = uniform inward thermal power flux at the source (typ. 0.1 to 1.0 W/cm2),

σ = Boltzmann’s constant (5.67× 10−8 W/m2K4),

ε = surface emissivity (typ. 0.8),

qs = solar inward thermal power flux (0.01 W/cm2).

The optimal design problem consists in minimizing the volume

J(Ω) def= π

∫ L

0R(z)2 dz (8.5)

over all length L > 0 and shape function R subject to the constraint

T (x, y, z) ≤ Tf (typ. 50C), ∀(x, y, z) ∈ Σ1. (8.6)

In this analysis we shall drop the requirement (ii) in the introduction.


8.2 Scaling of the ProblemAs in the case of the diffuser, it is convenient to introduce the following dimensionlesscoordinates and temperature y (see Figure 1.8):

x .→ ξ1 =x

R0, y .→ ξ2 =

y

R0, z .→ ζ =

z

L, 0 ≤ ζ ≤ 1,

L =L

R0, R(ζ) =

R(Lζ)R0

,

y(ξ1, ξ2, ζ) =(σεR0

k

)1/3

T (R0ξ1, R0ξ2, Lζ),

Ddef=

(ξ1, ξ2, ζ) : 0 < ζ < 1, ξ21 + ξ22 < R(z)2

.

The parameter L now appears as a coefficient in the partial differential equation

L2(∂2y

∂ξ21+∂2y

∂ξ22

)+∂2y

∂ζ2 = 0 in D

with the following boundary conditions on the boundary S = S1 ∪ S2 ∪ S3 of D:

∂y

∂νA= L

[(σεR0

k

)1/3 qin

kR0

]on S1,

∂y

∂νA= 0 on S2,

∂y

∂νA+ Ly|y|3 = L

[(σεR0

k

)1/3 qin

kR0

]qs

qinon S3,

(8.7)

where ν denotes the outward normal to the boundary surface S and ∂y/∂νA is theconormal derivative to the boundary surface Σ,

∂y

∂νA= L2

(ν1∂y

∂ξ1+ ν2

∂y

∂ξ2

)+ ν3

∂y

∂ζ.

ζ ζ

1

0

S3

S2

S1

10rr

1

1

A

Σ3

Σ2

Σ1

Figure 1.8. Volume Ω and its generating surface A.

9. A Glimpse into Segmentation of Images 21

9 A Glimpse into Segmentation of ImagesThe study of the problem of linguistic or visual perceptions was initiated by severalpioneering authors, such as H. Blum [1] in 1967, D. Marr and E. Hildreth [1] in1980, and D. Marr [1] in 1982. It involves specialists of psychology, artificialintelligence, and experimentalists such as D. H. Hubel and T. N. Wiesel [1] in1962 and F. W. Campbell and J. G. Robson [1] in 1968.

In the first part of this section, we revisit the pioneering work of D. Marrand E. Hildreth [1] in 1980 on the smoothing of the image by convolution with asufficiently differentiable normalized function as a function of the scaling parameter.We extend the space-frequency uncertainty principle to N -dimensional images. Itis the analogue of the Heisenberg uncertainty principle of quantum mechanics. Werevisit the Laplacian filter and we generalize the linearity assumption of D. Marrand E. Hildreth [1] from linear to curved contours.

In the second part, we show how shape analysis methods and the shape andtangential calculuses can be applied to objective functionals defined on the wholecontour of an image. We anticipate Chapters 9 and 10 on shape derivatives by thevelocity method and show how they can be applied to snakes, active geodesic con-tours, and level sets (cf., for instance, the book of S. Osher and N. Paragios [1]).It shows that the Eulerian shape semiderivative is the basic ingredient behind thoserepresentations, including the case of the oriented distance function. In all cases,the evolution equation for the continuous gradient descent method is shown to havethe same structure. For more material along the same lines, the reader is referredto M. Dehaes [1] and M. Dehaes and M. C. Delfour [1].

9.1 Automatic Image Processing

The first level of image processing is the detection of the contours or the boundariesof the objects in the image. For an ideal image I : D → R defined in an opentwo-dimensional frame D with values in an interval of greys continuously rangingfrom white to black (see Figure 1.9), the edges of an object correspond to the loci ofdiscontinuity of the image I (cf. D. Marr and E. Hildreth [1]), also called “stepedges” by D. Marr [1]. As can be seen from Figure 1.9, the loci of discontinuitymay only reveal part of an object hidden by another one and a subsequent anddifferent level of processing is required. A more difficult case is the detection ofblack curves or cracks in a white frame (cf. Figure 1.10), where the function Ibecomes a measure supported by the curves rather than a function.

In practice, the frame D of the image is divided into periodically spaced cellsP (square, hexagon, diamond) with a quantized value or pixel from 256 grey lev-els. For small squares a piecewise linear continuous interpolation or higher degreeC1-interpolation can be used to remove the discontinuities at the intercell bound-aries. In addition, observations and measurements introduce noise or perturbationsand the interpolated image I needs to be further smoothed or filtered. When thecharacteristics of the noise are known, an appropriate filter can do the job (see,for instance, the use of low pass filters in A. Rosenfeld and M. Thurston [1],A. P. Witkin [1], and A. L. Yuille and T. Poggio [1]).


frame D

levels of grey I open set Ω

Figure 1.9. Image I of objects and their segmentation in the frame D.

frame D

Figure 1.10. Image I containing black curves or cracks in the frame D.

Given an ideal grey level image I defined in a fixed bounded open frame Dwe want to identify the edges or boundaries of the objects contained in the imageas shown in Figure 1.9. The edge identification or segmentation problem is a low-level processing of a real image which should also involve higher-level processings ortasks (cf. M. Kass, M. Witkin, and D. Terzopoulos [1]). Intuitively the edgescoincide with the loci of discontinuities of the image function I. At such points thenorm |∇I| of the gradient ∇I is infinite. Roughly speaking the segmentation of anideal image I consists in finding the loci of discontinuity of the function I. The ideais now to first smooth I by convolution with a sufficiently differentiable function.This operation will be followed and/or combined with the use of an edge detectoras the zero crossings of the Laplacian. In the literature the term filter often appliesto both the filter and the detector. In this section we shall make the distinctionbetween the two operations.

9.2 Image Smoothing/Filtering by Convolutionand Edge Detectors

In this section we revisit the work of D. Marr and E. Hildreth [1] in 1980 on thesmoothing of the image I by convolution with a sufficiently differentiable normalizedfunction ρ as a function of the scaling parameter ε > 0. In the second part of this


section we revisit the Laplacian filter and we generalize the linearity assumption ofD. Marr and E. Hildreth [1] from linear to curved contours.

9.2.1 Construction of the Convolution of I

Let ρ : RN → R, N ≥ 1, be a sufficiently smooth function such that

ρ ≥ 0 and∫

RNρ(x) dx = 1.

Associate with the image I, ρ, and a scaling parameter ε > 0 the normalizedconvolution

Iε(x) def= (I ∗ ρε)(x) =1εN

∫

RNI(y) ρ

(x− y

ε

)dy, (9.1)

where for x ∈ RN

ρε(x) def=1εN

ρ(x

ε

)and

∫

RNρε(x) dx = 1. (9.2)

The function ρε plays the role of a probability density and ρε(x) dx of a probabilitymeasure. Under appropriate conditions Iε converges to the original image I asε → 0. For a sufficiently small ε > 0 the loci of discontinuity of the function Iare transformed into loci of strong variation of the gradient of the convolution Iε.When ρ has compact support, the convolution acts locally around each point in aneighborhood whose size is of order ε. A popular choice for ρ is the Gaussian withintegral normalized to one in RN defined by

GN (x) def=1

(√

2π)Ne− 1

2 |x|2 and∫

RNGN (x) dx = 1

and the normalized Gaussian of variance ε

GNε (x) =

1εN

1(√

2π)Ne− 1

2 | xε |2 and

∫

RNGNε (x) dx = 1. (9.3)

As noted in L. Alvarez, P.-L. Lions, and J.-M. Morel [1] in dimension N = 2,the function u(t) = G2√

2t∗ I is the solution of the parabolic equation

∂u

∂t(t, x) = ∆u(t, x), u(0, x) = I(x).

9.2.2 Space-Frequency Uncertainty Relationship

It is interesting to compute the Fourier transform F(GNε ) of GN

ε to make explicit therelationship between the mean square deviations of GN

ε and its Fourier transformF(GN

ε ). For ω ∈ RN, define the Fourier transform of a function f by

F(f)(ω) def=1

(√

2π)N

∫

RNf(x) e−iω·x. (9.4)


In applications the integral of the square of f often corresponds to an energy. Weshall refer to the L2-norm of f as the energy norm and the function f2(x) as theenergy density. The Fourier transform (9.4) of the normalized Gaussian (9.3) is

F(GNε )(ω) def=

1(√

2π)N

∫

RNGNε (x) e−iω·xdx =

1(√

2π)Ne− 1

2 |εω|2 . (9.5)

D. Marr and E. Hildreth [1] notice that there is an uncertainty relationshipbetween the mean square deviation with respect to the energy density f(x)2 and themean square deviation with respect to the energy density F(f)(ω)2 of its Fouriertransform in dimension 1 using a result of R. Bracewell [1, pp. 160–161], where heuses the constant 1/(2π) instead of 1/

√2π in the definition of the Fourier transform

that yields 1/(4π) instead of 1/2 as a lower bound to the product of the two meansquare deviations.

This uncertainty relationship generalizes to dimension N . First define thenotions of centroıd and variance with respect to the energy density f(x)2. Given afunction f ∈ L1(RN) ∩ L2(RN) and x ∈ RN, define the centroıd as

xdef=∫RN x f(x)2 dx∫RN f(x)2 dx

(9.6)

and the variance as

⟨∆x⟩2 def= ⟨x− x⟩2 =∫RN |x|2 f(x)2 dx∫

RN f(x)2 dx− |x|2.

Theorem 9.1 (Uncertainty relationship). Given N ≥ 1 and f ∈ W 1,1(RN) ∩W 1,2(RN) such that −ix f ∈ L1(RN) ∩ L2(RN),

⟨∆x⟩ ⟨∆ω⟩ ≥ N

2. (9.7)

For f = GNε

f(x) = GNε (x) =

1εN

1(√

2π)Ne− 1

2 | xε |2 ⇒ ⟨∆x⟩2 =

Nε2

2, (9.8)

f(ω) = F(GNε )(ω) =

1(√

2π)Ne− 1

2 |εω|2 ⇒ ⟨∆ω⟩2 =N

2 ε2(9.9)

⇒ ⟨∆x⟩⟨∆ω⟩ =N

2for GN

ε and F(GNε ). (9.10)

So the normalized Gaussian filter is indeed an optimal filter since it achieves thelower bound in all dimensions as stated by D. Marr and E. Hildreth [1] indimension 1 in 1980. In the context of quantum mechanics, this relationship is theanalogue of the Heisenberg uncertainty principle.


9.2.3 Laplacian Detector

One way to detect the edges of a regular object is to start from the convolution-smoothed image Iε. Given a direction v, |v| = 1, and a point x ∈ R2 an edgepoint will correspond to a local minimum or maximum of the directional derivativef(t) def= ∇Iε(x+ tv) ·v with respect to t. Denote by t such a point. Then a necessarycondition for an extremal is

D2Iε(x + tv)v · v = 0.

The point x = x + tv is a zero crossing following the terminology of D. Marr andE. Hildreth [1] of the second-order directional derivative in the direction v. Thuswe are looking for the pairs (x, v) verifying the necessary condition

D2Iε(x)v · v = 0 (9.11)

and, more precisely, lines or curves C such that

∀x ∈ C, ∃v(x), |v(x)| = 1, D2Iε(x)v(x) · v(x) = 0. (9.12)

This condition is necessary in order that, in each point x ∈ C, there exists a directionv(x) such that ∇Iε(x) · v is extremal.

In order to limit the search to points x rather than to pairs (x, v), the Laplaciandetector was introduced in D. Marr and E. Hildreth [1] under two assumptions:the linear variation of the intensity along the edges C of the object and the conditionof zero crossing of the second-order derivative in the direction normal to C.

Assumption 9.1 (Linear variation condition).The intensity Iε in a neighborhood of lines parallel to C is locally linear (affine).

Assumption 9.2.The zero-crossing condition is verified in all points of C in the direction of the normaln at the point, that is,

D2Iε n · n = 0 on C. (9.13)

Under those two assumptions and for a line C, it is easy to show that thepoints of C verify the necessary condition

∆Iε = 0 on C. (9.14)

Such conditions can be investigated for edges C that are the boundary Γ of asmooth domain Ω ⊂ RN of class C2 by using the tangential calculus developed insection 5.2 of Chapter 9 and M. C. Delfour [7, 8] for objects in RN, N ≥ 1, andnot just in dimension N = 2. Indeed we want to find points x ∈ Γ such that thefunction∇Iε(x)·n(x) is an extremal of the function f(t) = ∇Iε(x+t∇bΩ(x))·∇bΩ(x)in t = 0. This yields the local necessary condition D2Iε∇bΩ ·∇bΩ = D2Iε n · n = 0on Γ of Assumption 9.2. As for Assumption 9.1, its analogue is the following.


Assumption 9.3.The restriction to the curve C of the gradient of the intensity Iε is a constantvector c, that is,

∇Iε = c, c is a constant vector on C. (9.15)

Indeed the Laplacian of Iε on C can be decomposed as follows (cf. section 5.2of Chapter 9):

∆Iε = div Γ∇Iε + D2Iε n · n,

where div Γ∇Iε is the tangential divergence of ∇Iε that can be defined as follows:

div Γ∇Iε = div (∇Iε pC)|C

and pC is the projection onto C. Hence, under Assumption 9.3, div Γ(∇Iε) = 0 onC since

div Γ(∇Iε) = div (∇Iε pΓ)|Γ = div (c)|Γ = 0,

and, in view of (9.13) of Assumption 9.2, D2Iε n · n = 0 on C. Therefore

∆Iε = div Γ(∇Iε) + D2Iε n · n = 0 on C,

and we obtain the necessary condition

∆Iε = 0 on C. (9.16)

9.3 Objective Functions Defined on the Whole Edge

With the pioneering work of M. Kass, M. Witkin, and D. Terzopoulos [1]in 1988 we go from a local necessary condition at a point of the edge to a globalnecessary condition by introducing objective functionals defined on the entire edgeof an object. Here many computations and analytical studies can be simplified byadopting the point of view that a closed curve in the plane is the boundary of a setand using the whole machinery developed for shape and geometric analysis and thetangential and shape calculuses in Chapter 9.

9.3.1 Eulerian Shape Semiderivative

In this section we briefly summarize the main elements of the velocity method.

Definition 9.1.Given D, ∅ = D ⊂ RN, consider the set P(D) = Ω : Ω ⊂ D of subsets of D.The set D is the holdall or the universe. A shape functional is a map J : A → Efrom an admissible family A in P(D) with values in a topological vector space E(usually R).


Given a velocity field V : [0, τ ] ×RN → RN (the notation V (t)(x) = V (t, x)will often be used), consider the transformations

T (t, X) def= x(t, X), t ≥ 0, X ∈ RN, (9.17)

where x(t, X) = x(t) is defined as the flow of the differential equation

dx

dt(t) = V (t, x(t)), t ≥ 0, x(0, X) = X (9.18)

(here the notation x .→ Tt(x) = T (t, x) : RN → RN will be used). The Eulerianshape semiderivative of J in Ω in the direction V is defined as


J(Ωt)− J(Ω)t

(9.19)

(when the limit exists in E), where Ωt = Tt(Ω) = Tt(x) : x ∈ Ω. Under appro-priate assumptions on the family V (t), the transformations Tt are homeomor-phisms that transport the boundary Γ of Ω onto the boundary Γt of Ωt and theinterior Ω onto the interior of Ωt.

9.3.2 From Local to Global Conditions on the Edge

For simplicity, drop the subscript ε of the convolution-smoothed image Iε and as-sume that the edge of the object is the boundary Γ of an open domain Ω of class C2.Following V. Caselles, R. Kimmel, and G. Sapiro [1], it is important to chooseobjective functionals that are intrinsically defined and do not depend on an arbitraryparametrization of the boundary. For instance, given a frame D = ]0, a[× ]0, b[ anda smoothed image I : D → R, find an extremum of the objective functional

E(Ω) def=∫

Γ

∂I

∂ndΓ, (9.20)

where the integrand is the normal derivative of I. Using the velocity method theEulerian shape directional semiderivative is given by the expression (cf. Chapter 9)

dE(Ω; V ) =∫

Γ

[H∂I

∂n+

∂

∂n

(∂I

∂n

)]V · n dΓ, (9.21)

where H = ∆bΩ is the mean curvature and n = ∇bΩ is the outward unit normal.Proceeding in a formal way a necessary condition would be

H∂I

∂n+

∂

∂n

(∂I

∂n

)= 0 on Γ

⇒ ∆bΩ∇I ·∇bΩ +∇(∇I ·∇bΩ) ·∇bΩ = 0

⇒ ∆bΩ∇I ·∇bΩ + D2I∇bΩ ·∇bΩ + D2bΩ∇I ·∇bΩ = 0

⇒ D2I n · n + H∂I

∂n= 0 on Γ. (9.22)


This global condition is to be compared with the local condition (9.13). It can alsobe expressed in terms of the Laplacian and the tangential Laplacian of I as

∆I −∆ΓI = 0 on Γ, (9.23)

which can be compared with the local condition (9.16), ∆I = 0. This arises from thedecomposition of the Laplacian of I with respect to Γ using the following identityfor a smooth vector function U :

div U = div ΓU + DU n · n, (9.24)

where the tangential divergence of U is defined as

div ΓU = div (U pΓ)|Γ

and pΓ is the projection onto Γ. Applying this to U = ∇I and recalling the definitionof the tangential Laplacian

∆I = div Γ∇I + D2I n · n = ∆ΓI + H∂I

∂n+ D2I n · n,

where the tangential gradient ∇ΓI and the Laplace–Beltrami operator ∆ΓI aredefined as

∇ΓI = ∇(I pΓ)|Γ and ∆ΓI = div Γ(∇ΓI).

9.4 Snakes, Geodesic Active Contours, and Level Sets9.4.1 Objective Functions Defined on the Contours

In the literature the objective or energy functional is generally made up of twoterms: one (image energy) that depends on the image and one (internal energy)that specifies the smoothness of Γ. A general form of objective functional is

E(Ω) def=∫

Γg(I) dΓ, g(I) is a function of I. (9.25)

The directional semiderivative with respect to a velocity field V is given by

dE(Ω; V ) =∫

Γ

[Hg(I) +

∂

∂ng(I)

]n · V dΓ. (9.26)

This gradient will make the snakes move and will activate the contours.

9.4.2 Snakes and Geodesic Active Contours

If a gradient descent method is used to minimize (9.25) starting from an initial curveC0 = C, the iterative process is equivalent to following the evolution Ct (boundaryof the smooth domain Ωt) of the closed curve C given by the equation

∂Ct

∂t= − [Ht gI + (∇gI · nt)] nt on Ct, (9.27)


where Ht is the mean curvature, nt is the unit exterior normal, and the right-handside of the equation is formally the “derivative” of (9.25) given by (9.26). Equa-tion (9.27) is referred to as the geodesic flow. For gI = 1 it is the motion by meancurvature

∂Ct

∂t= −Ht nt on Ct. (9.28)

9.4.3 Level Set Method

The idea to represent the contours Ct by the zero-level set of a function ϕt(x) =ϕ(t, x) for a function ϕ : [0, τ ]×RN → R by setting

Ct =x ∈ RN : ϕ(t, x) = 0

= ϕ−1

t 0

and to replace (9.27) by an equation for ϕ is due to S. Osher and J. A. Sethian [1]in 1988. This approach seems to have been simultaneously introduced in imageprocessing by V. Caselles, F. Catte, T. Coll, and F. Dibos [1] in 1993 underthe name “geometric partial differential equations” with, in addition to the meancurvature term, a “transport” term and by R. Malladi, J. A. Sethian, andB. C. Vemuri [1] in 1995 under the name “level set approach” combined with thenotion of “extension velocity.”

Let (t, x) .→ ϕ(t, x) : [0, τ ]×RN → R be a smooth function and Ω be a subsetof RN of boundary Γ = Ω ∩ !Ω such that

int Ω =x ∈ RN : ϕ(0, x) < 0

and Γ =

x ∈ RN : ϕ(0, x) = 0

. (9.29)

Let V : [0, τ ]×RN → RN be a sufficiently smooth velocity field so that the transfor-mations Tt are diffeomorphisms. Moreover, assume that the images Ωt = Tt(Ω)verify the following properties: for all t ∈ [0, τ ]

int Ωt =x ∈ RN : ϕ(t, x) < 0

and Γt =

x ∈ RN : ϕ(t, x) = 0

. (9.30)

Assuming that the function ϕt(x) = ϕ(t, x) is at least of class C1 and that∇ϕt = 0 on ϕ−1

t 0, the total derivative with respect to t of ϕ(t, Tt(x)) for x ∈ Γyields

∂

∂tϕ(t, Tt(x)) +∇ϕ(t, Tt(x)) · d

dtTt(x) = 0. (9.31)

By substituting the velocity field in (9.18), we get

∂

∂tϕ(t, Tt(x)) +∇ϕ(t, Tt(x)) · V (t, Tt(x)) = 0, ∀Tt(x) ∈ Γt

⇒ ∂

∂tϕt +∇ϕt · V (t) = 0 on Γt, t ∈ [0, τ ]. (9.32)

This last equation, the level set evolution equation, is verified only on the boundariesor fronts Γt, 0 ≤ t ≤ τ . Can we find a representative ϕ in the equivalence class

[ϕ]Ω,V =ϕ : ϕ−1

t 0 = Γt and ϕ−1t < 0 = int Ωt,∀t ∈ [0, τ ]


of functions ϕ that verify conditions (9.29) and (9.30) in order to extend (9.32) fromΓt to the whole RN or at least almost everywhere in RN? Another possibility is toconsider the larger equivalence class

[ϕ]Γ,V =ϕ : ϕ−1

t 0 = Γt,∀t ∈ [0, τ ]

.

9.4.4 Velocity Carried by the Normal

In D. Adalsteinsson and J. A. Sethian [1], J. Gomes and O. Faugeras [1],and R. Malladi, J. A. Sethian, and B. C. Vemuri [1], the front moves underthe effect of a velocity field carried by the normal with a scalar velocity that dependson the curvatures of the level set or the front. So we are led to consider velocityfields V of the form

V (t)|Γt = v(t) nt, x ∈ Γt, (9.33)

for a scalar velocity (t, x) .→ v(t)(x) : [0, τ ] × RN → R. By using the expression∇ϕt/|∇ϕt| of the exterior normal nt as a function of ∇ϕt, (9.32) now becomes

∂

∂tϕt + v(t) |∇ϕt| = 0 on Γt. (9.34)

For instance, consider the example of the minimization of the total lengthof the curve C of (9.27) for the metric g(I) dC as introduced by V. Caselles,R. Kimmel, and G. Sapiro [1]. By using the computation (9.26) of the shapesemiderivative with respect to the velocity field V of the objective functional (9.25),a natural direction of descent is given by

v(t)nt, v(t) = −[Ht g(I) +

∂

∂ntg(I)

]on Γt. (9.35)

The normal nt and the mean curvature Ht can be expressed as a function of ∇ϕt:

nt =∇ϕt

|∇ϕt|and Ht = div Γtnt = div Γt

(∇ϕt

|∇ϕt|

)= div

(∇ϕt

|∇ϕt|

)∣∣∣∣Γt

. (9.36)

By substituting in (9.34), we get the following evolution equation:

∂ϕt

∂t−(

Ht g(I) +∂

∂ntg(I)

)∇ϕt · ∇ϕt

|∇ϕt|= 0 on Γt (9.37)

and⎧⎪⎨

⎪⎩

∂ϕt

∂t−(

Ht g(I) +∂

∂ntg(I)

)|∇ϕt| = 0 on Γt,

ϕ0 = ϕ0 on Γ0.

(9.38)


By substituting expressions (9.36) for nt and Ht in terms of ∇ϕt, we finally get

⎧⎪⎨

⎪⎩

∂ϕt

∂t−[div

(∇ϕt

|∇ϕt|

)g(I) +∇g(I) · ∇ϕt

|∇ϕt|

]|∇ϕt| = 0 on Γt,

ϕ0 = ϕ0 on Γ0.

(9.39)

The negative sign arises from the fact that we have chosen the outward rather thanthe inward normal. The main references to the existence and uniqueness theoremsrelated to (9.39) can be found in Y. G. Chen, Y. Giga, and S. Goto [1].

9.4.5 Extension of the Level Set Equations

Equation (9.39) on the fronts Γt is not convenient from both the theoretical andthe numerical viewpoints. So it would be desirable to be able to extend (9.39) in asmall tubular neighborhood of thickness h

Uh(Γt)def=x ∈ RN : dΓt(x) < h

(9.40)

of the front Γt for a small h > 0 (dΓt , the distance function to Γt). In theory, thevelocity field associated with Γt is given by

V (t) = −[div

(∇ϕt

|∇ϕt|

)g(I) +∇g(I) · ∇ϕt

|∇ϕt|

]∇ϕt

|∇ϕt|on Γt. (9.41)

Given the identities (9.36), the expressions of nt and Ht extend to a neighborhoodof Γt and possibly to RN if ∇ϕt is sufficiently smooth and ∇ϕt = 0 on Γt.

Equation (9.39) can be extended to RN at the price of violating the assump-tions (9.29)–(9.30), either by loss of smoothness of ϕt or by allowing its gradient tobe zero:

⎧⎪⎨

⎪⎩

∂ϕt

∂t−[div

(∇ϕt

|∇ϕt|

)g(I) +∇g(I) · ∇ϕt

|∇ϕt|

]|∇ϕt| = 0 in RN,

ϕ0 = ϕ0 in RN .

(9.42)

In fact, the starting point of V. Caselles, F. Catte, T. Coll, and F. Dibos [1]was the following geometric partial differential equation:

⎧⎪⎨

⎪⎩

∂ϕt

∂t−[div

(∇ϕt

|∇ϕt|

)g(I) + ν g(I)

]|∇ϕt| = 0 in RN,

ϕ0 = ϕ0 in RN(9.43)

for a constant ν > 0 and the function g(I) = 1/(1+|∇I|2). They prove the existence,in dimension 2, of a viscosity solution unique for initial data ϕ0 ∈ C([0, 1]× [0, 1])∩W 1,∞([0, 1]× [0, 1]) and g ∈W 1,∞(R2).


9.5 Objective Function Defined on the Whole Image

9.5.1 Tikhonov Regularization/Smoothing

The convolution is only one approach to smoothing an image I ∈ L2(D) defined ina frame D. Another way is to use the Tikhonov regularization: given ε > 0, find afunction Iε ∈ H1(D) that minimizes the objective functional

Eε(ϕ) def=12

∫

Dε|∇ϕ|2 + |ϕ− I|2 dx,

where the value of ε can be suitably adjusted to get the desired degree of smooth-ness. The minimizing function Iε ∈ H1(D) is solution of the following variationalequation:

∀ϕ ∈ H1(D),∫

Dε∇Iε ·∇ϕ+ (Iε − I)ϕ dx = 0. (9.44)

The price to pay now is that the computation is made on the whole image.This can be partly compensated by combining in a single operation or level of

processing the smoothing of the image with the detection of the edges. To do thatD. Mumford and J. Shah [1] in 1985 and [2] in 1989 introduced a new objectivefunctional that received a lot of attention from the mathematical and engineeringcommunities.

9.5.2 Objective Function of Mumford and Shah

Given a grey level ideal image in a fixed bounded open frame D we are looking for anopen subset Ω of D such that its boundary reveals the edges of the two-dimensionalobjects contained in the image of Figure 1.9.

Definition 9.2.Let D be a bounded open subset of RN with Lipschitzian boundary.

(i) An image in the frame D is specified by a function I ∈ L2(D).

(ii) We say that Ωjj∈J is an open partition of D if Ωjj∈J is a family of disjointconnected open subsets of D such that

mN (∪j∈JΩj) = mN (D) and mN (∂ ∪j∈J Ωj) = 0,

where mN is the N -dimensional Lebesgue measure. Denote by P(D) thefamily of all such open partitions of D.

The idea behind the formulation of D. Mumford and J. Shah [1] in 1985is to do a Tikhonov regularization Iε,j ∈ H1(Ωj) of I on each element Ωj of thepartition and then minimize the sum of the local minimizations over Ωj over all open


partitions of the image D: find an open partition P = Ωjj∈J in P(D) solution ofthe minimization problem

infP∈P(D)

∑

j∈J

infϕj∈H1(Ωj)

∫

Ωj

ε|∇ϕj |2 + |ϕj − I|2 dx (9.45)

for some fixed constant ε > 0. Observe that without the condition mN (∪j∈JΩj) =mN (D) the empty set would be a solution of the problem.

The question of existence requires a more specific family of open partitions ora penalization term that controls the “length” of the interfaces in some sense:

infP∈P(D)

∑

j∈J

infϕj∈H1(Ωj)

∫

Ωj

ε|∇ϕj |2 + |ϕj − I|2 dx + c HN−1(∂ ∪j∈J Ωj) (9.46)

for some c > 0, where HN−1 is the Hausdorff (N − 1)-dimensional measure. Equiv-alently, we are looking for

Ω = ∪j∈JΩj open in D such that χΩ = χD a.e. in D

(χΩ and χD, characteristic functions) solution of the following minimization problem:

infΩ open ⊂DχΩ=χD a.e.

infϕ∈H1(Ω)

∫

Ωε|∇ϕ|2 + |ϕ− I|2 dx + c HN−1(∂Ω). (9.47)

In general, the Hausdorff measure is not a lower semicontinuous functional. Soeither the (N − 1)-Hausdorff measure HN−1 is relaxed to a lower semicontinuousnotion of perimeter or the minimization problem is reformulated with respect toa more suitable family of open domains and/or a space of functions larger thanH1(Ω).

Another way of looking at the problem would be to minimize the number J ofconnected subsets of the open partition, but this seems more difficult to formalize.

9.5.3 Relaxation of the (N − 1)-Hausdorff Measure

The choice of a relaxation of the (N − 1)-Hausdorff measure HN−1 is critical. Herethe finite perimeter of Caccioppoli (cf. section 6 in Chapter 5) reduces to theperimeter of D since the characteristic function of Ω is almost everywhere equalto the characteristic function of D. However, the relaxation of HN−1(∂Ω) to the(N−1)-dimensional upper Minkowski content by D. Bucur and J.-P. Zolesio [8]is much more interesting and in view of its associated compactness theorem (cf.section 12 in Chapter 7) it yields a unique solution to the problem.

9.5.4 Relaxation to BV-, Hs-, and SBV-Functions

The other avenue to explore is to reformulate the minimization problem with respectto a space of functions larger than H1(Ω).

One possibility is to use functions with possible jumps, such as functions ofbounded variations. In dimension 1, the BV-functions can be decomposed into an


absolutely continuous function in W 1,1(0, 1) plus a jump part at a countable numberof points of discontinuity. Such functions have their analogue in dimension N . Withthis in mind one could consider the penalized objective function

JBV(ϕ) def=∫

D|ϕ− I|2 dx + ε∥ϕ∥2BV(D), (9.48)

which is similar to the Tikhonov regularization when rewritten in the form

JH1(ϕ) def=∫

D|ϕ− I|2 dx + ε∥ϕ∥2H1(D) =

∫

D(1 + ε)|ϕ− I|2 dx + ε|∇ϕ|2 dx,

where we square the penalization term to make it differentiable. Unfortunately, thesquare of the BV-norm is not differentiable.

Going back to Figure 1.9, assume that the triangle T has an intensity of 1,the circle C has an intensity of 2/3, and the remaining part of the square S hasan intensity of 1/3. The remainder of the image is set equal to 0. Then the imagefunctional is precisely

I(x) = χT (x) +23χC +

13χS\(T∪C).

We know from Theorem 6.9 in section 6.3 of Chapter 5 that the characteristicfunction of a Lipschitzian set is a BV-function. Therefore, if we minimize theobjective functional (9.48) over all ϕ ∈ BV(D), we get exactly ϕ = I. If we insiston formulating the minimization problem over a Hilbert space as for the Tikhonovregularization, the norm of the penalization term can be chosen in the space Hs(D)for some s ∈ (0, 1]:

JHs(ϕ) def=∫

D|ϕ− I|2 dx + ε ∥ϕ∥2Hs(D).

For 0 < s < 1/2, the norm is given by the expression

∥ϕ∥Hs(D)def=∫

Ddx

∫

Ddy

|ϕ(y)− ϕ(x)|2

|y − x|N+2s

and the relaxation is very close to the one in BV(D) since BV(D)∩L∞(D) ⊂ Hs(D),0 < s < 1/2 (cf. Theorem 6.9 (ii) in Chapter 5).

However, as seducing as those relaxations can be, it is not clear that the ob-jectives of the original formulation are preserved. Take the BV-formulation. It isimplicitly assumed that there are clean jumps across the interfaces. This is true indimension 1, but not in dimension strictly greater than 1 as explained in L. Am-brosio [1], where he points out that the distributional gradient of a BV-functionwhich is a vector measure has three parts: an absolutely continuous part, a jumppart, and a nasty Cantor part. The space BV(D) contains pathological functions ofCantor–Vitali type that are continuous with an approximate differential 0 almosteverywhere, and this class of functions is dense in L2(D), making the infimum of


JBV over BV(D) equal to zero without giving information about the segmentation.To get around this difficulty, he introduces the smaller space SBV(D) of functionswhose distributional gradient does not have a Cantor part. He considers the math-ematical framework for the minimization of the objective functional

J(ϕ, K) def=∫

D\K|ϕ− I|2 + ε|∇ϕ|2 dx + c HN−1(K)

with respect to all closed sets K ⊂ Ω and ϕ ∈ H1(Ω\K), where I ∈ L∞(D). Thisproblem makes sense and has a solution in SBV(D) when K is replaced by the setSϕ of points where the function ϕ ∈ SBV(D) has a jump, that is,

J(ϕ) def=∫

D|ϕ− I|2 + ε|∇ϕ|2 dx + c HN−1(Sϕ),

that turns out to be well-defined on SBV(D).

9.5.5 Cracked Sets and Density Perimeter

The alternate avenue to explore is to reformulate the minimization problem withrespect to a more suitable family of open domains (cf. section 15 of Chapter 7).

An additional reason to do that would be to remove the term on the length ofthe interface. When a penalization on the length of the segmentation is included,long slender objects are not “seen” by the numerical algorithms since they havetoo large a perimeter. In order to retain a segmentation of piecewise H1-functionswithout a perimeter term, M. C. Delfour and J.-P. Zolesio [38] introduced thefamily of cracked sets (see Figure 1.11) that yields a compactness theorem in theW 1,p-topology associated with the oriented distance function (cf. Chapter 7).

Figure 1.11. Example of a two-dimensional strongly cracked set.

The originality of this approach is that it does not require a penalizationterm on the length of the segmentation and that, within the set of solutions,there exists one with minimum density perimeter as defined by D. Bucur andJ.-P. Zolesio [8].


Theorem 9.2. Let D be a bounded open subset of RN and α > 0 and h > 0 be realnumbers.3 Consider the families

F(D, h,α) def=

⎧⎨

⎩Ω ⊂ D :Γ = ∅ and ∀x ∈ Γ, ∃d, |d| = 1,

such that inf0<t<h

dΓ(x + td)t

≥ α

⎫⎬

⎭ ,

Ch,αb (D) def= bΩ : Ω ∈ F(D, h,α) ,

(9.49)

Fs(D, h,α) def=

⎧⎨

⎩Ω ⊂ D :Γ = ∅ and ∀x ∈ Γ, ∃d, |d| = 1,

such that inf0<|t|<h

dΓ(x + td)|t| ≥ α

⎫⎬

⎭ ,

(Ch,αb )s(D) def= bΩ : Ω ∈ Fs(D, h,α) .

(9.50)

Then Ch,αb (D) and (Ch,α

b )s(D) are compact in W 1,p(D), 1 ≤ p < ∞, where dΓ isthe distance function to the boundary of Ω.

In view of the above compactness, it can be shown that there exists a solutionto the following minimization problem:

infΩ∈F(D,h,α)

Ω open ⊂D, mN (Ω)=mN (D)

infϕ∈H1(Ω)

∫

Ωε |∇ϕ|2 + |ϕ− f |2 dx. (9.51)

Cracked sets form a very rich family of sets with a huge potential that is notyet fully exploited in the image segmentation problem. Indeed, they can be usednot only to partition the frame of an image but also to detect isolated cracks andpoints provided an objective functional sharper than the one of Mumford and Shahis used. For instance, in view of the connection between image segmentation andfracture theory hinted at in J. Blat and J.-M. Morel [1], the theory may havepotential applications in problems related to the detection of fractures or cracksor fracture branching and segmentation in geomaterials (cf. K. B. Broberg [1]),but this is way beyond the scope of this book. Some initial considerations aboutthe numerical approximation of cracked sets can be found in M. C. Delfour andJ.-P. Zolesio [41].

10 Shapes and Geometries: Backgroundand Perspectives

10.1 Parametrize Geometries by Functions or Functionsby Geometries?

In the buckling of the column and in the design of the thermal diffuser and thethermal radiator the geometry was a volume of revolution generated by rotationaround the z-axis of the hypograph between the axis and the graph of the function

3In view of the fact that the distance function dΓ is Lipschitzian with constant 1, we necessarilyhave 0 < α ≤ 1.

10. Shapes and Geometries: Background and Perspectives 37

defined in a variable interval [0, L]. They are examples of sets parametrized by func-tions or scalars. A well-documented weakness of this approach is that numericalcomputations of optimal shapes often yield boundary oscillations. This is typicalof the parametrization of the boundary by a function. It provides a good controlof the displacement in the direction normal to the graph but little control in thetangential direction.

Extending this approach to sets locally defined by several graphs becomestricky. For instance, it is not possible to represent a sphere or a torus from a bi-Lipschitzian mapping from some domain in the plane unless we replace the plane bya surface with facets as illustrated in Figure 1.12 for a ball (cf. M. C. Delfour [8]for the construction of a surface with facets from a C1,1-surface). Moreover, thesurface with facets of Figure 1.12 is closely associated with the ball and would notbe appropriate for a torus.

Figure 1.12. Example of a surface with facets associated with a ball.

Another family of function-parametrized sets is the family of images of a fixedset by a family of homeomorphisms or diffeomorphisms. This can be advantageousin problems where it is desirable to work with a fixed mesh of the original do-main, thus avoiding remeshing the domain at each step of the optimization process.Function-parametrized sets are also used in the identification of objects (cf., for in-stance, the school of R. Azencott, A. Trouve, and L. Younes). This approach can betraced back to R. Courant and D. Hilbert [1] in 1953 and the construction ofCourant metrics and complete metric spaces of images of a fixed closed or crack-freeopen set by A. M. Micheletti [1] in 1972. This approach is less interesting inoptimization problems where the topology of the set is part of the unknown fea-tures of the set. Indeed, the images of a fixed set by an homeomorphism cannotchange the topology of the fixed set. For instance, it cannot create holes that arenot present in the fixed set.

The example of the distribution of two materials in a fixed domain is genericof problems where the topology of the set is part of the unknowns. The variablechosen to represent the set is the characteristic function χA, a function parametrizedby the set A. As such, arbitrary geometries that are only measurable can be used asindependent variables. The topology (for instance, the number of holes) is not fixeda priori and extremely complex design can be obtained. Moreover, the underlying


Lp-spaces where the characteristic functions naturally live induce a metric

ρ([A1], [A2]) = ∥χA2 − χA2∥Lp(D)

on the family of equivalence classes of Lebesgue measurable subsets of a boundedmeasurable holdall D. This type of topology has been used in the proof of theMaximum Principle by Pontryagin and later by Ekeland.

Other examples of set-parametrized functions include the uniformly Lipschitz-ian distance function dA to a set A. The family of such functions for subsets of aholdall D, D bounded open, can be considered as a family of functions in C0(D) orW 1,p(D). For instance, the metric

ρ([A1], [A2]) = ∥dA2 − dA2∥C0(D)

([A] is the equivalence class of all subsets of D with the same closure) coincideswith the ecart mutuel between two sets introduced by D. Pompeiu [1] in his thesispresented in Paris in March 1905 that was studied in more detail by F. Hausdorff[2, “Quellenangaben,” p. 280, and Chap. VIII, sect. 6] in 1914. Unfortunately thatmetric does not preserve the volume functional, but the norm can be changed toobtain the new metric

ρ([A1], [A2]) = ∥dA2 − dA2∥W 1,p(D)

that preserves the volume functional.Another example of a set-parametrized function is the support function

σA(x) def= supa∈A

a · x

of convex analysis. Here the equivalence classes are

[A] def=B ⊂ D : co B = co A

,

that is, sets with the same closed convex hull. The associated metric is

ρ([A1], [A2]) = ∥σA2 − σA2∥C0(B),

where B is the unit open ball in RN (cf. M. C. Delfour and J.-P. Zolesio [17]).The general pattern that emerges from the last three examples is the following

one. Choose a fixed holdall D and let A denote a variable set in D. Select a familyof set parametrized functions fA and let

[A] def=B ⊂ RN : fA = fB

be the associate equivalence class. Finally, assume that the family of functions fAis contained in a Banach space B(D). Choose as a metric

ρ([A1], [A2]) = ∥fA2 − fA2∥B(D).

This approach will be detailed in Chapters 5, 6, and 7 for the characteristic functionχA, the distance function dA, and the oriented distance function bA, but many otherset-parametrized functions can be used according to the requirements of the problemat hand.


10.2 Shape Analysis in Mechanics and Mathematics

The terminology shape analysis has been introduced independently in at least twodifferent contexts: continuum mechanics and mathematical theory of partial differ-ential equations (cf., for instance, E. J. Haug and J. Cea [1]).

In continuum mechanics, shape analysis encompasses contributions to struc-tural mechanics of elastic bodies such as beams, plates, shells, arches, and trusses.In such problems, the objective is to optimize the compliance, e.g., the work of theapplied loadings, by choosing the design parameters of the structure. Many of theearly contributions are in dimension 2 and complete analytical solutions are oftenprovided. Yet, it is not always easy to distinguish between a shape optimizationproblem, such as the shape of a two-dimensional plate, and a distributed param-eter problem, such as the optimal thickness of that plate. The thickness that isoften considered as a shape parameter is really a distributed parameter over thetwo-dimensional domain, which also specifies the plate. When the thickness goes tozero in some parts of the plate, holes are created and induce changes in the topologyand the shape of the associated two-dimensional domain. This naturally leads totopological optimization, which deals with the connectivity of a domain, the numberof holes, the fractal dimension of the boundary, and ultimately the appearance ofa microstructure. This was exemplified by K.-T. Cheng and N. Olhoff [2]’scelebrated optimization of the compliance of the circular plate with respect to itsthickness under prescribed loading and a constraint on the volume of material.Such questions have received a lot of attention in specific cases and have beenanalyzed by homogenization methods or Γ-convergence (cf., for instance, F. Mu-rat and L. Tartar [1], the conference proceedings edited by M. Bendsøe andC. A. Mota Soares [1], the book by M. Bendsøe [1], and the book edited byA. Cherkaev and R. Kohn [1] that contains a selection of translations of keypapers written in French or Russian). However, many fundamental questions stillremain open. For instance, how does such an analysis affect the validity of theunderlying mechanical or physical models?

For convenience we shall refer to this viewpoint as the compliance analysis thatgenerally involves the extremum of the minimum of an energy or work functionalwith respect to some design parameters.

In the mathematical theory of partial differential equations, the analysis dealtwith the sensitivity of the solution of boundary value problems with respect tothe shape of the geometric domain on which the partial differential equation isdefined. This was done for different applications, including free boundary problems,noncylindric problems,4 and shape identification problems. This shape sensitivityanalysis was simultaneously developed for the solution of the partial differentialequation and for shape functionals depending on their solution.

In that context, the compliance analysis becomes a special case that, quiteremarkably, does not require the shape sensitivity analysis of the solution of theassociated partial differential equation (cf. section 2 of Chapter 10). This important

4A cylindric problem is a partial differential equation where the geometric domain is fixed andindependent of the time variable. A noncylindric problem is a partial differential equation wherethe geometric domain changes with time.


simplification arises from the fact that the compliance is the minimum of an energyor work functional. An historical example that also benefits from this property isthe shape derivative of the first eigenvalue of the plate studied at the beginning ofthe 20th century by J. Hadamard [1]. As for the compliance, the shape sensitivityanalysis of the first eigenfunction is not required even when the first eigenvalue isrepeated. This follows from the fact that the first eigenvalue can be expressed asa minimum through Rayleigh’s quotient or G. Auchmuty [1]’s dual variationalprinciple.

Shape sensitivity analysis deals with a larger class of shape functionals (e.g.,minimal drag, noise reduction) and partial differential equations (e.g., the waveequation, viscous or non-Newtonian fluids), where variational energy functionalsare usually not available and for which the compliance analysis is no longer appli-cable. Yet, the shape sensitivity analysis of the solution of the partial differentialequation can again be avoided by incorporating the partial differential equationinto a Hamiltonian or Lagrangian formulation. As in control theory, this yieldsa partial differential equation for the adjoint state that is coupled with the initialpartial differential equation or state equation. A precise mathematical justificationof this approach can be given when the Lagrangian has saddle points and the shapederivative can be obtained from theorems on the differentiability of saddle pointswith respect to a parameter even when the saddle point solution is not unique(cf. section 5 in Chapter 10).

The shape sensitivity analysis through a family of diffeomorphisms which pre-serve the smoothness of the images of a fixed domain is primarily a local analysis.It is used to establish continuity, to define derivatives, or to optimize in a narrowclass of domains with fixed regularities and topologies: it cannot create holes orsingularities that were not present in the initial domain. Consider the problem offinding the best location and shape of a hole of given volume in a homogeneouselastic plate to optimize the compliance or some other criterion under a given load-ing. The expectation is that the presence of the hole would improve the complianceover a homogeneous plate without holes. This problem has its analogue in controltheory, for instance the optimal placement of sensors and actuators for the con-trol and stabilization of large flexible space structures or flexible arms of robots.Another important example is the localization of sensors and actuators to achievenoise reduction in structures. The optimal placement is usually an integral part ofthe control synthesis.

In the early 1970s J. Cea, A. J. Gioan, and J. Michel [1] proposed tointroduce relaxed problems in which the optimal domain (here an optimal hole, theoptimal location of the support of the optimal control) was systematically replacedby a density function ranging between zero and one and that would hopefully be acharacteristic function (a bang bang control) in the optimal regime. Furthermore,in order to work on a fixed domain D without holes, they replaced the holes bya very weak elastic material. This changed the original topological optimizationproblem into the identification of distributed coefficients over the fixed domain D.Equivalently, this identification problem reduces to finding the optimal distributionof two materials for a transmission equation in D under a volume constraint on oneof the two materials. Under appropriate conditions, the solution of this problem isa characteristic function, even when the space of distributed parameters is relaxed


to the closed convex hull of the set of characteristic functions (cf. Chapter 5).This general technique can also be used to study the continuity of the solution ofthe homogeneous Dirichlet boundary value problems with respect to the domainΩ, by introducing a transmission problem over a fixed domain D and letting thedistributed coefficient go to infinity in the complement of Ω with respect to D. Ofcourse, the coefficient over the complement of Ω could be replaced by a Lagrangemultiplier, thus providing a formulation over the fixed domain D. This is one ofthe many ways to make the domain fictitious and avoid dealing directly with thegeometry (cf. R. Glowinski, T.-W. Pan, and J. Periaux [1]).

10.3 Characteristic Functions: Surface Measure andGeometric Measure Theory

The main advantage of the relaxation to a characteristic function in the formulationof problems involving a domain integral and/or a volume constraint on Lebesguemeasurable sets is that the unknown set is completely described by a single functioninstead of a family of local diffeomorphisms.

Yet, such measurable sets can be quite unstructured. For example, in problemsinvolving a surface tension on the free boundary of a fluid or on the interface be-tween two fluids, the domain must have a locally finite boundary measure. Anotherexample is when the objective functional is a function of the normal derivative of thestate variable along the boundary (e.g., a flow or a thermal power flux through theboundary). In both cases the use of characteristic functions is limited by the factthat they are not differentiable on the boundary of the set and cannot be readilyused to describe very smooth domains.

Fortunately, there is enough latitude to make sense of a locally finite boundarymeasure for sets for which the characteristic function is of bounded variation; thatis, its gradient is a vector of bounded measures. Such sets are known as Caccioppolior finite perimeter sets. They were a key ingredient in the contribution of E. DeGiorgi to the theory of minimal surfaces in the 1950s, since the norm of that vectormeasure (which is the total variation of the characteristic function) turns out to be arelaxation of the boundary measure or “perimeter” of the set. To get a compactnessresult for a family of Caccioppoli sets it is sufficient to put a uniform bound on theirperimeters. This field of activities is known as geometric measure theory, and itstools have been used very successfully in the theory of free and moving boundaryproblems. Even though it came rather late to shape analysis, this material is bothimportant and fundamental.

10.4 Distance Functions: Smoothness, Normal, and Curvatures

Another set-parametrized function that plays a role similar to the characteristicfunction is the distance function between closed subsets of a fixed holdall5 D ofRN. The “ecart mutuel” between two sets was introduced by D. Pompeiu [1] in

5This set D will play several roles in this book. It is the universe in which the variable subsetslive. It will often be referred to as the underlying holdall. In other circumstances it will have apurely technical role, much like the control volume in fluid mechanics.


his thesis presented in Paris in March 1905. This is the first example of a metric be-tween two sets in the literature. It was studied in more detail by F. Hausdorff [2,“Quellenangaben,” p. 280, and Chap. VIII, sect. 6] in 1914. The Pompeiu–Hausdorffmetric between two sets corresponds to the uniform norm of the difference of theirrespective distance functions. For most applications it is not a very interestingtopology since the volume functional is not continuous in that topology. The con-tinuity of the volume functional can be recovered by replacing the uniform normby the W 1,p-norm since the characteristic function of the closure of the set can beexpressed in terms of the gradient of the distance function.

In 1951, H. Federer [1] introduced the family of sets of positive reach andgave a first access to curvatures from the distance function in much the same spiritas the perimeter from the characteristic function. They are sets for which the pro-jection onto the set is unique in a neighborhood of the set. Since the projectioncan be expressed in terms of the gradient of the square of the distance function,this is equivalent to requiring that the square of the distance function be C1,1 ina neighborhood of the set. Since the gradient of the distance function has a jumpdiscontinuity at the boundary of the set, he managed to recover from the distancefunction the curvature measures of the boundary and made sense of the classicalSteiner formula for sets with positive reach.

The jump discontinuity of the characteristic function across the boundarycan be bypassed by going to the oriented distance function. For smooth domainsthis function is quite remarkable since it inherits the same degree of smoothnessin a neighborhood of the boundary as the boundary itself. Then the gradient,the Hessian matrix, and the higher-order derivatives in the neighborhood of theboundary can be used to characterize and compute normals, curvatures, and theirderivatives along the boundary. This correspondence remains true for domainsof class C1,1. The gradient of the oriented distance function coincides with theoutward unit normal on the boundary. This implicit orientation is at the origin ofthe terminology oriented distance function that was introduced by M. C. Delfourand J.-P. Zolesio [17] in 1994 to distinguish it from the algebraic distance functionto a submanifold that can be defined only in terms of some discriminating criterionthat distinguishes between what is above and what is below the submanifold. Thisis not always possible, while the oriented distance function to a set is always well-defined. The restriction of the Hessian matrix of the oriented distance function tothe boundary coincides with the second fundamental form of differential geometry.Its eigenvalues are zero and the principal curvatures of the boundary.

A nice relaxation of the curvatures is obtained by considering sets for whichthe elements of the Hessian matrix of the oriented distance function are boundedmeasures. They are called sets of locally bounded curvature (cf. M. C. Delfourand J.-P. Zolesio [17, 32]). To get a compactness result, it is sufficient to put auniform bound on the total variation of the gradient (the elements of the Hessianmatrix are bounded measures).

The oriented distance function also provides a framework to compare thesmoothness of sets ranging from arbitrary sets with a nonempty boundary to setsof class C∞ via the smoothness of the oriented distance function in a neighborhoodof the boundary of the set. As a result Sobolev spaces can be used to introduce the


notion of a Sobolev domain, which becomes intertwined with the classical notion ofCk-domains.

10.5 Shape Optimization: Compliance Analysisand Sensitivity Analysis

As in the vector space case, optimization and control problems with respect togeometry are of various degrees of difficulty. When the objective functional does notdepend on the solution of a state equation or variational inequality defined on thevariable domain, it is sufficient to invoke compactness and continuity arguments. Inspecial cases such as the optimization of the compliance or of the first eigenvalue, theproblem can be transformed into the optimization of an objective functional thatis itself the minimum of some appropriate functional defined on a fixed functionspace. There a direct study of the dependence of the solution of the state equationwith respect to the underlying domain can be bypassed. In the general case, a stateequation constraint has to be handled very carefully from both the mathematicaland the application viewpoints. When the analysis can be restricted to families ofLipschitzian or convex domains, it is usually possible to give a meaning and provethe continuity of the solution of the state equation with respect to the underlyingvarying domain.

When families of arbitrary bounded open domains are considered, new phe-nomena can occur. As the domains converge in some sense, the correspondingsolutions may only converge in a weak sense to the solution of a different type ofstate equation over the limit domain: boundary conditions may no longer be sat-isfied, strange terms6 may occur on the right-hand side of the equation, etc. Insuch cases, it is often a matter of modeling of the physical or technological phe-nomenon. Is it natural to accept a generalized solution or a relaxed formulationof the state equation or should the family of domains be sufficiently restricted topreserve the form of the original state equation and maintain the continuity of thesolution with respect to the domain? The most suitable relaxation is not necessarilythe most general mathematical relaxation. It must always be compatible with thephysical or technological problem at hand. Too general a relaxation of the problemor too restrictive conditions on the varying domains can yield completely unsatis-factory solutions however nice the underlying mathematics are. A good balance ofmathematical, physical, and engineering intuitions is essential.

The study of the shape continuity of the solution of the partial differentialequation is also of independent mathematical interest. In the literature, this issuehas been addressed in various ways. Some authors simply introduce a stabilityassumption that essentially says that the limiting domain is such that continuityoccurs. Others introduce a set of more technical assumptions that correspond to aminimal set of conditions to make the crucial steps of the proof of the continuitywork. On the constructive side the really challenging issue is to characterize thefamilies of domains for which the continuity holds.

6Cf., for instance, D. Cioranescu and F. Murat [1, 2].


Many authors have constructed compact families for that purpose. For in-stance the Courant metric topology was used by A. M. Micheletti [1] in 1972 forCk-domains, the uniform cone condition by D. Chenais [1] in 1973 for uniformlyLipschitzian domains, and other metric topologies by F. Murat and J. Simon [1]in 1976 for Lipschitzian domains.

More general capacity conditions were introduced after 1994 by D. Bucur andJ.-P. Zolesio [5] in order to obtain compact subfamilies of domains with respect tothe complementary Hausdorff topology and control the curvature of the boundariesof the domains. In dimension 2 they recover the nice result of V. Sverak [2]in 1993, which involves a uniform bound on the number of connected componentsof the complement of the sets. Intuitively the capacity conditions are such that,locally, the complement of the domains in the chosen family has “enough capacity”to preserve the homogeneous Dirichlet boundary condition in the limit. Yet, thecapacity conditions might not be easy to use in a specific example. So D. Bucurand J.-P. Zolesio [1, 5] introduced a simpler geometric constraint, called the flatcone condition, under which the continuity and compactness results still hold. Thisgeneralizes the uniform cone property to a much larger class of open domains. Allthis will be further generalized in section 9 of Chapter 8.

10.6 Shape DerivativesFor functionals defined on a family of domains, it is important to distinguish betweena function with values in a topological vector space over a fixed domain and afunction such as the solution of a partial differential equation that lives in a Sobolevspace defined on the varying domain. In the latter case special techniques have tobe used to transport the solution onto a fixed domain or to first embed the varyingdomains into a fixed holdall D, extend the solutions to D, and enlarge the Sobolevspace to a large enough space of functions defined over the fixed holdall D. In bothcases the function is defined over a family of domains or sets belonging to someshape space which is generally nonlinear and nonconvex. Thus, defining derivativeson those spaces is more related to defining derivatives on differentiable manifoldsthan in vector spaces.

It is perhaps for groups of diffeomorphisms that the most complete theory ofshape derivatives is available. This material will be covered in detail in Chapters 2,3, and 9, but it is useful to briefly introduce the main ideas and definitions here.Given a Banach space Θ of mappings from a fixed open holdall D ⊂ RN into RN,first consider the group of diffeomorphisms

F(Θ) def=F : D → D : F − I ∈ Θ and F−1 − I ∈ Θ

.

Then consider the images of a fixed domain Ω0 in D by F(Θ):

X (Ω0)def= F (Ω0) : ∀F ∈ F(Θ) .

They can be identified with the quotient group

F(Θ)/G(Ω0), G(Ω0)def= F ∈ F(Θ) : F (Ω0) = Ω0


of diffeomorphisms of D. For the specific choices of D, Θ, and Ω0 that are ofinterest, this quotient group can be endowed with the so-called Courant metric tomake it a complete metric space. This metric space is neither linear nor convex.

For unconstrained domains (D = RN) and “sufficiently small” elements θ ∈ Θ,transformations of the form F = I + θ belong to F(Θ) and hence perturbations ofthe identity (i.e., of Ω0) can be chosen in the vector space Θ. This makes it possibleto define directional derivatives and speak of Gateaux and Frechet differentiabilitywith respect to Θ as in the classical case of functions defined on vector spaces.However, this approach does not extend to submanifolds D of RN or to domainsthat are constrained in one way or another (e.g., constant volume, perimeter, etc.).

In the unconstrained case, we get an infinite-dimensional differentiable man-ifold structure on the quotient group F(Θ)/G(Ω0) and the tangent space to F(Θ)is the whole linear space Θ. As a result ideas and techniques from differential ge-ometry are readily applicable. For instance, the continuity in I (i.e., at Ω0) canbe characterized by the continuity along one-dimensional flows of velocity fields inF(Θ) through the point I (that is, I(Ω0) = Ω0). This suggests defining a notion ofdirectional derivative along one-dimensional flows associated with “velocity fields”V which keep the flows inside D. When D is a smooth submanifold of RN suchvelocities are tangent to D and the set of all such velocities has a vector space struc-ture. This key property generalizes to other types of holdalls. So it is possible todefine directional derivatives and speak of shape gradient and shape Hessian withrespect to the associated vector space of velocities. This second approach has beenknown in the literature as the velocity method. Another very nice property of thatmethod is that perturbations of the identity, as previously defined in the uncon-strained case, can be recovered by a special choice of velocity field, thus creating acertain unity in the methodology.

The concept of topological derivative was introduced by J. Soko!lowski andA. Zochowski [1] for problems where the knowledge of the optimal topology ofthe domain is important. It gives some sensitivity of a shape functional to thepresence of a small hole at a point of the domain as the size of that hole goes tozero. For a domain integral, that is, the integral of a locally Lebesgue integrablefunction, this derivative is the negative of the classical set-derivative which is equalto the function at almost every point. This is a direct consequence of the Lebesguedifferentiation theorem. So, at least in its simplest form, this approach aims atextending the classical concept of set-differentiation as the inverse of integrationover sets.

This type of derivative becomes more intricate as we look at the sensitivity ofthe solution of a boundary value problem or at shape functionals which are functionsof that solution. For functions that are the integrals of an integrable function withrespect to a Radon measure, the Lebesgue differentiation extends in the form of theLebesgue–Besicovitch differentiation theorem, which says that the set-derivative ofthat integral is again equal to the function almost everywhere with respect to theRadon measure. So one could try to determine the class of shape functionals thatcan be expressed as an integral with respect to some Radon measure. Since thismaterial requires extensive technical results to fully appreciate its impact, we didnot include a chapter on this topic in the book, but the approach is an important


addition to the global arsenal! Fortunately, the theoretical and numerical work iswell documented in the literature and books by J. Soko(lowski and his coauthorsshould be available in the near future.

10.7 Shape Calculus and Tangential Differential CalculusThe velocity method has been used in various applications and contexts, and a verycomplete shape calculus is now available (for instance, the reader is referred to thebook of J. Soko!lowski and J.-P. Zolesio [9]). In the computations of derivatives,and especially of second-order derivatives, an intensive use is made of the tangentialdifferential calculus. In order to avoid parametrizations and local bases, tangentialderivatives are defined through extensions of functions from the boundary to somesmall Euclidean neighborhood. Their importance should not be underestimatedfrom both the theoretical and the computational points of view. The use of anintrinsic tangential gradient, divergence, or Laplacian can considerably simplify thecomputation and the final form of the expressions making more apparent their finestructure. Too many computations using local coordinates, Christoffel symbols, orintricate parametrizations are often difficult to decipher or to use effectively.

This book provides the latest and most important developments of that calcu-lus. They arises from the systematic use of the oriented distance function in thetheory of thin and asymptotic shells.7 In that context, it has been realized thatextending functions defined on the boundary Γ of a domain Ω by composition withthe projection onto Γ results in sweeping simplifications in the tangential calculus.This is due to the fact that the projection can be expressed in terms of the gradientof the oriented distance function. Computing derivatives on Γ becomes as easy ascomputing derivatives in the Euclidean space. Curvature terms, when they occur,appear in the right place and in the right form through the Hessian matrix of theoriented distance function that coincides with the second fundamental form of Γ.Chapter 9 will provide a self-contained introduction to these techniques and showhow the combined strengths of the shape calculus and the tangential differentialcalculus considerably simplify computations and expand our capability to tacklehighly complex and challenging problems.

10.8 Shape Analysis in This BookProblems in which the design, control, or optimization variable is no longer a vectorof parameters or functions, but is the shape of a geometric domain, a set, or evena “fuzzy entity,” cover a much broader range of applications than those for whichthe compliance or the shape sensitivity analysis have been used. Yet their analysismakes use of common mathematical techniques: partial differential equations, func-tional analysis, geometry, modern optimization and control theories, finite elementanalysis, large scale constrained numerical optimization, etc.

In this book the terminology shape will be used for domains ranging fromunstructured sets to C∞-domains. Relaxations of their geometric characteristicssuch as the volume, perimeter, connectivity, curvatures, and their derivatives will

7Cf. M. C. Delfour and J.-P. Zolesio [19, 20, 25] and M. C. Delfour [3, 4, 6, 7].

11. Shapes and Geometries: Second Edition 47

be considered. Shape spaces (often metric and complete) corresponding to differentlevels of smoothness or degrees of relaxation of the geometry will be systematicallyconstructed. For smooth domains we will emphasize the use of groups of diffeo-morphisms endowed with the Courant metric. For more general domains we willuse a generic construction based on the use of set-parametrized functions. Thecharacteristic function will be associated with metric spaces of equivalence classesof Lebesgue measurable sets. The distance function will be associated not onlywith the uniform Hausdorff metric topology in the space of continuous functionsbut also with W 1,p-topologies for which the characteristic function and hence thevolume function are continuous. The oriented distance function will be used in thesame way to generate new metric topologies. In each case, the metric is constructedfrom the norm of one of the set-parametrized functions in an appropriate functionspace. The construction is generic and applies to other choices of set-parametrizedfunctions and function spaces. For instance, the support function of convex analysiscan be used to generate a complete metric topology on equivalence classes of setswith the same closed convex hull (cf. M. C. Delfour and J.-P. Zolesio [17]).

The nice property of the characteristic and distance functions over classicallocal diffeomorphisms is that the set is globally described in terms of the analyti-cal properties of a single function. For instance, the gradient of the characteristicfunction yields a relaxed definition of the perimeter, and the Hessian of the dis-tance function yields the boundary measure and curvature terms. But the charac-teristic function and the gradient of the distance function are both discontinuous atthe boundary of the set. This seriously limits their use in the description of smoothdomains.

In contrast, the oriented distance function can describe a broad spectrum ofsets ranging from arbitrary sets with nonempty boundary to open C∞-domainsaccording to its degree of smoothness in a neighborhood of the boundary of theset. It readily combines the advantages of local diffeomorphisms and the character-istic and distance functions that are readily obtained from it. This provides, as infunctional analysis, a common framework for the classification and comparison ofdomains according to their relative degree or lack of smoothness. As in geometricmeasure theory, compact families of sets will be introduced based on the degree ofdifferentiability. One interesting family is made up of the sets with locally boundedcurvature, which provide a sufficient degree of relaxation for most applications andfor which nice compactness theorems are available (cf. sections 5 and 11 in Chap-ter 7). This family includes Federer’s sets of positive reach and hence closed convexand semiconvex sets.

11 Shapes and Geometries: Second EditionExcept for the last two chapters, this second edition is almost a new book. Chapters2, 3, and 4 give the classical description of sets and domains from the point of view ofdifferential geometry. Special attention is paid to domains that verify some segmentproperties and have a local epigraph representation and to domains that are theimage of a fixed set by a family of diffeomorphisms of the Euclidean space.


Chapters 5, 6, and 7 give a function analytic description of sets and domainsvia the set-parametrized characteristic functions, distance functions, and orienteddistance functions. It emphasizes the fact that we are now dealing with equivalenceclasses of sets that may or may not have an invariant open or closed set representa-tive in the class. In particular, they include Lebesgue measurable sets and Federer’ssets of positive reach. Many of the classical properties of sets can be recovered fromthe regularity or function analytic properties of those functions. We concentrate onthe basic properties, the construction of spaces of domains, metrics, and topologies,and the characterization of compact families.

Chapter 8 deals with problem formulations and shape continuity and optimiza-tion for some generic examples.

Chapters 9 and 10 are devoted to a modern version of the shape calculus, anintroduction to the tangential differential calculus, and the shape derivatives undera state equation constraint.

11.1 Geometries Parametrized by Functions

Chapter 2 gives several classical characterizations and properties of sets or domainsfrom the point of view of differential geometry : sets locally described in a neighbor-hood of each point of their boundary by an homeomorphism or a diffeomorphism(e.g., Ck or Holderian diffeomorphisms), by the epigraph of a C0 function (e.g., Lip-schitzian or Holderian domains), or by a geometric property (e.g., segment, cone,cusp), or sets globally described by the level sets of a C1-function.

The sections on sets that are locally the epigraph of a C0 function and setshaving one of the segment properties have been completely reorganized, rewritten,and enriched with new results and older ones that are difficult to find in the literatureother than in the form of folk theorems8 or theorems without satisfying proofs.Special attention is given to the uniform fat segment property that was introducedin the first edition of the book under the name uniform cusp property. Severalequivalent properties are given for domains that satisfy a uniform segment propertyand the stronger uniform fat segment property that can be expressed in terms of adominating function and its modulus of continuity. All this is specialized to domainsverifying a uniform cone or cusp property.

Chapter 3 adopts another point of view by considering families of sets thatare the images of a fixed subset of RN by some family of transformations of RN.The structure and the topology of the images can then be specified via the natu-ral algebraic and topological structures of transformations or equivalence classes oftransformations for which the full power of function analytic methods is available.In 1972, A. M. Micheletti [1] introduced the so-called Courant metric and gavewhat seems to be the first construction of a complete metric topology on the images

8As the term is understood by mathematicians, folk mathematics or mathematical folkloremeans theorems, definitions, proofs, or mathematical facts or techniques that are found by in-vestigation and may circulate among mathematicians by word-of-mouth but have not appearedin print, either in books or in scholarly journals. Knowledge of folklore is the coin of the realmof academic mathematics, showing the relative insight of investigators (http://en.wikipedia.org/wiki/Mathematical folklore).


of a fixed set. It is the nonlinear and nonconvex character of such shape spaces thatwill make the differential calculus and the analysis of shape optimization problemsmore challenging than their counterparts in topological vector spaces. The con-struction of A. M. Micheletti [1] is generic and readily extends to many familiesof transformations of RN.

The new version of Chapter 3 considerably expands the material and ideasof the first edition by extracting the fundamental assumptions behind the genericframework of A. M. Micheletti that leads to the Courant metrics on the quotientspace of families of transformations by subgroup of isometries such as identities,rotations, translations, or flips. Constructions are given for a large spectrum oftransformations of the Euclidean space and for arbitrary closed subgroups. Newcomplete metrics on the whole spaces of homeomorphisms and Ck-diffeomorphismsare also introduced. They extend classical results for transformations of compactmanifolds to general unbounded closed sets and open sets that are crack-free. Thismaterial is central in classical mechanics and physics and in modern applicationssuch as imaging and detection.

The former Chapter 7 on transformations versus flows of velocities has beenmoved right after the Courant metrics as Chapter 4 and considerably expanded.It now specializes the results of Chapter 3 to spaces of transformations that aregenerated by the flow of a velocity field over a generic time interval. The mainmotivation is to introduce a notion of semiderivatives as well as a tractable criterionfor continuity with respect to Courant metrics. The velocity point of view was alsoadopted by R. Azencott and A. Trouve starting in 1994 to construct completemetrics and geodesic paths in spaces of diffeomorphisms generated by a velocityfield with applications to imaging.

Chapter 4 also gives general equivalences between transformations and flowsof velocity fields for unconstrained and constrained families of domains and furthersharpens the results to the specific families of transformations associated with theCourant metrics studied in Chapter 2. Several examples of transformations andvelocities associated with widely used families of domains are given: C∞-domains,Ck-domains, Cartesian graphs, polar coordinates and star-shaped domains, andlevel sets. The chapter clarifies the long-standing issue of the equivalence of thecontinuity of shape functionals with respect to the Courant metrics and along theflow of velocity fields.

This chapter also prepares the ground for and motivates the definition of shapesemiderivatives that will be given in Chapter 9. As in the case of the continuitythe equivalent characterizations via transformations and flows of velocities are verymuch in the background and at the origin of the many seemingly different definitionswhich can be found in the literature. Preliminary considerations are first given tothe definition of a shape functional and to two candidates for the definition of adirectional shape semiderivative. They respectively correspond to perturbations ofthe identity associated with any one of the metric spaces constructed in Chapter 2and to the velocity method associated with the flow of a generally nonautonomousvector field. The first one seems to be limited to domains in RN, while the secondone naturally extends to domains living in a fixed smooth submanifold of RN.Moreover, the shape directional derivative obtained by perturbations of the identity


can be recovered by a special choice of velocity field. Most definitions of shapederivatives which can be found in the literature can be brought down to one of thetwo approaches.

11.2 Functions Parametrized by Geometries

Almost all compactness theorems for families of sets are specified by function an-alytic conditions on special set-parametrized families of functions. In this bookwe single out the characteristic function, the distance function, and the orienteddistance function. In the last case, a complete and convivial tangential differentialcalculus on the boundary of smooth sets will be obtained without local bases orChristoffel symbols and will be exploited in the computation of shape derivativeslater in Chapters 9 and 10.

Chapter 5 relaxes the family of classical domains to the equivalence classes ofLebesgue measurable sets. Using the characteristic function associated with a set,complete metric groups of equivalence classes of characteristic functions are con-structed. As for Courant metrics on groups, they are nonlinear and nonconvex. Onone hand, this type of relaxation is desirable in optimization problems where thetopology of the optimal set is not a priori specified; on the other hand, it necessi-tates the relaxation of the theory of partial differential equations on a smooth opendomain to measurable sets that have no smoothness and may not even be open.Furthermore some optimization problems yield optimal solutions where the char-acteristic function is naturally relaxed to a function between zero and one. Suchsolutions can be interpreted as microstructures, fuzzy sets, probability measures,etc. As a first illustration of the use of characteristic functions in optimization,the solution of the original problem of J. Cea and K. Malanowski [1] for theoptimization of the compliance with respect to the distribution of two materials isgiven with complete details. A second example deals with the buckling of columns,which is one of the very early optimal design problems formulated by Lagrange in1770. This is followed by the construction of the nice representative of an equiva-lence class of measurable functions: the measure theoretic representative. The lastsection is devoted to the Caccioppoli or finite perimeter sets, which have been in-troduced to solve the Plateau problem of minimal surfaces (J. A. F. Plateau [1]).Even if, by nature, a characteristic function is discontinuous at the boundary ofthe associated set, the characteristic function of Caccioppoli sets has some smooth-ness: it belongs to W ϵ,p(D), 1 ≤ ϵ < 1/p, p ≥ 1. This is sufficient to obtaincompact families of sets by putting a uniform bound on the perimeter. One suchfamily is the set of (locally) Lipschitzian (epigraph) domains contained in a fixedbounded holdall and satisfying a uniform cone property. This property puts a uni-form bound on the perimeter of the sets. The use of the theory of finite perimetersets is illustrated by an application to a free boundary problem in fluid mechanics:the modeling of the Bernoulli wave where the surface tension of the water entersvia the perimeter of the free boundary. The chapter concludes with an approxima-tion of the Dirichlet problem by transmission problems over a fixed larger space inorder to study the continuity of its solution with respect to the underlying movingdomains.


Chapter 6 moves on to the classical Pompeiu–Hausdorff metric topology whichis associated with the space of equivalence classes of distance functions of sets withthe same closure. As in Chapter 2, the construction of the metric on equivalenceclasses of sets with the same closure is generic. By going to the distance functionof the complement of the set in the uniform topology of the continuous functions,we get the complementary Hausdorff topology. These uniform topologies are oftentoo coarse for applications to physical or technological systems. But, since thedistance function is uniformly Lipschitzian, it can also be embedded into W 1,p-Sobolev spaces, and finer metric topologies can be generated. They offer definiteadvantages over the uniform Hausdorff topology in the sense that they preserve thecontinuity of the volume of sets since the characteristic function is continuous withrespect to W 1,p-topologies. Yet, we lose the compactness of the family of subsets ofa fixed bounded holdall of RN. Compact families are recovered by imposing somesmoothness on the Hessian matrix as in the case of the characteristic functionsof Chapter 5. Sets for which the gradient of the distance function is a vectorof functions of bounded variation are said to be of bounded curvature since theirHessian matrix is intimately connected with the curvatures of the boundary. Thisclass of sets is sufficiently large for applications and at the same time sufficientlystructured to obtain interesting theoretical results. For instance, such sets turnout to be Caccioppoli sets. Furthermore, the squared distance function is directlyrelated to the projection of a point onto the set and is used to characterize Federer’ssets of positive reach and construct compact families. Closed convex sets thatare completely characterized by the convexity of their distance function are alsoof locally bounded curvature. To complete the list of families of sets that areassociated with the distance function we introduce Federer’s sets of positive reachand a first compactness theorem. The chapter is complemented with a generalcompactness theorem for families of sets of global or local bounded curvature in atubular neighborhood of their boundary.

The use of the distance function of Chapter 6 to characterize the smoothnessof sets is limited by the fact that its gradient presents a jump discontinuity at theboundary. This is similar to the jump discontinuity of the characteristic function.To get around this difficulty we introduce the oriented distance function that isobtained by subtracting from the distance function to a set the distance functionto its complement, providing a level set description of the set. One remarkableproperty of this function is the fact that a set is of class C1,1 (resp., Ck, k ≥ 2) ifand only if its oriented distance function is locally C1,1 (resp., Ck) in a neighborhoodof its boundary. It also provides an orientation of the boundary since its gradientcoincides with the unit outward normal. This is why we use the terminology orienteddistance function. As in Chapter 6, Hausdorff and W 1,p-topologies and sets ofglobal or locally bounded curvature can be introduced. We now have a continuousclassification of sets ranging from sets with a nonempty boundary to C∞-sets, muchas in the theory of functions. Closed convex sets are characterized by the convexityof their oriented distance function. Convex sets and semiconvex sets are of locallybounded curvature. This property extends to Federer’s sets of positive reach.

Several compactness theorems in the W 1,p-topology of the oriented distancefunction are presented in this chapter for families of subsets of a bounded open


holdall with either bounded curvature in tubular neighborhoods of their boundary,or with a bound on the density perimeter of D. Bucur and J.-P. Zolesio [8], orwith a uniform fat segment property or equivalently the uniform boundedness andequicontinuity of all the local graphs of the sets in the family. The theorem on thecompactness under the uniform fat segment property is specialized to the family ofsubsets of a bounded holdall satisfying a uniform cone or cusp property. It is moregeneral than the one we obtained for Lipschitzian domains in Chapter 5, where theperimeter of each set was finite and uniformly bounded for all subsets of the holdall.Holderian domains do not necessarily have a locally finite boundary measure. Yet,the fact that the boundary of a Holderian domain may have cusp does not meanthat all Holderian domains do not have a locally finite boundary measure. In orderto include applications where the perimeter is bounded, the general compactnesstheorem is specialized to families of subsets of a holdall that verify the uniform fatsegment property with a uniform bound on either the De Giorgi perimeter or thedensity perimeter. A last section 15 deals with the family of cracked sets. They havebeen used in M. C. Delfour and J.-P. Zolesio [38] in the context of the imagesegmentation problem of D. Mumford and J. Shah [2] that will be detailed inthis section. Cracked sets are more general than sets which are locally the epigraphof a continuous function in the sense that they include domains with cracks, andsets that can be made up of components of different codimensions. The Hausdorff(N − 1) measure of their boundary is not necessarily finite. Yet, compact families(in the W 1,p-topology) of such sets can be constructed.

11.3 Shape Continuity and Optimization

Chapter 8 deals with problem formulations, continuity, or semicontinuity of shapefunctionals or of the solutions of boundary value problems with respect to theirunderlying domain of definition for selected generic examples. Combined with com-pact families of sets studied in the previous chapters, they are all essential to gettingthe existence of optimal shapes. This is illustrated to some extent in Chapter 5 forthe modeling of the transmission problem with the help of the characteristic func-tion that occurs both in the model and in the specification of the metric on theequivalence classes of measurable sets.

This chapter first reviews the continuity of the transmission problem usingcharacteristic functions. It characterizes the upper semicontinuity of the first eigen-value of the generalized Laplacian with respect to the domain using the comple-mentary Hausdorff topology. Then it studies the continuity of the solution of thehomogeneous Dirichlet and Neumann boundary value problems with respect totheir underlying domain of definition since they require different constructions andtopologies that are generic of the two types of boundary conditions even for morecomplex nonlinear partial differential equations. In problems where the objectivefunctional depends explicitly on the domain and the solution of an elliptic equationdefined on the same domain, the strong continuity of the solution with respect tothe underlying domain is the key element in the proof of the existence of optimaldomains. To get that continuity, some extra conditions have to be imposed on thefamily of open domains, such as the uniform fat segment property.


The second part extends some of the results to a larger family of domains sat-isfying capacity conditions which turn out to be important to obtain the continuityof solutions of partial differential equations with homogeneous Dirichlet boundaryconditions with respect to their underlying domain of definition. One special caseof a capacity condition is the flat cone condition that generalizes the condition ofV. Sverak [2] involving a bound on the number of connected components of thecomplement of the sets.

11.4 Derivatives, Shape and Tangential Differential Calculuses,and Derivatives under State Constraints

Chapter 9 is devoted to the essential shape calculus and the no less essential tan-gential or boundary calculuses since traces, normals, or tangential gradients willnaturally occur in the final expressions.

After a self-contained review of differentiation in topological vector spaces thatemphasizes Gateaux and Hadamard differentials, it introduces basic definitions offirst- and second-order Eulerian shape semiderivatives and derivatives by the ve-locity method. General structure theorems are given for Eulerian semiderivativesof a shape functional. They arise from the fact that shape functionals are usuallydefined over equivalence classes of sets, and hence only the normal part of the ve-locity along the boundary really affects the shape functional. Bridges are providedwith the method of perturbations of the identity. A section is devoted to a modernversion of the shape calculus. It gives the general formulae for the shape deriva-tive of domain and boundary integrals. From these formulae several examples areworked out, including the semiderivative of the boundary integral of the square ofthe normal derivative. For the computation of a broader range of shape derivativesthe reader is referred to the book by J. Soko!lowski and J.-P. Zolesio [9]. Inmost cases, both domain and boundary expressions are available for derivatives.The boundary expression usually contains more information on the structure of thederivative than its domain counterpart. Finally, to effectively deal with the differ-ential calculus in boundary integrals, we provide the latest version of the tangentialcalculus on C2-submanifolds of codimension 1 which has been developed in the con-text of the theory of shells (cf. M. C. Delfour and J.-P. Zolesio [28, 33] andM. C. Delfour [3, 7]). This calculus has been significantly simplified by using theprojection associated with the oriented distance function studied in Chapter 7. Thispowerful tool combined with the shape calculus makes it possible to obtain cleanexplicit expressions of second-order shape derivatives of domain integrals along witha better understanding of their fine structure.

Chapter 10, the final chapter, completes the shape calculus by introducingthe basic theoretical results and computational tools for the shape derivative offunctionals that depend on a state variable that is usually the solution of a partialdifferential equation or inequality defined over the varying underlying domain. Thefirst section concentrates on shape functionals that are of compliance type; that is,they are the minimum of an energy functional associated with the state equationor inequality. Such functionals are very nice in the sense that they do not generatean adjoint state equation, and their derivative can be obtained by theorems on the


differentiability of a minimum with respect to a parameter even when the minimiz-ers are not unique. A detailed generic example is provided to illustrate how to usethe function space parametrization to transport the functions in Sobolev spaces overvariable domains to a Sobolev space over the fixed larger holdall. These techniquesextend to more complex situations. For instance, Sobolev spaces of vector functionswith zero divergence can be transported by the so-called Piola transformation. Do-main and boundary expressions are provided. The main theorem is applied to theexample of the buckling of columns. An explicit expression of the semiderivativeof Euler’s buckling load with respect to the cross-sectional area is obtained fromthe main theorem, and a necessary and sufficient analytical condition is given tocharacterize the maximum Euler’s buckling load with respect to a family of cross-sectional areas. The theory is further illustrated by providing the semiderivative ofthe first eigenvalue of several boundary value problems over a bounded open do-main: Laplace equation, bi-Laplace equation, linear elasticity. In general, the firsteigenvalue is not simple over an arbitrary bounded open domain and the eigenvalueis not differentiable; yet the main theorem provides explicit domain and boundaryexpressions of the semiderivatives.

For general shape functionals, a Lagrangian formulation is used to incorporatethe state equation and to avoid the study of the derivative of the state equationwith respect to the domain. The computation of the shape derivative of a state-constrained functional reduces to the computation of the derivative of a saddle pointwith respect to a parameter even when the saddle point solution is not unique. Ityields an expression that depends on the associated adjoint state equation, muchlike in control theory, but here the domains play the role of the controls. The tech-nique is illustrated on both the homogeneous Dirichlet and the Neumann boundaryvalue problems by function space parametrization. An alternative to this methodis the function space embedding combined with the use of Lagrange multipliers. Itconsists in extending solutions of the boundary value problems over the variabledomains to a larger fixed holdall, rather than transporting them. This approachoffers many technical advantages over the other one. The computations are easierand they apply to larger classes of problems. This is illustrated on the nonhomoge-neous Dirichlet boundary value problem. Again domain and boundary expressionsfor the shape gradient are obtained. Yet the relative advantages of one method overthe other are very much problem and objective dependent. Finally, it is importantto acknowledge that the above techniques seem quite robust and are systematicallyused for nonlinear state equations and in contexts where optimization or saddlepoint formulations are not available.

Chapter 2

Classical Descriptions ofGeometries and TheirProperties

1 IntroductionThis chapter is devoted to the classical descriptions of nonempty subsets of thefinite-dimensional Euclidean space that are characterized by the smoothness orproperties of their boundary.

A first approach is to assume that we can associate with each point of theboundary a diffeomorphism from a neighborhood of that point that locally flattensthe boundary. Another way is to assume that the set is the union of the positivelevel sets of a continuous function and that the zero level is its boundary. A thirdway is to assume that in each point of the boundary the set is locally the epigraph ofa function. The smoothness of the set is then characterized by the smoothness of thecorresponding diffeomorphism, level function, or graph function. Those definitionsare equivalent for sufficiently smooth sets. Domains that are locally the epigraphof a continuous function in the third category also belong to the first category, butthe converse is generally not true.

The basic definitions and constructions for sets locally described by an isomor-phism or a diffeomorphism are given in section 3, for sets locally described by thelevel sets of a function in section 4, and for sets locally described by the epigraphof a function in section 5. Section 6 deals with sets characterized by geometricsegment properties. A set satisfying the basic segment property has the propertythat its boundary ∂Ω be locally a C0 epigraph (cf. section 6.2). The strongeruniform segment property is further strengthened by introducing the uniform fatsegment property and the concept of a dominating function that will be necessaryto construct compact families of sets later in Chapter 7. Complete equivalencesare provided between the three C0 epigraph properties and the three geometricsegment properties in sections 6.2 and 6.3. For sets with a compact boundary, thesix properties are shown to be equivalent in section 6.2. Section 6.4 specializesthe fat segment property to the geometric uniform cusp and cone properties thatrespectively characterize Holderian and Lipschitzian domains. The uniform coneproperty has provided one of the early examples of a compact family of Lipschitziandomains (cf. section 6.4 in Chapter 5) in 1973.

55

56 Chapter 2. Classical Descriptions of Geometries and Their Properties

2 Notation and Definitions2.1 Basic NotationN is the set of integers 1, 2, . . . and R the field of real numbers. The interior andthe closure of a subset A in RN will be denoted, respectively, by intA and A. Therelative complement of A in B will be written as

!BA (or B\A) def= x ∈ B : x /∈ A.

When B = RN, we also write !A or Ac. The boundary ∂A of A is defined as A∩!A.Since the superscript t will often appear in the book, the transpose of a vector

v and a matrix A will be denoted, respectively, by ∗v and ∗A. The inverse of Awill be denoted by A−1. The inner product and norm in RN will be written as

x · ydef=

N∑

i=1

xi yi, |x| =√

x · x.

For a linear transformation A : RN → RK , the norm of A is defined as

|A| def= max|x|RN≤1

|Ax|RK ,

and if Aij and Bij are the matrix representations of A and B with respect to somebases a1, . . . , aN and b1, . . . , bK of RN and RK , respectively, the double innerproduct is defined as

A · ·B def=N∑

i=1

K∑

j=1

AijBij (2.1)

and the associated norm√

A · ·A is equivalent to the norm |A| of A.

2.2 Abelian Group Structures on Subsets of a Fixed Holdall D

Given a nonempty set D, consider the power set of D

P(D) def= A : A ⊂ D .

The power set is closed under the operation of union, intersection, and complement,but those operations do not have an inverse with respect to the neutral element ∅:

B ∪A = A ∪B, A ∪∅ = A,

A ∪B = ∅ ⇐⇒ A = B = ∅.

Similarly, with the neutral element RN

A ∩B = B ∩A, A ∩RN = A,

A ∩B = RN ⇐⇒ A = B = RN .

2. Notation and Definitions 57

2.2.1 First Abelian Group Structure on (P(D), ∆)

Denote by the symmetric difference of two sets in P(D):

A Bdef=[A ∩ !B

]∪[B ∩ !A

]= [A ∪B] ∩

[!A ∪ !B

]= [A ∪B] ∩ ! [A ∩B] .

By definition, is commutative and associative:

A B = B A, (A B) C = A (B C).

∅ is a neutral element :

A ∅ = A.

An inverse must verify

A B = ∅ ⇐⇒[A ∩ !B

]∪[B ∩ !A

]= ∅

⇐⇒ B ∩ !A = ∅ = A ∩ !B ⇐⇒ A = B.

Therefore A A = ∅, every element A of P(D) is its own inverse, A−1 = A, and(P(D),) is an Abelian group. This yields a kind of triangle inequality :

(A B) (B C) = A C.

The symmetric difference of A and C is contained in the union of the symmetricdifference of A and B and that of B and C. (But note that for the diameter of thesymmetric difference the triangle inequality does not hold.)

Because every element in this group is its own inverse, (P(D),) is in fact avector space over the field Z2 with two elements.

Finally, the intersection distributes over the symmetric difference:

A ∩ (B C) = (A ∩B) (A ∩ C).

Hence the power set P(D) becomes a ring with symmetric difference as additionand intersection as multiplication. It is the prototypical example of a Boolean ring.

For operations involving the complement

!A !B = A B,

!(A B) = [A ∩B] ∪ [!A ∩ !B] = B !A = A !B

⇒ A RN = !A, A !A = RN.

For operations involving the closure

∂A = !(A !A

)= A !!A = !A !A

⇒ ∂A !!A = A and ∂A !A = !A.


2.2.2 Second Abelian Group Structure on (P(D), ∇)

If for some reason ∅ is not acceptable as the multiplicative neutral element, RN canplay that role with the following new multiplication:

ABdef= !

([A ∩ !B

]∪[B ∩ !A

])= !

([A ∪B] ∩ ! [A ∩B]

)= ! [A ∪B] ∪ [A ∩B] .

By definition

AB = BA, ARN = A.

A multiplicative inverse must verify

AB = RN ⇔[A ∩ !B

]∪[B ∩ !A

]= ∅

⇔ B ∩ !A = ∅ = A ∩ !B ⇔ A = B.

Therefore A−1 = A and (P(D),) are an Abelian group.When D = RN, some identities are interesting:

!A!B = AB,

A!B = !([A ∩B] ∪

[!B ∩ !A

])= !(AB) = !AB,

⇒ A∅ = !A, A!A = ∅.

2.3 Connected Space, Path-Connected Space,and Geodesic Distance

Definition 2.1. (i) A connected space is a topological space which cannot berepresented as the disjoint union of two or more nonempty open subsets.

(ii) A topological space X is said to be path-connected (or pathwise connected or0-connected) if for any two points x and y in X there exists a continuousfunction f from the unit interval [0, 1] to X with f(0) = x and f(1) = y.(This function is called a path in X from x to y.)

(iii) Given a path-connected subset S of RN, the geodesic distance distS(x, y)between two points x and y of S is the infimum of the lengths of the paths inS joining x and y.

The definitions of a connected space and a path-connected space are com-patible with the convention that the empty set ∅ be both connected and path-connected.

Theorem 2.1.

(i) The closure of a connected set is connected. But the closure of a path-connectedset need not be path-connected.

(ii) Every path-connected space is connected. But a connected space need not bepath-connected.


(iii) Y is path-connected if and only if it is connected, and each y ∈ Y has a path-connected neighborhood. In particular, open subsets of RN are connected ifand only if they are path-connected.

Proof. Cf. J. Dugundji [1]: (i) Thm. 1.4 and Ex. 4, p. 108. (ii) Thm. 5.3, p. 115.The topologist’s sine curve (x, y) : y = sin 1/x, x > 0 ∪ (0, y) : |y| ≤ 1 in R2 isan example of a connected space that is not path-connected. (iii) Thm. 5.5, p. 116,Cor. 5.6, p. 116.

Remark 2.1.Subsets of the real line R are connected if and only if they are path-connected;these subsets are the intervals of R.

Definition 2.2.Given a subset A of RN,

#(A) def=

number of connected components of A, if A = ∅,

0, if A = ∅.

We say that the set A is hole-free if !A is connected or A = RN.

If we adopt the convention that the empty set ∅ is both connected and path-connected, then a set A is connected if and only if its complement !A is hole-free.

Theorem 2.2. Let A and Ω be subsets of RN.

(i) A set A = ∅ is connected if and only if #(A) = 1; a set A is hole-free if andonly if #(!A) ≤ 1.

(ii) An open set Ω = ∅ is path-connected if and only if #(Ω) = 1; an open set Ωis hole-free if and only if #(!Ω) ≤ 1.

Remark 2.2.Observe that for a nonempty open subset Ω of an open holdall D in RN, #(!Ω∩D)can be strictly greater than #(!Ω). As an example, let D be the open square ofthe side equal to 2 centered in (0, 0) and Ω be the open ball of center (0, 0) andradius 1: #(!Ω ∩D) = 4 > 1 = #(!Ω).

2.4 Bouligand’s Contingent Cone, Dual Cone,and Normal Cone

Definition 2.3.Let ∅ = Ω ⊂ RN and let x ∈ Ω. The dual cone associated with Ω is defined as

Ω∗ def=x∗ ∈ RN : ∀x ∈ Ω, x∗ · x ≥ 0

. (2.2)

Ω∗ is a closed convex cone in 0.


Definition 2.4.Let ∅ = Ω ⊂ RN and let x ∈ Ω.

(i) The vector h ∈ RN is an admissible direction for Ω in x if there exists asequence εn > 0, εn 0 as n→∞, such that

∀n, ∃xn ∈ Ω and limn→∞

xn − x

εn= h. (2.3)

TxΩ will denote the set of all admissible directions for Ω in x and it will bereferred to as Bouligand’s contingent cone1 to Ω in x.

(ii) The dual cone (−TxΩ)∗ associated with Ω and a point x ∈ Ω will be referredto as the normal cone to Ω in x ∈ Ω.

In general, TxΩ is a closed cone in 0. If Ω is convex, then TxΩ = R(Ω− x) isa closed convex cone in 0. When x ∈ int Ω, TΩ(x) = RN.

2.5 Sobolev Spaces2.5.1 Definitions

For a detailed analysis of Sobolev spaces the reader is referred to R. A. Adams [1],J. Necas [1], and Vo-Khac Khoan [1]. Let Ω be a bounded open subset of RN.D(Ω) is the space of infinitely continuously differentiable functions with compactsupport in Ω endowed with Schwartz’s topology. Its topological dual D(Ω)∗ is calledthe space of distributions. Denote by NN the set of all N -tuples α = (α1, . . . ,αn) ∈NN . An element of NN will be called a multi-index. For each α ∈ NN , define theorder |α| of α and the partial derivative ∂α as follows:

|α| =N∑

i=1

αi, ∂α =∂|α|

∂xα11 . . . ∂xαN

N

. (2.4)

The Sobolev space Wm,p(Ω), 1 ≤ p ≤ ∞, m ∈ N, is the space of all distribu-tions T ∈ Lp(Ω)∗ with distributional partial derivatives DαT ∈ Lp(Ω)∗ for all α,|α| ≤ m. For p = 2 we shall use the notation Hm(Ω) = Wm,2(Ω). By density ofD(Ω) in Lp(Ω) for 1 ≤ p < ∞, we shall often identify a distribution T ∈ Lp(Ω)∗

with a function f ∈ Lp(Ω), p−1 + q−1 = 1:

∀ϕ ∈ D(Ω), ⟨T,ϕ⟩ =∫

Ωf ϕ dx.

Endowed with the norm

∥v∥m,p,Ωdef=

⎡

⎣∑

|α|≤m

∥∂αv∥pLp(Ω) dx

⎤

⎦1/p

, ∥v∥m,∞,Ωdef= max

|α|≤m∥∂αv∥L∞(Ω) dx (2.5)

1The contingent cone TΩ(x) was introduced in the 1930s by G. Bouligand [1]. It naturallyoccurs in the viability theory of differential equations (cf. M. Nagumo [1] or J.-P. Aubin andA. Cellina [1, p. 174 and p. 180]).


the space Wm,p(Ω) is a Banach space. We shall also use the seminorm

|v|m,p,Ω =

⎡

⎣∑

|α|=m

∫

Ω|∂αv|pLp(Ω) dx

⎤

⎦1/p

, |v|m,∞,Ωdef= ∥∂mv∥L∞(Ω) dx. (2.6)

When p = 2 we drop the index p and write ∥v∥m,Ω and |v|m,Ω. By the Meyers andSerrin theorem Wm,p(Ω) coincides with the completion of ϕ ∈ Cm(Ω) : ∥ϕ∥m,p <∞ with respect to the norm ∥ϕ∥m,p for 1 ≤ p <∞.2

The reader is referred to R. A. Adams [1] for details and extension of thisdefinition to the case where m is not an integer. When f is a vector function fromΩ to Rm, the corresponding spaces will be denoted by W s,p(Ω)m or W s,p(Ω,Rm).

2.5.2 The Space W m,p0 (Ω)

Define the Sobolev space

Wm,p0 (Ω) = D(Ω)

W m,p

, (2.7)

where the closure is with respect to the norm ∥ ·∥m,p,Ω of (2.5). When Ω is boundedand 1 ≤ p <∞, there exists a constant C(diam Ω, m, p) such that

∀ v ∈Wm,p0 (Ω), ∥v∥m,p,Ω ≤ C(diam Ω, m, p) |v|m,p,Ω (2.8)

and the seminorm | · |m,p,Ω is a norm on the space Wm,p0 (Ω) equivalent to the norm

∥ · ∥m,p,Ω (cf. W. P. Ziemer [1, Thm. 4.4.1, p. 188]). We have the chain ofcontinuous embeddings

Wm,p0 (Ω)→Wm,p(Ω)→ Lp(Ω).

For p = 2, m ≥ 1, and a bounded open Ω, the injection of Hm0 (Ω) into Hm−1

0 (Ω)is compact (cf. R. A. Adams [1, Rellich–Kondrachov compactness Theorem 6.2,p. 144] for other embedding theorems).

Now assume that the set Ω is Lipschitzian in the sense of Definition 5.2 (ii)in section 5.2. There exists a constant C(Ω) such as

∀ v ∈ C∞(Ω), ∥v∥L2(Γ) ≤ C(Ω) ∥v∥1,Ω. (2.9)

But, by Theorem 6.3 in section 6.1 C∞(Ω) = H1(Ω), where the closure is withrespect to the norm∥ · ∥1,Ω. Hence there exists a continuous linear map

v ∈ H1(Ω) 5→ v ∈ L2(Γ)

that will be referred to as the trace operator . From this we get the following char-acterization:

H10 (Ω) =

v ∈ H1(Ω) : v = 0 on Γ

. (2.10)

2Cm(Ω) is defined in section 2.6.1.


Since the exterior unit normal ν = (ν1, . . . , νN ) exists almost everywhere along Γ,the normal derivative operator

∂

∂ν=

N∑

i=1

νi∂

∂xi(2.11)

is also defined almost everywhere along Γ and the following characterization alsoholds:

H20 (Ω) =

v ∈ H2(Ω) : v = 0 and

∂v

∂ν= 0 on Γ

. (2.12)

Given two functions u, v in H1(Ω), we get the following Green formula:∫

Ωu∂v

∂xidx = −

∫

Ω

∂u

∂xiv dx +

∫

Γuv νi dΓ (2.13)

is true for 1 ≤ i ≤ N . From that formula, other Green formulae can be obtained.For instance, by replacing u by ∂u/∂xi and summing from 1 to N , we get

∫

Ω∇u ·∇v dx =

∫

Ω

N∑

i=1

∂u

∂xi

∂v

∂xidx = −

∫

Ω∆u v dx +

∫

Γ

∂u

∂νv dΓ (2.14)

for all u ∈ H2(Ω) and v ∈ H1(Ω).

2.5.3 Embedding of H10(Ω) into H1

0(D)

Let Ω and D be two open subsets of RN such that Ω ⊂ D. Denote by e0(ϕ) theextension by zero of an element ϕ of D(Ω) to D and consider the linear injection

ϕ 5→ e0(ϕ) : D(Ω)→ D(D).

By definition, for k ≥ 1,

∥ϕ∥Hk(Ω) = ∥e0(ϕ)∥Hk(D)

and e0 extends by continuity and density to a linear isometric map

e0 : Hk0 (Ω) def= D(Ω)

Hk

→ Hk0 (D) def= D(D)

Hk

.

Denote by Hk0 (Ω; D) the image of Hk

0 (Ω) by e0.

Theorem 2.3. Let Ω ⊂ D be two open subsets of RN such that D is bounded.The linear subspace Hk

0 (Ω; D) of Hk0 (D) is closed and isometrically isomorphic to

Hk0 (Ω) with the following properties: for all ψ ∈ Hk

0 (Ω; D)

ψ|Ω ∈ Hk0 (Ω) and ∀α, |α| ≤ k, ∂αψ = 0 a.e. in D\Ω. (2.15)

Any convergent sequence in Hk0 (Ω)-weak converges in Hk−1

0 (Ω)-strong.


Proof. (i) It is sufficient to prove that Hk0 (Ω; D) is closed. The other properties are

easy to check. Pick a sequence ϕn in Hk0 (Ω) such that e0(ϕn) is Cauchy in

Hk0 (D) and denote by Φ its limit in Hk

0 (D). Since e0 is an isometry, then ϕn isalso Cauchy in Hk

0 (Ω). Denote by ϕ its limit in Hk0 (Ω). Then

∥Φ− e0(ϕ)∥Hk(D) ≤ ∥Φ− e0(ϕn)∥Hk(D) + ∥e0(ϕn − ϕ)∥Hk(D)

= ∥Φ− e0(ϕn)∥Hk(D) + ∥ϕn − ϕ∥Hk(Ω) → 0

and there exists ϕ ∈ Hk0 (Ω) such that Φ = e0(ϕ) in Hk

0 (D). Hence Φ ∈ Hk0 (Ω; D).

(ii) Since Ω is bounded, there exists a sufficiently large open ball B such thatΩ ⊂ B. By the embedding of Hk

0 (Ω) into Hk0 (B), if a sequence ϕn converges to

ϕ in Hk0 (Ω)-weak, then eΩ(ϕn) converges to eΩ(ϕ) in Hk

0 (B)-weak. Since the ballis sufficiently smooth, by Rellich’s theorem, the sequence converges in Hk−1

0 (B)-strong, and in view of the linear isometric isomorphism, ϕn converges to ϕ inHk−1

0 (Ω)-strong.

2.5.4 Projection Operator

Given u ∈ H10 (D), let y = y(Ω) ∈ H1

0 (Ω; D) be the solution of the variationalproblem

∀ϕ ∈ H10 (Ω; D),

∫

D∇y ·∇ϕ dx =

∫

D∇u ·∇ϕ dx.

The projection operator PΩ of H10 (D) onto H1

0 (Ω) or H10 (Ω; D) is the mapping

u 5→ PΩudef= y(Ω) : H1

0 (D)→ H10 (Ω; D)

that is linear and continuous since

∥∇PΩu∥L2(D) ≤ ∥∇u∥L2(D).

A fundamental issue will be the continuity of the projection PΩ with respect to Ω.

2.6 Spaces of Continuous and Differentiable Functions

2.6.1 Continuous and Ck Functions

Let Ω be an open subset of RN. Denote by C(Ω) or C0(Ω) the space of continuousfunctions from Ω to R. For a multi-index α ∈ NN , let |α| be the order of α and ∂αbe the partial derivative as defined in (2.4) of section 2.5.1. For an integer k ≥ 1,

Ck(Ω) def=f ∈ Ck−1(Ω) : ∂αf ∈ C(Ω), ∀α, |α| = k

.

By convention ∂0f will be the function f in order to make sense of the case α =0. When |α| = 1, we also use the standard notation ∂if or ∂f/∂xi. Dk(Ω) orCk

c (Ω) (resp., D(Ω) or C∞c (Ω)) will denote the space of all k-times (resp., infinitely)


continuously differentiable functions with compact support contained in the openset Ω.

Denote by B0(Ω) the space of bounded continuous functions from Ω to R,and, for an integer k ≥ 1, the space

Bk(Ω) def=f ∈ Bk−1(Ω) : ∂αf ∈ B0(Ω), ∀α, |α| = k

,

that is, the space of all functions in B0(Ω) whose derivatives of order less than orequal to k are continuous and bounded in Ω. Endowed with the norm

∥f∥Ck(Ω)def= max

0≤|α|≤ksupx∈Ω

|∂αf(x)|, (2.16)

Bk(Ω) is a Banach space.When D is open but not necessarily bounded, the space C0(D) of continuous

functions on D is endowed with the Frechet topology of uniform convergence oncompact subsets K of D, which is defined by the family of seminorms

∀K compact ⊂ D, qK(f) def= maxx∈K

|f(x)|. (2.17)

It is metrizable since the topology induced by the family of seminorms qK isequivalent to the one generated by the subfamily qKkk≥1, where the compact setsKkk≥1 are chosen as follows:

Kkdef=

x ∈ D : d!D(x) ≥ 1k

and |x| ≤ k

, k ≥ 1 (2.18)

(cf., for instance, J. Horvath [1, Ex. 3, p. 116]). Thus the Frechet topology onC0(D) is equivalent to the topology defined by the metric

δ(f, g) def=∞∑

k=1

12k

qKk(f − g)1 + qKk(f − g)

. (2.19)

When C0(D) is endowed with that topology, it will be denoted by C0loc(D).

If a function f is bounded and uniformly continuous3 on Ω, it possesses aunique, continuous extension to the closure Ω of Ω. Denote by Ck(Ω) the space offunctions f in Ck(Ω) for which ∂αf is bounded and uniformly continuous on Ω forall α, 0 ≤ |α| ≤ k. A function f in Ck(Ω) is said to vanish at the boundary of Ω iffor every α, 0 ≤ |α| ≤ k, and ε > 0 there exists a compact subset K of Ω such that,for all x ∈ Ω ∩ !K, |∂αf(x)| ≤ ε. Denote by Ck

0 (Ω) the space of all such functions.Clearly Ck

0 (Ω) ⊂ Ck(Ω) ⊂ Bk(Ω) ⊂ Ck(Ω). Endowed with the norm (2.16), Ck0 (Ω),

Ck(Ω), and Bk(Ω) are Banach spaces. Finally

C∞(Ω) def=⋂

k≥0

Ck(Ω), C∞0 (Ω) def=

⋂

k≥0

Ck0 (Ω), and B(Ω) def=

⋂

k≥0

Bk(Ω).

3A function f : Ω → R is uniformly continuous if for each ε > 0 there exists δ > 0 such thatfor all x and y in Ω such that |x − y| < δ we have |f(x) − f(y)| < ε.


When f is a vector function from Ω to Rm, the corresponding spaces will be denotedby Ck

0 (Ω)m or Ck0 (Ω,Rm), Ck(Ω)m or Ck(Ω,Rm), Bk(Ω)m or Bk(Ω,Rm), Ck(Ω)m

or Ck(Ω,Rm), etc.We quote the following classical compactness theorem.

Theorem 2.4 (Ascoli–Arzela theorem). Let Ω be a bounded open subset of RN.A subset K of C(Ω) is precompact in C(Ω) provided the following two conditionshold:

(i) there exists a constant M such that for all f ∈ K and x ∈ Ω, |f(x)| ≤M ;

(ii) for every ε > 0 there exists δ > 0 such that if f ∈ K, x, y in Ω, and |x−y| < δ,then |f(x)− f(y)| < ε.

2.6.2 Holder (C0,ℓ) and Lipschitz (C0,1) Continuous Functions

Given λ, 0 < λ ≤ 1, a function f is (0,λ)-Holder continuous in Ω if

∃c > 0, ∀x, y ∈ Ω, |f(y)− f(x)| ≤ c |x− y|λ.

When λ = 1, we also say that f is Lipschitz or Lipschitz continuous. Similarly fork ≥ 1, f is (k,λ)-Holder continuous in Ω if

∀α, 0 ≤ |α| ≤ k, ∃c > 0, ∀x, y ∈ Ω, |∂αf(y)− ∂αf(x)| ≤ c |x− y|λ.

Denote by Ck,λ(Ω) the space of all (k.λ)-Holder continuous functions on Ω. Definefor k ≥ 0 the subspaces4

Ck,λ(Ω) def=

f ∈ Ck(Ω) :

∀α, 0 ≤ |α| ≤ k, ∃c > 0, ∀x, y ∈ Ω

|∂αf(y)− ∂αf(x)| ≤ c |x− y|λ

(2.20)

of Ck(Ω). By definition for each α, 0 ≤ |α| ≤ k, ∂αf has a unique, bounded,continuous extension to Ω. Endowed with the norm

∥f∥Ck,λ(Ω)def= max

⎧⎪⎨

⎪⎩∥f∥Ck(Ω), max

0≤|α|≤ksup

x,y∈Ωx=y

|∂αf(y)− ∂αf(x)||x− y|λ

⎫⎪⎬

⎪⎭(2.21)

Ck,λ(Ω) is a Banach space. Finally denote by Ck,λ0 (Ω) the space Ck,λ(Ω) ∩ Ck

0 (Ω).

2.6.3 Embedding Theorem

In general Ck+1(Ω) ⊂ Ck,1(Ω), but the inclusion is true for a large class of domains,including convex domains.5 We first quote the following embedding theorem.

4The notation Ck,λ(Ω) should not be confused with the notation Ck,λ(Ω) for (k,λ)-Holdercontinuous functions in Ω without the uniform boundedness assumption in Ω. In particular,Ck,λ(RN) is contained in but not equal to Ck,λ(RN).

5For instance the convexity can be relaxed to the more general condition: there exists M > 0such that for all x and y in Ω, there exists a path γx,y in Ω (that is, a C1 injective map from theinterval [0, 1] into Ω with γ(0) = x, γ(1) = y, and

∫ 10

∥γ′(t)∥ dt ≤ M |x − y|).


Theorem 2.5 (R. A. Adams [1]). Let k ≥ 0 be an integer and 0 < ν < λ ≤ 1 bereal numbers. Then the following embeddings exist:

Ck+1(Ω)→ Ck(Ω), (2.22)

Ck,λ(Ω)→ Ck(Ω), (2.23)

Ck,λ(Ω)→ Ck,ν(Ω). (2.24)

If Ω is bounded, then the embeddings (2.23) and (2.24) are compact. If Ω is convex,we have the further embeddings

Ck+1(Ω)→ Ck,1(Ω), (2.25)

Ck+1(Ω)→ Ck,ν(Ω). (2.26)

If Ω is convex and bounded, then embeddings (2.22) and (2.26) are compact.

As a consequence of the second part of the theorem, the definition of Ck,λ(Ω)simplifies when Ω is convex:

Ck,λ(Ω) def=f ∈ Bk(Ω) : ∀α, |α| = k, ∃c > 0, ∀x, y ∈ Ω,

|∂αf(y)− ∂αf(x)| ≤ c |x− y|λ,

(2.27)

and its norm is equivalent to the norm

∥f∥Ck,λ(Ω)def= ∥f∥Ck(Ω) + max

|α|=ksup

x,y∈Ωx=y

|∂αf(y)− ∂αf(x)||x− y|λ . (2.28)

When f is a vector function from Ω to Rm, the corresponding spaces will bedenoted by Ck,λ(Ω)m or Ck,λ(Ω,Rm).

2.6.4 Identity Ck,1(Ω) = W k+1,∞(Ω): From Convex to Path-ConnectedDomains via the Geodesic Distance

By Rademacher’s theorem (cf., for instance, L. C. Evans and R. F. Gariepy [1])

Ck,1(Ω) ⊂W k+1,∞loc (Ω)

and, when Ω is convex,

Ck,1(Ω) = W k+1,∞(Ω).

This last convexity assumption can be relaxed as follows. From Rademacher’stheorem and the inequality (cf. H. Brezis [1, p. 154])

∀u ∈W 1,∞(Ω) and ∀x, y ∈ Ω, |u(x)− u(y)| ≤ ∥∇u∥L∞(Ω) distΩ(x, y)

we have the following theorem.

3. Sets Locally Described by an Homeomorphism or a Diffeomorphism 67

Theorem 2.6. Let Ω be a bounded, path-connected, open subset of RN such that

∃c, ∀x, y ∈ Ω, distΩ(x, y) ≤ c |x− y|. (2.29)

Then we have W 1,∞(Ω) = C0,1(Ω) algebraically and topologically, and there existsa constant c′ such that

∀u ∈W 1,∞(Ω) and ∀x, y ∈ Ω, |u(x)− u(y)| ≤ c′ ∥∇u∥L∞(Ω)|x− y|.

Hence for all integers k ≥ 0, W k+1,∞(Ω) = Ck,1(Ω) algebraically and topologically.

This is true for Lipschitzian domains (D. Gilbarg and N. S. Trudinger [1]).

Corollary 1. Let Ω be a bounded, open, path-connected, and locally Lipschitzian6

subset of RN. Then property (2.29) and Theorem 2.6 are verified.

Proof. By Theorem 5.8 later in section 5.4 of this chapter.

3 Sets Locally Described by an Homeomorphism ora Diffeomorphism

Open domains in the Euclidean space RN are classically described by introduc-ing at each point of their boundary a local diffeomorphism (that is, defined in aneighborhood of the point) that locally flattens the boundary. The smoothness ofthe boundary is determined by the smoothness of the local diffeomorphism. Afterintroducing the main definition in section 3.1, we briefly recall the definition of an(embedded) submanifold of codimension greater than or equal to 1 in section 3.2in order to make sense of the associated boundary integral and, more generally, theintegral over smooth submanifolds of arbitrary dimension in RN. The definition ofthe integral can be generalized by introducing the Hausdorff measures that extendthe integration theory from smooth submanifolds to arbitrary Hausdorff measura-ble subsets of RN, thus making the writing of the integral completely independentof the choice of the local diffeomorphisms. Section 3.3 completes the section bydefining the fundamental forms and curvatures for a smooth submanifold of RN ofcodimension 1.

3.1 Sets of Classes Ck and Ck,ℓ

Let e1, . . . , eN be the standard unit orthonormal basis in RN. We use the notationζ = (ζ ′, ζN ) for a point ζ = (ζ1, . . . , ζN ) in RN, where ζ ′ = (ζ1, . . . , ζN−1). Denoteby B the open unit ball in RN and define the sets

B0def= ζ ∈ B : ζN = 0 , (3.1)

B+def= ζ ∈ B : ζN > 0 , B−

def= ζ ∈ B : ζN < 0 . (3.2)

The main elements entering Definition 3.1 are illustrated in Figure 2.1 for N = 2.6Cf. Definition 5.2 of section 5 and Theorem 5.3, which says that a locally Lipschitzian domain

is equi-Lipschitzian when ∂Ω is bounded.


gx

hx = g−1x

set Ω

U(x)∂Ω x

hx(x) = (0, 0)B0

B− ball B

B+

ζN

ζ ′

Figure 2.1. Diffeomorphism gx from U(x) to B.

Definition 3.1.Let Ω be a subset of RN such that ∂Ω = ∅, 0 ≤ k be an integer or +∞, and0 ≤ ℓ ≤ 1 be a real number.

(i) - Ω is said to be locally of class Ck at x ∈ ∂Ω if there exist

(a) a neighborhood U(x) of x, and(b) a bijective map gx : U(x)→ B with the following properties:

gx ∈ Ck(U(x);B

), hx

def= g−1x ∈ Ck

(B;U(x)

), (3.3)

int Ω ∩ U(x) = hx(B+), (3.4)Γx

def= ∂Ω ∩ U(x) = hx(B0), B0 = gx(Γx). (3.5)

- Given 0 < ℓ < 1, Ω is said to be locally (k, ℓ)-Holderian at x ∈ ∂Ω ifconditions (a) and (b) are satisfied with a map gx ∈ Ck,ℓ

(U(x), B

)with

inverse hx = g−1x ∈ Ck,ℓ

(B,U(x)

).

- Ω is said to be locally k-Lipschitzian at x ∈ ∂Ω if conditions (a) and (b)are satisfied with a map gx ∈ Ck,1(U(x), B

)with inverse hx = g−1

x ∈Ck,1(B,U(x)

).

(ii) - Ω is said to be locally of class Ck if, for each x ∈ ∂Ω, Ω is locally of classCk at x.

- Given 0 < ℓ < 1, Ω is said to be locally (k, ℓ)-Holderian if, for eachx ∈ ∂Ω, Ω is locally (k, ℓ)-Holderian at x.

- Ω is said to be locally k-Lipschitzian if, for each x ∈ ∂Ω, Ω is locallyk-Lipschitzian at x.

Remark 3.1.By definition int Ω = ∅ and int !Ω = ∅. The above definitions are usually givenfor an open set Ω called a domain. This terminology naturally arises in partialdifferential equations, where this open set is indeed the domain in which the solution


of the partial differential equation is defined.7 At this point it is not necessary toassume that the set Ω is open. The family of diffeomorphisms gx : x ∈ ∂Ωcharacterizes the equivalence class of sets

[Ω] = A ⊂ RN : intA = int Ω and ∂A = ∂Ω

with the same interior and boundary. The sets int Ω and ∂Ω are invariants for theequivalence class [Ω] of sets of class Ck,ℓ. The notation Γ for ∂Ω and the standardterminology domain for the unique (open) set int Ω associated with the class [Ω]will be used. In what follows, a Ck,ℓ-mapping gx with Ck,ℓ-inverse will be called aCk,ℓ-diffeomorphism.

Consider the solution y = y(Ω) of the Dirichlet problem on the bounded opendomain Ω:

−∆y = f in Ω, y = g on ∂Ω. (3.6)

When Ω is of class C∞, the solution y(Ω) can be sought in any Sobolev spaceHm(Ω), m ≥ 1, by choosing sufficiently smooth data f and g and appropriatecompatibility conditions. However, when Ω is only of class Ck, 1 ≤ k < ∞, mcannot be made arbitrary large by choosing smoother data f and g and appropriatecompatibility conditions. The reader is referred to R. Dautray and J.-L. Lions[1, Chap. VII, sect. 3, pp. 1271–1304] for classical smoothness results of solutionsto elliptic problems in domains of class Ck. It is important to understand that weshall consider shape problems involving the solution of a boundary value problemover domains with minimal smoothness. Since we know that, at least for ellipticproblems, the smoothness of the solution depends not only on the smoothness ofthe data but also on the smoothness of the domain. This issue will be of paramountimportance to make sure that the shape problems are well posed.

Remark 3.2.We shall see in section 5 that domains that are locally the epigraph of a Ck,ℓ, k ≥ 1(resp., Lipschitzian), function are of class Ck,ℓ (resp., C0,1), but domains which areof class C0,1 are not necessarily locally the epigraph of a Lipschitzian function (cf.Examples 5.1 and 5.2). Nevertheless globally C0,1-mappings with a C0,1 inverse areimportant since they transport Lp-functions onto Lp-functions and W 1,p-functionsonto W 1,p-functions (cf. J. Necas [1, Lems. 3.1 and 3.2, pp. 65–66]).

For sets of class C1, the unit exterior normal to the boundary Γ = ∂Ω canbe characterized through the Jacobian matrices of gx and hx. By definition of B0,e1, . . . , eN−1 ⊂ B0 and the tangent space TyΓ, Γ = ∂Ω, at y to Γx is the vectorspace spanned by the N − 1 vectors

Dhx(ζ ′, 0)ei : 1 ≤ i ≤ N − 1 , (ζ ′, 0) = gx(y) ∈ B0, (3.7)

7Classically for k ≥ 1 a bounded open domain Ω in RN is said to be of class Ck if its boundaryis a Ck-submanifold of RN of codimension 1 and Ω is located on one side of its boundary Γ = ∂Ω(cf. S. Agmon [1, Def. 9.2, p. 178]).


where Dhx(ζ) is the Jacobian matrix of hx at the point ζ:

(Dhx)ℓmdef= ∂m(hx)ℓ.

So from (3.7) a normal vector field to Γx at y ∈ Γx is given by

mx(y) = − ∗(Dhx)−1(ζ ′, 0) eN = − ∗Dgx(y) eN , hx(ζ ′, 0) = y, (3.8)

since

−mx(y) · Dhx(ζ ′, 0) ei = eN · ei = δiN , 1 ≤ i ≤ N.

Thus the outward unit normal field n(y) at y ∈ Γx is given by

∀hx(ζ ′, 0) = y ∈ Γx, n(y) = −∗(Dhx)−1(ζ ′, 0) eN

| ∗(Dhx)−1(ζ ′, 0) eN | ,

∀y ∈ Γx, n(y) = −∗(Dhx)−1(h−1

x (y)) eN

| ∗(Dhx)−1(h−1x (y)) eN |

.

(3.9)

It can be verified that n is uniquely defined on Γ by checking that for y ∈ Γx ∩Γx′ ,n is uniquely defined by (3.9).

3.2 Boundary Integral, Canonical Density,and Hausdorff Measures

3.2.1 Boundary Integral for Sets of Class C1

The family of neighborhoods U(x) associated with all the points x of Γ is an opencover of Γ. If Γ = ∂Ω is assumed to be compact, then there exists a finite opensubcover: that is, there exists a finite sequence of points xj : 1 ≤ j ≤ m of Γ suchthat Γ ⊂ U1 ∪ · · · ∪ Um, where Uj = U(xj). For simplicity, index all the previoussymbols by j instead of xj .

The boundary integration on Γ is obtained by using a partition of unity rj :1 ≤ j ≤ m for the family of open neighborhoods Uj : 1 ≤ j ≤ m of Γ:

⎧⎪⎪⎨

⎪⎪⎩

rj ∈ D(Uj), 0 ≤ rj(x) ≤ 1,m∑

j=1

rj(x) = 1 in a neighborhood U of Γ,(3.10)

such that U ⊂ ∪mj=1Uj , where D(Uj) is the set of all infinitely continuously differ-

entiable functions with compact support in Uj . If f ∈ C(Γ), then

(frj) hj ∈ C(B0), 1 ≤ j ≤ m. (3.11)

Define the boundary integral of frj on Γj as∫

Γj

frj dΓ def=∫

B0

(frj) hj(ζ ′, 0)ωj(ζ ′) dζ ′, Γj = U(xj) ∩ Γ, (3.12)


where ωj = ωxj and ωx is the density term

ωx(ζ ′) = |mx(hx(ζ ′, 0))| | det Dhx(ζ ′, 0)|, (3.13)

mx(y) = − ∗(Dhx)−1(h−1x (y)) eN = − ∗Dgx(y) eN .

From this define the boundary integral of f on Γ as∫

Γf dΓ def=

m∑

j=1

∫

Γj

frj dΓ. (3.14)

The expression on the right-hand side of (3.12) results from the parametrization ofthe boundary Γj by B0 through the diffeomorphism hj .

3.2.2 Integral on Submanifolds

In differential geometry there is a general procedure to define the canonical densityfor a d-dimensional submanifold V in RN parametrized by a Ck-mapping, k ≥ 1(cf., for instance, M. Berger and B. Gostiaux [1, Def. 2.1.1, p. 48 (56) andProp. 6.62, p. 214 (239)]).

Definition 3.2.Let ∅ = S ⊂ RN, let k ≥ 1 and 1 ≤ d ≤ N be integers, and let 0 ≤ ℓ ≤ 1 be a realnumber.

(i) Given x ∈ S, S is said to be locally a Ck- (resp., Ck,ℓ-) submanifold ofdimension d at x in RN if there exists an open subset U(x) of RN containingx and a Ck- (resp., Ck,ℓ-) diffeomorphism gx from U(x) onto its open imagegx

(U(x)

), such that

gx

(U(x) ∩ S

)= gx

(U(x)

)∩Rd,

where Rd =

(x1, . . . , xd, 0, . . . , 0) ∈ RN : ∀(x1, . . . , xd) ∈ Rd

.

(ii) S is said to be a Ck- (resp., Ck,ℓ-) submanifold of dimension d in RN if, foreach x ∈ S, it is locally a Ck- (resp., Ck,ℓ-) submanifold of dimension d at xin RN.

For k ≥ 1 the canonical density ωx on the submanifold S at a point y ∈U(x) ∩ S is given by

ωx =√

|det B|,

where the d× d matrix B is given by

(B)ij = (Dhx ei) · (Dhx ej), 1 ≤ i, j ≤ d, hx = g−1x on gx(U(x)).

In the special case d = N − 1, it is easy to verify that

∗Dhx Dhx =[ ∗C C ∗C c

∗c C ∗c c

], B = ∗C C,


where C is the (N × (N − 1))-matrix and c is the N -vector defined by

Cij = Dhxij , 1 ≤ i ≤ N, 1 ≤ j ≤ N − 1, c = Dhx eN .

Denote by M(A) the matrix of cofactors associated with a matrix A: M(A)ij isequal to the determinant of the matrix obtained after deleting the ith row and thejth column times (−1)i+j . Then M(A) = (detA) ∗A−1, M( ∗A) = ∗M(A), and fortwo invertible matrices A1 and A2, M(A1A2) = M(A1)M(A2). As a result

det B = M( ∗Dhx Dhx)NN = eN · M( ∗Dhx Dhx)eN ,

where M( ∗Dhx Dhx)NN is the NN -cofactor of the matrix ∗Dhx Dhx. Then

M( ∗Dhx Dhx)NN = eN · M( ∗Dhx Dhx)eN

= eN · M( ∗Dhx) M(Dhx)eN = |M(Dhx)eN |2.

In view of the previous considerations and (3.13)√

det B = |M(Dhx)eN | = |det Dhx| | ∗(Dhx)−1 eN | = ωx.

3.2.3 Hausdorff Measures

Definition 3.2 gives the classical construction of a d-dimensional surface measureon the boundary of a C1-domain. In 1918, F. Hausdorff [1] introduced a d-dimensional measure in RN which gives the same surface measure for smooth sub-manifolds but is defined on all measurable subsets of RN. When d = N , it is equalto the Lebesgue measure. To complete the discussion we quote the definition fromF. Morgan [1, p. 8] or L. C. Evans and R. F. Gariepy [1, p. 60].

Definition 3.3.For any subset S of RN, define the diameter of S as

diam (S) def= sup|x− y| : x, y ∈ S.

Let α(d) denote the Lebesgue measure of the unit ball in Rd. The d-dimensionalHausdorff measure Hd(A) of a subset A of RN is defined by the following process.For δ small, cover A efficiently by countably many sets Sj with diam (Sj) ≤ δ, addup all the terms

α(d) (diam (Sj)/2)d ,

and take the limit as δ → 0:

Hd(A) def= limδ0

infA⊂∪Sj

diam (Sj)≤δ

∑

j

α(d)(

diam (Sj)2

)d

,

where the infimum is taken over all countable covers Sj of A whose members havediameter at most δ.


For 0 ≤ d < ∞, Hd is a Borel regular measure, but not a Radon measurefor d < N since it is not necessarily finite on each compact subset of RN. TheHausdorff dimension of a set A ⊂ RN is defined as

Hdim(A) def= inf0 ≤ s <∞ : Hs(A) = 0. (3.15)

By definition Hdim(A) ≤ N and

∀k > Hdim(A), Hk(A) = 0.

If a submanifold S of dimension d, 1 ≤ d < N , of Definitions 3.2 and 3.3 is charac-terized by a single C1-diffeomorphism g, that is,

g(S) = Rd and S = h(Rd), h = g−1,

then for any Lebesgue-measurable set E ⊂ Rd

∫

Eω dx = Hd(h(E)).

This is a generalization to submanifolds of codimension greater than 1 of formula(3.12).

The definition of the Hausdorff measure and the Hausdorff dimension extendfrom integers d to reals s, 0 ≤ s ≤ ∞, by modifying Definition 3.3 as follows.

Definition 3.4.For any real s, 0 ≤ s ≤ ∞, the s-dimensional Hausdorff measure Hs(A) of a subsetA of RN is defined by the following process. For δ small, cover A efficiently bycountably many sets Sj with diam (Sj) ≤ δ, and add all the terms

α(s) (diam (Sj)/2)d ,

where

α(s) def=πs/2

Γ(s/2 + 1), Γ(t) def=

∫ ∞

0e−x xt−1 . (3.16)

Take the limit as δ → 0:

Hs(A) def= limδ0

infA⊂∪Sj

diam (Sj)≤δ

∑

j

α(s)(

diam (Sj)2

)d

,

where the infimum is taken over all countable covers Sj of A whose membershave diameter at most δ. The Hausdorff dimension is defined by the same formula(3.15).

3.3 Fundamental Forms and Principal CurvaturesConsider a set Ω locally of class C2 in RN. Its boundary Γ = ∂Ω is an (N − 1)-dimensional submanifold of RN of class C2. At each point x ∈ Γ there is a C2-diffeomorphism gx from a neighborhood U(x) of x onto B. Denoting its inverse by


hx = g−1x , the covariant basis at a point y ∈ U(x) ∩ Γ is defined as

aα(y) def=∂hx

∂ζα(ζ ′, 0), α = 1, . . . , N − 1, hx(ζ ′, 0) = y,

and aN is chosen as the inward unit normal :

aN (y) def=∗(Dhx(h−1

x (y)))−1eN

| ∗(Dhx(h−1x (y)))−1eN |

.

The standard convention that the Greek indices range from 1 to N − 1 and theRoman indices from 1 to N will be followed together with Einstein’s rule of sum-mation over repeated indices. The associated contravariant basis ai = ai(y) isdefined from the covariant one ai = ai(y) as

ai · aj = δij ,

where δij is the Kronecker index function. The first, second, and third fundamentalforms a, b, and c are defined as

aαβdef= aα · aβ , bαβ

def= −aα · aN,β , cαβdef= bλα bλβ ,

where

aN,β =∂aN

∂ζβ, bλα = aλ · aµ bµα.

The above definitions extend to sets of class C1,1 for which hx ∈ C1,1(B) and hencehx ∈ C1,1(B0). So by Rademacher’s theorem in dimension N − 1 (cf., for instance,L. C. Evans and R. F. Gariepy [1]), hx ∈ W 2,∞(B0) and the definitions of thesecond and third fundamental forms still make sense HN−1 almost everywhere on Γ.The eigenvalues of bαβ are the (N−1) principal curvatures κi, 1 ≤ i ≤ N−1, of thesubmanifold Γ. The mean curvature H and the Gauss curvature K are defined as

Hdef=

1N − 1

N−1∑

α=1

κα and Kdef=

N−1∏

α=1

κα.

The choice of the inner normal for aN is necessary to make the principal curvaturesof the sphere (boundary of the ball) positive. The factor 1/(N − 1) is used to makethe mean curvature of the unit sphere equal to 1 in all dimensions. The reader shouldkeep in mind the long-standing differences in usage between geometry and partialdifferential equations, where the outer unit normal is used in the integration-by-parts formulae for the Euclidean space RN. For integration by parts on submanifoldsof RN the sum of the principal curvatures will naturally occur rather than the meancurvature. It will be convenient to redefine H as the sum of the principal curvaturesand introduce the notationH for the classical mean curvature:

Hdef=

N−1∑

α=1

κα, Hdef=

1N − 1

N−1∑

α=1

κα. (3.17)

4. Sets Globally Described by the Level Sets of a Function 75

4 Sets Globally Described by the Level Setsof a Function

From the definition of a set of class Ck,ℓ, k ≥ 1 and 0 ≤ ℓ ≤ 1, the set Ω can alsobe locally described by the level sets of the Ck,ℓ-function

fx(y) def= gx(y) · eN , (4.1)

since by definition

int Ω ∩ U(x) = y ∈ U(x) : fx(y) > 0,

∂Ω ∩ U(x) = y ∈ U(x) : fx(y) = 0.

The boundary ∂Ω is the zero level set of fx and the gradient

∇fx(y) = ∗Dgx(y)eN = 0

is normal to that level set. Thus the exterior normal to Ω is given by

n(y) = − ∇fx(y)|∇fx(y)| = −

∗Dgx(y)eN

| ∗Dgx(y)eN | = −∗(Dhx(gx(y)))−1eN

| ∗(Dhx(gx(y)))−1eN | .

From this local construction of the functions fx : x ∈ ∂Ω, a global function onRN can be constructed to characterize Ω.

Theorem 4.1. Given k ≥ 1, 0 ≤ ℓ ≤ 1, and a set Ω in RN of class Ck,ℓ withcompact boundary, there exists a Lipschitz continuous f : RN → R such that

int Ω = y ∈ RN : f(y) > 0 and ∂Ω = y ∈ RN : f(y) = 0 (4.2)

and a neighborhood W of ∂Ω such that

f ∈ Ck,ℓ(W ) and ∇f = 0 on W and n = − ∇f

|∇f | , (4.3)

where n is the outward unit normal to Ω on ∂Ω.

Proof. Construction of the function f . Fix x ∈ ∂Ω and consider the function fx

defined by (4.1). By continuity of ∇fx, let V (x) ⊂ U(x) be a neighborhood of xsuch that

∀y ∈ V (x), |∇fx(y)−∇fx(x)| ≤ 12|∇fx(x)|.

Furthermore, let W (x) be a bounded open neighborhood of x such that W (x) ⊂V (x). For ∂Ω compact there exists a finite subcover Wj = W (xj) : 1 ≤ j ≤ mof ∂Ω for a finite sequence xj : 1 ≤ j ≤ m ⊂ ∂Ω. Index all the previous symbolsby j instead of xj and define W = ∪m

j=1Wj . Let rj : 1 ≤ j ≤ m be a partition ofunity for Vj = V (xj) such that

⎧⎪⎪⎨

⎪⎪⎩

rj ∈ D(Vj), 0 ≤ rj(x) ≤ 1, 1 ≤ j ≤ m,m∑

j=1

rj(x) = 1 in W,(4.4)


where, by definition, W is an open neighborhood of ∂Ω such that W ⊂ V =∪m

j=1Vj and D(Vj) is the set of all infinitely continuously differentiable functionswith compact support in Vj . Define the function f : RN → R:

f(y) def= dW∪Ωc − dW∪Ω +m∑

j=1

rj(y)fj(y)

|∇fj(xj)|,

where dA(x) = inf|y − x| : y ∈ A is the distance function from a point x toa nonempty set A of RN. The function dA is Lipschitz continuous with Lipschitzconstant equal to 1. Since for all j, fj ∈ Ck,ℓ(Vj), rj ∈ D(Vj), and W j ⊂ Vj ,rjfj ∈ Ck,ℓ(Vj) with compact support in Vj . Therefore since k ≥ 1, rjfj is Lipschitzcontinuous in RN. So, by definition, f is Lipschitz continuous on RN as the finitesum of Lipschitz continuous functions on RN.

Properties (4.2). Introduce the index set

J(y) def= j : 1 ≤ j ≤ m, rj(y) > 0

for y ∈ V . For all y ∈W , J(y) = ∅, dW∪Ωc = 0 = dW∪Ω, and

f(y) =∑

j∈J(y)

rj(y)fj(y)

|∇fj(xj)|. (4.5)

For y ∈ ∂Ω = W ∩ ∂Ω, fj(y) = 0 for all j ∈ J(y) and hence f(y) = 0; fory ∈ int Ω ∩W , fj(y) > 0 and rj(y) > 0 for all j ∈ J(y) and hence f(y) > 0; fory ∈ int Ωc ∩ W , fj(y) < 0 and rj(y) > 0 for all j ∈ J(y) and hence f(y) < 0.For y ∈ Ω\W , fj ≥ 0, rj ≥ 0,

∑mj=1 rjfj ≥ 0, dW∪Ωc > 0, dW∪Ω = 0, and f > 0;

for y ∈ Ωc\W , fj ≤ 0, rj ≥ 0,∑m

j=1 rjfj ≤ 0, dW∪Ωc = 0, dW∪Ω > 0, and f < 0. Sowe have proven that ∂Ω ⊂ f−1(0), int Ω ⊂ f > 0, and int Ωc ⊂ f < 0. Hencef has the properties (4.2).

Properties (4.3). Recall that on W the function f is given by expression (4.5).It belongs to Ck,ℓ(W ), and, a fortiori, to C1(W ), as the finite sum of Ck,ℓ-functionson W since k ≥ 1. The gradient of f in W is given by

∇f(y) =∑

j∈J(y)

∇rj(y)fj(y)

|∇fj(xj)|+ rj(y)

∇fj(y)|∇fj(xj)|

.

It remains to show that it is nonzero on ∂Ω. On ∂Ω ∩ Vj , fj = 0 and

∀j ∈ J(y),∇fj(y)|∇fj(y)| = −n(y).

Therefore in W , ∇f can be rewritten in the form

∇f(y) = −n(y)∑

j∈J(y)

rj(y)|∇fj(y)||∇fj(xj)|

and

∇f(y) + n(y) = n(y)∑

j∈J(y)

rj(y)[1− |∇fj(y)|

|∇fj(xj)|

].

4. Sets Globally Described by the Level Sets of a Function 77

Finally, by construction of W ,

|∇f(y)|≥ |n(y)|− |n(y)|∑

j∈J(y)

rj(y)∣∣∣∣1−

|∇fj(y)||∇fj(xj)|

∣∣∣∣

≥ 1−∑

j∈J(y)

rj(y)12

=12

and ∇f = 0 in W . This proves properties (4.3).

This theorem has a converse.

Theorem 4.2. Associate with a continuous function f : RN → R the set

Ω def= y ∈ RN : f(y) > 0. (4.6)

Assume that

f−1(0) def= y ∈ RN : f(y) = 0 = ∅ (4.7)

and that there exists a neighborhood V of f−1(0) such that f ∈ Ck,ℓ(V ) for somek ≥ 1 and 0 ≤ ℓ ≤ 1 and that ∇f = 0 in f−1(0). Then Ω is a set of class Ck,ℓ,

int Ω = Ω and ∂Ω = f−1(0). (4.8)

Proof. By continuity of f , Ω is open, int Ω = Ω, !Ω is closed,

Ω ⊂ y ∈ RN : f(y) ≥ 0, !Ω = !Ω = y ∈ RN : f(y) ≤ 0,

and ∂Ω ⊂ f−1(0). Conversely for each x ∈ ∂Ω, define the function

g(t) def= f(x + t∇f(x)).

There exists δ > 0 such that for all |t| < δ, x + t∇f(x) ∈ V and the function g isC1 in (−δ, δ). Hence since g(0) = 0 and g′(t) = ∇f(x + t∇f(x)) ·∇f(x)

f(x + t∇f(x)) =∫ t

0∇f(x + s∇f(x)) ·∇f(x) ds.

By continuity of ∇f in V and the fact that ∇f(x) = 0, there exists δ′, 0 < δ′ < δ,such that

∀s, 0 ≤ |s| ≤ δ′, ∇f(x + s∇f(x)) ·∇f(x) ≥ 12|∇f(x)|2 > 0.

Hence for all t, 0 < t ≤ δ′, f(x + t∇f(x)) > 0. So any point in f−1(0) can beapproximated by a sequence xn = x + tn∇f(x) : n ≥ 1, 0 < tn ≤ δ′ in Ω,tn → 0, and f−1(0) ⊂ Ω. Similarly, using a sequence of negative tn’s, f−1(0) ⊂ !Ωand hence f−1(0) ⊂ ∂Ω. This proves (4.8).


Fix x ∈ ∂Ω = f−1(0). Since ∇f(x) = 0, define the unit vector eN (x) =∇f(x)/|∇f(x)|. Associate with eN (x) unit vectors e1(x), . . . , eN−1(x), which forman orthonormal basis in RN with eN (x). Define the map gx : V → RN as

gx(y) def=(

(y − x) · eα(x)N−1α=1 ,

f(y)|∇f(x)|

)⇒ gx ∈ Ck,ℓ(V ;RN).

The transpose of the Jacobian matrix of gx is given by∗Dgx(y) = (e1(x), . . . , eN−1(x),∇f(y)/|∇f(x)|)

and Dgx(x) = I, the identity matrix in the ei(x) reference system. By the inversemapping theorem gx has a Ck,ℓ inverse hx in some neighborhood U(x) of x in V .Therefore,

Ω ∩ U(x) = y ∈ U(x) : f(x) > 0 = int Ω ∩ U(x),y ∈ U(x) : f(x) = 0 = ∂Ω ∩ U(x),

and the set Ω is of class Ck,ℓ.

We complete this section with the important theorem of Sard.

Theorem 4.3 (J. Dieudonne [3, Vol. III, sect. 16.23, p. 167]). Let X and Ybe two differential manifolds, f : X → Y be a C∞-mapping, and E be the set ofcritical points of f . Then f(E) is negligible in Y , and Y − f(E) is dense in Y .

Combining this theorem with Theorem 4.1, this means that, for almost all tin the range of a C∞-function f : RN → R, the set

y ∈ RN : f(x) > t

is of class C∞ in the sense of Definition 3.1. The following theorem extends andcompletes Sard’s theorem.

Theorem 4.4 (H. Federer [5, Thm. 3.4.3, p. 316]). If m > ν ≥ 0 and k ≥ 1 areintegers, A is an open subset of Rm, B ⊂ A, Y is a normed vector space, and

f : A→ Y is a map of class k, dim Im Df(x) ≤ ν for x ∈ B,

then

Hν+(m−ν)/k(f(B)) = 0.

5 Sets Locally Described by the Epigraphof a Function

After diffeomorphisms and level sets, the local epigraph description provides a thirdpoint of view and another way to characterize the smoothness of a set or a domain.For instance, Lipschitzian domains, which are characterized by the property thattheir boundary is locally the epigraph of a Lipschitzian function, play a centralrole in the theory of Sobolev spaces and partial differential equations. They canequivalently be characterized by the geometric uniform cone property. We shall seethat sets that are locally the epigraph of Ck,ℓ-functions are equivalent to sets thatare locally of class Ck,ℓ for k ≥ 1.

5. Sets Locally Described by the Epigraph of a Function 79

5.1 Local C0 Epigraphs, C0 Epigraphs, and Equi-C0 Epigraphsand the Space H of Dominating Functions

Let eN in RN be a unit reference vector and introduce the following notation:

Hdef= eN⊥, ∀ζ ∈ RN, ζ ′ def= PH(ζ), ζN

def= eN · ζ (5.1)

for the associated reference hyperplane H orthogonal to eN , the orthogonal pro-jection PH onto H, and the normal component ζN of ζ. The vector ζ is equal toζ ′ + ζNeN and will often be denoted by (ζ ′, ζN ). In practice the vector eN is chosenas eN = (0, . . . , 0, 1), but the actual form of the unit vector eN is not important.

A direction in RN is specified by a unit vector d ∈ RN, |d| = 1. Alternatively,it can be specified by AeN for some matrix A in the orthogonal subgroup of N ×Nmatrices

O(N) def= A : ∗A A = A ∗A = I , (5.2)

where ∗A is the transposed matrix of A. Conversely, for any unit vector d in RN,|d| = 1, there exists8 A ∈ O(N) such that d = AeN . For all x ∈ RN and A ∈ O(N),it is easy to check that |Ax| = |x| = |A−1x| for all x ∈ RN.

The main elements of Definition 5.1 are illustrated in Figure 2.2 for N = 2.

set Ω

U(x)

∂Ω x

AxeN

ζNgraph of ax

Vx ⊂ H = eN⊥0

ζ ′

Figure 2.2. Local epigraph representation (N = 2).

Definition 5.1.Let eN be a unit vector in RN, H be the hyperplane eN⊥, and Ω be a subset ofRN with nonempty boundary ∂Ω.

(i) Ω is said to be locally a C0 epigraph if for each x ∈ ∂Ω there exist

(a) an open neighborhood U(x) of x;(b) a matrix Ax ∈ O(N);

8Complete the orthonormal basis with respect to d by adding N−1 unit vectors d1, d2, . . . , dN−1and construct the matrix whose columns are the vectors A = [d1 d2 . . . dN−1 d] for which AeN = d.


(c) a bounded open neighborhood Vx of 0 in H such that

U(x) ⊂ y ∈ RN : PH(A−1x (y − x)) ∈ Vx; (5.3)

(d) and a function ax ∈ C0(Vx) such that ax(0) = 0 and

U(x) ∩ ∂Ω = U(x) ∩

x + Ax(ζ ′ + ζNeN ) :ζ ′ ∈ Vx

ζN = ax(ζ ′)

, (5.4)

U(x) ∩ int Ω = U(x) ∩

x + Ax(ζ ′ + ζNeN ) :ζ ′ ∈ Vx

ζN > ax(ζ ′)

. (5.5)

(ii) Ω is said to be a C0 epigraph if it is locally a C0 epigraph and the neighbor-hoods U(x) and Vx can be chosen in such a way that Vx and A−1

x (U(x) − x)are independent of x: there exist bounded open neighborhoods V of 0 in Hand U of 0 in RN such that PH(U) ⊂ V and

∀x ∈ ∂Ω, Vx = V and ∃Ax ∈ O(N) such that U(x) = x + AxU.

(iii) Ω is said to be an equi-C0 epigraph if it is a C0 epigraph and the family offunctions ax : x ∈ ∂Ω is uniformly bounded and equicontinuous:

∃c > 0 such that ∀x ∈ ∂Ω, ∀ξ′ ∈ V, |ax(ξ′)| ≤ c,∀ε > 0, ∃δ > 0 such that∀x ∈ ∂Ω, ∀y,∀z ∈ V such that |z − y| < δ, |ax(z)− ax(y)| < ε.

(5.6)

Remark 5.1.Conditions (5.3), (5.4), and (5.5) are equivalent to the following three conditions:

U(x) ∩ ∂Ω ⊂

x + Ax(ζ ′ + ζNeN ) :ζ ′ ∈ Vx

ζN = ax(ζ ′)

, (5.7)

U(x) ∩ int Ω ⊂

x + Ax(ζ ′ + ζNeN ) :ζ ′ ∈ Vx

ζN > ax(ζ ′)

, (5.8)

U(x) ∩ int !Ω ⊂

x + Ax(ζ ′ + ζNeN ) :ζ ′ ∈ Vx

ζN < ax(ζ ′)

. (5.9)

In particular, Ω is locally a C0 epigraph if and only if !Ω is locally a C0 epigraph.As a result, conditions (5.3), (5.4), and (5.5) are also equivalent to conditions (5.3),(5.4), and the following condition:

U(x) ∩ int !Ω ⊂

x + Ax(ζ ′ + ζNeN ) :ζ ′ ∈ Vx

ζN < ax(ζ ′)

(5.10)

in place of condition (5.5).


Remark 5.2.Note that from (5.3) PH(A−1

x (U(x) − x)) ⊂ Vx and that this yields the conditionPH(U) ⊂ V in part (iii) of the definition.

Remark 5.3.It is always possible to redefine U(x)−x or Vx to be open balls or open hypercubescentered in 0. For instance, in the case of balls, for V ′

x = BH(0, ρ) ⊂ Vx, choose theassociated neighborhood U ′(x) = U(x) ∩ y ∈ RN : PH(A−1

x (y − x)) ∈ BH(0, ρ).Similarly, for U ′(x) = B(x, r) ⊂ U(x) choose the associated neighborhood V ′

x =PH(A−1

x (B(x, r)−x) = BH(0, r) ⊂ Vx, since, from (5.3), Vx ⊃ PH(A−1x (U(x)−x)) ⊃

PH(A−1x B(x, r)−x)) = BH(0, r). In both cases properties (5.3) to (5.5) are verified

for the new neighborhoods.

Remark 5.4.Sets Ω ⊂ RN that are locally C0 epigraphs have a boundary ∂Ω with zeroN -dimensional Lebesgue measure.

The three cases considered in Definition 5.1 differ only when the boundary ∂Ωis unbounded. But we first introduce some notation.

Notation 5.1.(i) Under condition (5.3), a point y in U(x) ⊂ RN is represented by (ζ ′, ζN ) ∈ Vx×R,where

ζ ′ def= PH(A−1x (y − x)) ∈ Vx ⊂ H, ζN

def= A−1x (y − x) · eN (x).

We can also identify H with RN−1 def= (z′, 0) ∈ RN : z′ ∈ RN−1.(ii) The graph, epigraph, and hypograph of ax : Vx → R will be denoted as

follows:

A0xdef= x + Ax(ζ ′ + ζNeN ) : ∀ζ ′ ∈ Vx, ζN = ax(ζ ′) ,

A+x

def= x + Ax(ζ ′ + ζNeN ) : ∀ζ ′ ∈ Vx, ∀ζN > ax(ζ ′) ,

A−x

def= x + Ax(ζ ′ + ζNeN ) : ∀ζ ′ ∈ Vx, ∀ζN < ax(ζ ′) .

The equi-C0 epigraph case is important since we shall see in Theorem 5.2 thatthe three cases of Definition 5.1 are equivalent when the boundary ∂Ω is compact.We now show that, for an arbitrary boundary, the equicontinuity of the graphfunctions ax : x ∈ ∂Ω can be expressed in terms of the continuity in 0 of adominating function h. For this purpose, define the space of dominating functionsas follows:

H def= h : [0,∞[→ [0,∞[ : h(0) = 0 and h is continuous in 0 . (5.11)

By continuity in 0, h is locally bounded in 0:

∀c > 0, ∃ρ > 0 such that ∀θ ∈ [0, ρ], |h(θ)| ≤ c.

The only part of the function h that will really matter is in a bounded neighborhoodof zero. Its extension outside to [0,∞[ can be relatively arbitrary.


Theorem 5.1. Let Ω be a C0-epigraph. The following conditions are equivalent:

(i) Ω is an equi-C0-epigraph.

(ii) There exist ρ > 0 and h ∈ H such that BH(0, ρ) ⊂ V and for all x ∈ ∂Ω

∀ζ ′, ξ′ ∈ BH(0, ρ) such that |ξ′ − ζ ′| < ρ, |ax(ξ′)− ax(ζ ′)| ≤ h(|ξ′ − ζ ′|). (5.12)

(iii) There exist ρ > 0 and h ∈ H such that BH(0, ρ) ⊂ V and for all x ∈ ∂Ω

∀ξ′ ∈ BH(0, ρ), ax(ξ′) ≤ h(|ξ′|). (5.13)

Inequalities (5.12) and (5.13) for h ∈ H are also verified with lim suph that alsobelongs to H.

Proof. Since the ax’s are uniformly continuous, inequalities (5.12) and (5.13) withh ∈ H imply the same inequalities with lim suph that also belongs to H by takingthe pointwise lim sup of both sides of the inequalities.

(ii)⇒ (i). For h ∈ H, there exists ρ1 > 0 such that 0 < ρ1 ≤ ρ/2 and |h(θ)| ≤ 1for all θ ∈ [0, ρ1]. By construction BH(0, ρ1) ⊂ V and from inequality (5.12)

∀ζ ′, ξ′ ∈ BH(0, ρ1), |ax(ξ′)− ax(ζ ′)| ≤ h(|ξ′ − ζ ′|),∀ ξ′ ∈ BH(0, ρ1), |ax(ξ′)| ≤ h(|ξ′|) ≤ 1

and the family ax : BH(0, ρ1) → R : x ∈ ∂Ω is equicontinuous and uniformlybounded.

(i) ⇒ (ii). By Definition 5.1 (iii), there exist a neighborhood V of 0 in H anda neighborhood U of 0 in RN such that for all x ∈ ∂Ω

Vx = V and ∃Ax ∈ O(N) such that U(x) = x + AxU,

and the family ax ∈ C0(V ) : x ∈ ∂Ω is uniformly bounded and equicontinuous.Choose r > 0 such that B(0, 3r) ⊂ U . From property (5.3) in Definition 5.1

U(x) ⊂y ∈ RN : PHA−1

x (y − x) ∈ V⇒ BH(0, 3r) ⊂ V.

Define the modulus of continuity

∀θ ∈ [0, 3r], h(θ) = supζ′∈H|ζ′|=1

supx∈∂Ω

supξ′∈V

ξ′+θζ′∈V

|ax(θζ ′ + ξ′)− ax(ξ′)|. (5.14)

The function h(θ), as the sup of the family θ 5→ |ax(θζ ′ + ξ′)−ax(ξ′)| of continuousfunctions with respect to (ζ ′, ξ′), is lower semicontinuous with respect to θ andbounded in [0, 3r] since the ax’s are uniformly bounded. By construction h(0) = 0and by equicontinuity of the ax, for all ε > 0, there exists δ > 0 such that for allθ ∈ [0, 3r], |θ| < δ,

∀x ∈ ∂Ω, ∀ζ ′ ∈ H, |ζ ′| = 1, |ax(θζ ′ + ξ′)− ax(ξ′)| < ε ⇒ |h(θ)| < ε,


where h is continuous in 0. In view of Remark 5.3, choose ρ = 3r and the newneighborhoods V ′ = BH(0, ρ) and U ′ = ξ ∈ B(0, ρ) : PHξ ∈ V ′ = B(0, ρ) andextend h by the constant h(ρ) to [ρ,∞[ . Therefore h ∈ H and, by definition of h,for all x ∈ ∂Ω and for all ξ′ and ζ ′ in BH(0, ρ) such that |ξ′ − ζ ′| < ρ

|a′x(ξ′)− a′

x(ζ ′)| =∣∣∣∣a

′x

(ζ ′ + |ξ′ − ζ ′| ξ

′ − ζ ′

|ξ′ − ζ ′|

)− a′

x(ζ ′)∣∣∣∣ ≤ h(|ξ′ − ζ ′|)

⇒ ∀x ∈ ∂Ω, ∀ξ′, ζ ′ ∈ BH(0, ρ), |ξ′ − ζ ′| < ρ, |a′x(ξ′)− a′

x(ζ ′)| ≤ h(|ξ′ − ζ ′|).

(ii) ⇒ (iii) Choose ζ ′ = 0 in inequality (5.12):

∀ξ′ ∈ BH(0, ρ) such that |ξ′| < ρ, ax(ξ′) ≤ |ax(ξ′)| ≤ h(|ξ′|).

(iii) ⇒ (ii). In view of Remark 5.3, choose the new neighborhoods V ′ =BH(0, ρ) and U ′ = U ∩ y ∈ RN : PH(y) ∈ BH(0, ρ) and the restriction of thefunctions ax : V → R : ∀x ∈ ∂Ω to V ′. Choose r > 0 such that B(0, 4r) ⊂ U ′

(hence 4r ≤ ρ) and consider the region

Odef= ζ ′ + ζNeN : ζ ′ ∈ BH(0, 2r) and h(|ζ ′|) < ζN < r .

For any x ∈ ∂Ω and ζ ∈ O, consider the point yζ = x + Ax(ζ ′ + ζNeN ). It is readilyseen that |yζ − x| = |ζ| <

√(2r)2 + r2 < 3r and that yζ ∈ U(x). In addition

yζ ∈ A+x . From (5.13) ax(ζ ′) ≤ h(|ζ ′|) for all ζ ′ ∈ V = BH(0, ρ) and hence ax(ζ ′) ≤

h(|ζ ′|) < ζN for all ζ ∈ O (2r < ρ). As a result yζ ∈ U(x) ∩ A+x = U(x) ∩ int Ω ⊂

int Ω and

∀x ∈ ∂Ω, x + AxO ⊂ int Ω. (5.15)

Given ζ ′ and ξ′ in BH(0, 2r), consider the two points

yζ′def= x + Ax(ζ ′ + ax(ζ ′)eN ) ∈ A0

x = U(x) ∩ ∂Ω,

yξ′def= x + Ax(ξ′ + ax(ξ′)eN ) ∈ A0

x = U(x) ∩ ∂Ω.

Since yξ′ and yζ′ belong to U(x) ∩ ∂Ω, yξ′ does not belong to yζ′ + AxO ⊂ int Ω orequivalently yξ′ − yζ′ /∈ AxO. Hence

ξ′ − ζ ′ + (ax(ξ′)− ax(ζ ′))eN = A−1x (yξ′ − yζ′) /∈ O.

For |ξ′ − ζ ′| < 2r, this is equivalent to saying that one of the following inequalitiesis verified:

h(|ξ′ − ζ ′|) ≥ AxeN · (yξ′ − yζ′) = ax(ξ′)− ax(ζ ′)or r ≤ AxeN · (yξ′ − yζ′) = ax(ξ′)− ax(ζ ′)

by definition of ax and the fact that ξ′ − ζ ′ ∈ BH(0, 2r) ⊂ BH(0, ρ) = V ′. But thesecond inequality contradicts the continuity of ax. Hence the first inequality holdsand, by interchanging the roles of ξ′ and ζ ′, we get inequality (5.12) with a new ρequal to 2r.


Theorem 5.2. When the boundary ∂Ω is compact, the three properties of Defini-tion 5.1 are equivalent:

locally C0 epigraph ⇐⇒ C0 epigraph ⇐⇒ equi-C0 epigraph.

Under such conditions, there exist neighborhoods V of 0 in H and U of 0 in RN

such that PHU ⊂ V and the neighborhoods Vx of 0 in H and the neighborhoods U(x)of x in RN can be chosen of the form

Vx = V and ∃Ax ∈ O(N) such that U(x) = x + AxU. (5.16)

Moreover, the family of functions ax : V → R : x ∈ ∂Ω can be chosen uniformlybounded and equicontinuous with respect to x ∈ ∂Ω. In addition, there exist ρ > 0such that BH(0, ρ) ⊂ V and a function h ∈ H such that

∀x ∈ ∂Ω, ∀ξ′, ζ ′ ∈ BH(0, ρ), |ξ′ − ζ ′| < ρ, |ax(ξ′)− ax(ζ ′)| ≤ h(|ξ′ − ζ ′|). (5.17)

Proof. It is sufficient to show that a locally C0 epigraph is an equi-C0 epigraph. Foreach x ∈ ∂Ω, there exists rx > 0 such that B(x, rx) ⊂ U(x). Since ∂Ω is compact,there exist a finite sequence xim

i=1 of points of ∂Ω and a finite subcover Bimi=1,

Bi = B(xi, Ri), Ri = rxi .(i) We first claim that

∃R > 0,∀x ∈ ∂Ω, ∃i, 1 ≤ i ≤ m, such that B(x, R) ⊂ Bi.

We proceed by contradiction. If this is not true, then for each n ≥ 1,

∃xn ∈ ∂Ω, ∀i, 1 ≤ i ≤ m, B(xn, 1/n

)⊂ Bi,

⇒ ∀i, 1 ≤ i ≤ m, ∃yin ∈ B(xn, 1/n

)such that yin /∈ Bi.

Since ∂Ω is compact, there exist a subsequence of xn and x ∈ ∂Ω such thatxnk → x as k goes to infinity. Hence for each i, 1 ≤ i ≤ m,

|yink − x| ≤ |yink − xnk | + |xnk − x| ≤ 1nk

+ |xnk − x|→ 0

and yink → x as k goes to infinity. For each i, the set !Bi is closed. Thus x ∈!Bi and

yink ∈ !Bi → x ∈ !Bi ⇒ ∃x ∈ ∂Ω such that x /∈m⋃

i=1

Bi.

But this contradicts the fact that Bimi=1 is an open cover of ∂Ω. This property is

obviously also satisfied for all neighborhoods N of 0 such that ∅ = N ⊂ B(0, R).(ii) Construction of the neighborhoods V ′

x and U ′(x) and the maps A′x and a′

x.From part (i) for each x ∈ ∂Ω, there exists i, 1 ≤ i ≤ m, such that

B(x, R) ⊂ Bi = B(xi, Ri) ⊂ U(xi)

⇒ PH(A−1xi

(B(x, R)− xi)) ⊂ PH(A−1xi

(U(xi)− xi)) ⊂ Vxi

⇒ PH(A−1xi

(B(x− xi, R))) ⊂ Vxi .


Since x ∈ ∂Ω ∩ U(xi), it is of the form

x = xi + Axi(ξ′i + axi(ξ

′i)eN ), ξ′

i = PH(A−1xi

(x− xi))

⇒ PH(A−1xi

(B(x− xi, R))) = PH(A−1xi

(x− xi)) + PH(A−1xi

(B(0, R)))= ξ′

i + BH(0, R)

⇒ ξ′i + BH(0, R) ⊂ Vxi . (5.18)

Choose the neighborhoods

Udef= B(0, R) and V

def= BH(0, R).

Associate with x ∈ ∂Ω the following new neighborhoods and functions:

V ′x

def= V, A′x

def= Axi , U(x) def= x + A′xU,

ζ ′ 5→ a′x(ζ ′) def= axi(ζ

′ + ξ′i)− axi(ξ

′i) : V → R .

It is readily seen from identity (5.18) that a′x is well-defined, a′

x(0) = 0, and a′x ∈

C0(V ). Since the family axi : 1 ≤ i ≤ m is finite,

∥a′x∥C0(V ) ≤ 2 ∥axi∥C0(Vxi )

≤ 2 max1≤i≤m

∥axi∥C0(Vxi )<∞

and the family a′x;x ∈ ∂Ω is uniformly bounded. Since each axi is uniformly

continuous, for all ε > 0 there exists δi > 0 such that

∀ζ ′1, ζ

′2 ∈ Vxi such that |ζ ′

1 − ζ ′2| < δi, |axi(ζ

′1)− axi(ζ

′2)| < ε.

And since the family axi : 1 ≤ i ≤ m is finite pick δ = min1≤i≤m δi to get thefollowing: for all ε > 0, there exists δ > 0 such that for all i ∈ 1, . . . , m,

∀ζ ′1, ζ

′2 ∈ Vxi such that |ζ ′

1 − ζ ′2| < δ, |axi(ζ

′1)− axi(ζ

′2)| < ε.

Therefore, for all ε > 0, there exists δ > 0 such that

∀x ∈ ∂Ω, ∀ζ ′1, ζ

′2 ∈ V such that |ζ ′

1 − ζ ′2| < δ, |a′

x(ζ ′1)− a′

x(ζ ′2)| < ε (5.19)

and the family a′x;x ∈ ∂Ω is uniformly equicontinuous. By construction, proper-

ties (a) and (b) of Definition 5.1 are verified for U ′(x) and A′x.

By construction, properties (a) and (b) are verified for U ′(x) and A′x. Always

by construction in x ∈ ∂Ω

U ′(x) = x + A′xB(0, R) = x + B(0, R) = B(x, R),

PH((A′x)−1(U ′(x)− x)) = PH(B(0, R)) = BH(0, R) = V = V ′

x

⇒ U ′(x) ⊂y ∈ RN : PH((A′

x)−1(y − x)) ∈ V ′x


and condition (c) of Definition 5.1 is verified. For condition (d) of Definition 5.1in x ∈ ∂Ω, let xi be the point associated with x such that B(x, R) ⊂ U(xi). SinceU ′(x) = B(x, R),

U ′(x) ∩ ∂Ω ⊂ U(xi) ∩ ∂Ω = xi + Axi(ζ′ + axi(ζ

′)eN ) : ∀ζ ′ ∈ Vxi ∩ ∂Ω

and for all y ∈ U ′(x) ∩ ∂Ω there exists ξ′ ∈ Vxi such that

y = xi + Axi(ξ′ + axi(ξ

′)eN ) = x + Axi(ξ′ + axi(ξ

′)eN )− (x− xi).

But x also has a representation with respect to xi

x = xi + Axi(ξ′i + axi(ξ

′i)eN ), ξ′

i ∈ Vxi

⇒ y = x + Axi(ξ′ − ξ′

i + (axi(ξ′)− axi(ξ

′i))eN )

that can be rewritten as

y = x + Axi(ξ′ − ξ′

i + (axi((ξ′ − ξ′

i) + ξ′i)− axi(ξ

′i))eN )

⇒ ζ ′ def= PH(A−1xi

(y − x)) = ξ′ − ξ′i ∈ V

since by (c) PH(A−1xi

(U ′(x) − x)) = PH((A′x)−1(U ′(x) − x)) ⊂ Vx = V . Finally,

there exists ζ ′ ∈ V such that

y = x + Axi(ζ′ + (axi(ζ


′i))eN ) = x + Axi(ζ

′ + a′x(ζ ′)eN )

and property (5.4) is verified.The constructions and the proof of property (5.5) are similar:

U ′(x) ∩ int Ω ⊂U(xi) ∩ int Ω

= U(xi) ∩

xi + Ax′i(ζ ′ + ζNeN ) : ∀ζ ′ ∈ Vxi , ∀ζN > axi(ζ

′)

and for all y ∈ U ′(x)∩int Ω there exists (ξ′, ξN ) ∈ Vxi×R such that ξN > axi(ξ′) and

y = xi + Axi(ξ′ + ξNeN ) = x + Axi(ξ

′ + ξNeN )− (x− xi)= x + Axi(ξ

′ − ξ′i + (ξN − axi(ξ

′i))eN )

⇒ ζ ′ def= PH(A−1xi

(y − x)) = ξ′ − ξ′i ∈ V, ζN

def= (A−1xi

(y − x)) · eN = ξN − axi(ξ′i)

since by (c) PH(A−1xi

(U ′(x) − x)) = PH((A′x)−1(U ′(x) − x)) ⊂ Vx = V . Finally,

there exists (ζ ′, ζN ) ∈ V ×R such that

y = x + Axi(ζ′ + a′

x(ζ ′)eN ),ζN = ξN − axi(ξ

′i) = ξN − axi(ξ

′) + axi(ξ′)− axi(ξ

′i)

= ξN − axi(ξ′) + axi(ζ


′i)

= ξN − axi(ξ′) + a′

x(ζ ′) > a′x(ζ ′)

since ξN > axi(ξ′) and property (5.5) in (d) is verified.(iii) The other properties and (5.17) follow from Theorem 5.1.


5.2 Local Ck,ℓ-Epigraphs and Holderian/Lipschitzian SetsWe extend the three definitions of C0 epigraphs to Ck,ℓ epigraphs. C0,1 is theimportant family of Lipschitzian sets and C0,ℓ, 0 < ℓ < 1, of Holderian sets.

Definition 5.2.Let Ω be a subset of RN such that ∂Ω = ∅. Let k ≥ 0, 0 ≤ ℓ ≤ 1.

(i) Ω is said to be locally a Ck,ℓ epigraph if for each x ∈ ∂Ω there exist

(a) an open neighborhood U(x) of x;(b) a matrix Ax ∈ O(N);(c) a bounded open neighborhood Vx of 0 in H such that

U(x) ⊂ y ∈ RN : PH(A−1x (y − x)) ∈ Vx; (5.20)

(d) and a function ax ∈ Ck,ℓ(Vx) such that ax(0) = 0 and

U(x) ∩ ∂Ω = U(x) ∩

x + Ax(ζ ′ + ζNeN ) :ζ ′ ∈ Vx

ζN = ax(ζ ′)

, (5.21)

U(x) ∩ int Ω = U(x) ∩

x + Ax(ζ ′ + ζNeN ) :ζ ′ ∈ Vx

ζN > ax(ζ ′)

, (5.22)

where ζ ′ = (ζ1, . . . , ζN−1) ∈ RN−1.

We shall say that Ω is locally Lipschitzian in the C0,1 case and locally Holderianof index ℓ in the C0,ℓ, 0 < ℓ < 1, case.

(ii) Ω is said to be a Ck,ℓ epigraph if it is locally a Ck,ℓ epigraph and the neigh-borhoods U(x) and Vx can be chosen in such a way that Vx and A−1

x (U(x)−x)are independent of x: there exist bounded open neighborhoods V of 0 in Hand U of 0 in RN such that PH(U) ⊂ V and

∀x ∈ ∂Ω, Vxdef= V and ∃Ax ∈ O(N) such that U(x) def= x + AxU.

We shall say that Ω is Lipschitzian in the C0,1 case and Holderian of index ℓin the C0,ℓ, 0 < ℓ < 1, case.

(iii) Given (k, ℓ) = (0, 0), Ω is said to be an equi-Ck.ℓ epigraph if it is a Ck,ℓ

epigraph and

∃c > 0, ∀x ∈ ∂Ω, ∥ax∥Ck,ℓ(V ) ≤ c, (5.23)

where ∥f∥Ck,ℓ(V ) is the norm on the space Ck,ℓ(V ) as defined in (2.21):

∥f∥Ck,ℓ(V )def= ∥f∥Ck(V ) + max

0≤|α|≤ksup

x,y∈Vx=y

|∂αf(y)− ∂αf(x)||y − x|ℓ , if 0 < ℓ ≤ 1,

Ck,0(V ) def= Ck(V ) and ∥f∥Ck,0(V )def= ∥f∥Ck(V ).


Ω is said to be an equi-C0,0 epigraph if it is an equi-C0 epigraph in the senseof Definition 5.1 (iii).We shall say that Ω is equi-Lipschitzian in the C0,1 case and equi-Holderianof index ℓ in the C0,ℓ, 0 < ℓ < 1, case.

Remark 5.5.In Definition 5.1 (iii) for (k, ℓ) = (0, 0) it follows from condition (5.23) that thefunctions ax : x ∈ ∂Ω are uniformly equicontinuous on V . In the case (k, ℓ) =(0, 0) this is no longer true. So we have to include it by using Definition 5.1 (iii).

Theorem 5.2 readily extends to the Ck,ℓ case.

Theorem 5.3. When the boundary ∂Ω is compact, the three types of sets of Defi-nition 5.2 are equi-Ck,ℓ epigraphs in the sense of Definitions 5.2 (iii) and 5.1 (iii).

Proof. The proof is the same as the one of Theorem 5.2 for the C0 case. The equi-Ck,ℓ property is obtained by choosing in the proof of Theorem 5.2 the commonconstant as c = max1≤i≤m cxi , where cxi is the constant associated with axi on Vxi ,∥axi∥Ck(Vxi )

≤ cxi .

Sets Ω that are locally a Ck,ℓ epigraph are Lebesgue measurable in RN, their“volume” is locally finite, and their boundary has zero “volume.”

Theorem 5.4. Let k ≥ 0, let 0 ≤ ℓ ≤ 1, and let Ω be locally a Ck,ℓ epigraph.

(i) The complement !Ω is also locally a Ck,ℓ epigraph, and

int Ω = ∅, int Ω = Ω, and ∂(int Ω) = ∂Ω,

int !Ω = ∅, int !Ω = !Ω, and ∂(int !Ω) = ∂Ω.(5.24)

Moreover, for all x ∈ ∂Ω

m(U(x) ∩ ∂Ω

)= 0 ⇒ m

(∂Ω)

= 0, (5.25)

where m is the N -dimensional Lebesgue measure.

(ii) If Ω = ∅ is open, then Ω = int Ω.

Proof. (i) The fact that the complement of Ω is locally a Ck,ℓ epigraph followsfrom Remark 5.1. To complete the proof, it is sufficient to establish the first twoproperties of the first line of (5.24). The second line is the first line applied to !Ω.

Since U(x) is a neighborhood of x, there exists r > 0 such that B(x, 3r) ⊂U(x). Since ax ∈ C0(Vx), ax(0) = 0, and Vx is a neighborhood of 0, there existsρ, 0 < ρ < r, such that BH(0, ρ) ⊂ Vx and for all ζ ′, |ζ ′| < ρ, |ax(ζ ′)| ≤ r. Givenζ ′ ∈ BH(0, ρ), the point yζ′ = x + A(ζ ′ + ax(ζ ′)eN ) ∈ ∂Ω ∩ U(x). Given a realα, |α| < r, consider the point zα = x + A(ζ ′ + (ax(ζ ′) + α)eN ). By construction|zα − x| = |A(ζ ′ + (ax(ζ ′) + α)eN )| = |ζ ′ + (ax(ζ ′) + α)eN | < 3r and zα ∈ U(x).Hence the segment (yζ′ , yζ′ + rAxeN ) ⊂ A+ ∩ U(x) = int Ω ∩ U(x) and int Ω = ∅.Also the segment (yζ′ , yζ′ − rAxeN ) ⊂ A− ∩ U(x) = int !Ω ∩ U(x) and int !Ω = ∅.


Finally, associate with yζ′ ∈ ∂Ω and the sequence αn = r/n the points zαn ∈int Ω. As n goes to infinity, zn → yζ′ and ∂Ω ⊂ int Ω and hence Ω = int Ω. Fromthe following Lemma 5.1 we get the third identity in the first line of (5.24).

(ii) From part (i) ∂(int !Ω) = ∂Ω and by Lemma 5.1 (ii) int !Ω = !Ω. SinceΩ is open, !Ω = int !Ω = !Ω and int Ω = !!Ω = Ω.

Lemma 5.1.

(i) ∂(int Ω) = ∂Ω ⇐⇒ int Ω = Ω.

(ii) ∂(int !Ω) = ∂Ω ⇐⇒ int !Ω = !Ω.

Proof. It is sufficient to prove (i). (⇒) By definition of the closure,

Ω = int Ω ∪ ∂Ω = int Ω ∪ ∂(int Ω) = int Ω.

Conversely (⇐) By definition ∂Ω = Ω ∩ !Ω,

Ω ∩ !Ω = int Ω ∩ !Ω = int Ω ∩ !(int Ω) = int Ω ∩ !(int Ω) = ∂(int Ω).

5.3 Local Ck,ℓ-Epigraphs and Sets of Class Ck,ℓ

Sets or domains which are locally the epigraph of a Ck,ℓ-function, k ≥ 0, 0 ≤ ℓ ≤ 1(resp., locally Lipschitzian), are sets of class Ck,ℓ (resp., C0,1). However, we shallsee from Examples 5.1 and 5.2 that a domain of class C0,1 is generally not locallythe epigraph of a Lipschitzian function.

Theorem 5.5. Let ℓ, 0 ≤ ℓ ≤ 1, be a real number.

(i) If Ω is locally a Ck,ℓ epigraph for k ≥ 0, then it is locally of class Ck,ℓ.

(ii) If Ω is locally of class Ck,ℓ for k ≥ 1, then it is locally a Ck,ℓ epigraph.

Proof. (i) For each point x ∈ ∂Ω, let U(x), Vx, and ax be the associated neighbor-hoods and the Ck,ℓ-function. Define the Ck,ℓ-mappings for ξ = (ξ′, ξN ) ∈ Vx ×R

hx(ξ) def= x + Ax(ξ′ + [ξN + ax(ξ′)] eN ), (5.26)

gx(y) def= (PHA−1x (y − x), eN · A−1

x (y − x)− ax(PHA−1x (y − x))). (5.27)

It is easy to verify that hx is the inverse of gx,

hx(gx(y)) = y in U(x) and gx(hx(ξ)) = ξ in gx(U(x)),

and that since for y ∈ U(x) ∩ ∂Ω, eN · A−1x (y − x) = ax(PHA−1

x (y − x)

gx(U(x) ∩ ∂Ω) ∈ H = (ξ′, ξN ) ∈ RN : ξN = 0,

and that since for y ∈ U(x) ∩ int Ω, eN · A−1x (y − x) > ax(PHA−1

x (y − x)

gx(U(x) ∩ int Ω) ⊂ (ξ′, ξN ) ∈ RN : ξN > 0.

The set Ω is of class Ck,ℓ.


(ii) In the discussion preceding Theorem 4.1, we showed that a set of classCk,ℓ is locally the level set of a Ck,ℓ-function. From the definition of a set of classCk,ℓ, k ≥ 1 and 0 ≤ ℓ ≤ 1, the set Ω can be locally described by the level sets ofthe Ck,ℓ-function

fx(y) def= gx(y) · eN ,

since by definition

int Ω ∩ U(x) = y ∈ U(x) : fx(y) > 0,

∂Ω ∩ U(x) = y ∈ U(x) : fx(y) = 0.

The boundary ∂Ω is the zero level set of fx and the gradient

∇fx(y) = ∗Dgx(y)eN = 0

is normal to that level set. The exterior normal to Ω is given by

n(y) = − ∇fx(y)|∇fx(y)| = −

∗Dgx(y)eN

| ∗Dgx(y)eN | = −∗(Dhx(gx(y)))−1eN

| ∗(Dhx(gx(y)))−1eN | .

Since fx is C1 and ∇fx(x) = 0, choose a smaller neighborhood of x, still denotedby U(x), such that

∀y ∈ U(x), ∇fx(y) ·∇fx(x) > 0.

To construct the graph function around the point x ∈ Γ = ∂Ω, choose Ax ∈ O(N)such that AxeN = −n(x) and H = eN⊥. Consider the Ck,ℓ-function, k ≥ 1,

λ(ζ ′, ζN ) def= fx (x + Ax(ζ ′ + ζNeN )) .

By construction, λ(0, 0) = 0 and

0 = ∇λ(0, 0) · (0, ξN ) = ∇fx(x) · (0, ξN ) = ξN |∇fx(x)| ⇒ ξN = 0.

Therefore, by the implicit function theorem, there exist a neighborhood Vx ⊂ B0 of(0, 0) and a function ax ∈ Ck.ℓ(Vx) such that λ(0, 0) = 0 and

∀ζ ′ ∈ Vx, λ(ζ ′, ax(ζ ′)) = fx(x + Ax(ζ ′ + ax(ζ ′)eN ) = 0.

By construction, x + Ax(ζ ′ + ax(ζ ′)eN ) ∈ Γ ∩ U(x). Choose

U(x) def= U(x) ∩ x + Ax(ζ ′ + ζNeN ) : ζ ′ ∈ Vx and ζN ∈ R.

By construction, Vx and U(x) satisfy condition (5.3). Moreover, from (3.4)–(3.5)

int Ω ∩ U(x) = hx(B+) and ∂Ω ∩ U(x) = hx(B0)

⇒

∣∣∣∣∣int Ω ∩ U(x) = y ∈ U(x) : fx(y) > 0,

∂Ω ∩ U(x) = y ∈ U(x) : fx(y) = 0,


and since U(x) ⊂ U(x) we get conditions (5.4) and (5.5):

int Ω ∩ U(x) = y ∈ U(x) : fx(y) > 0 = A+ ∩ U(x),

∂Ω ∩ U(x) = y ∈ U(x) : fx(y) = 0 = A0 ∩ U(x).

Recall that, by construction of U(x), for all (ζ ′, ζN ) ∈ A−1x (U(x)− x),

ζN > ax(ζ ′) ⇐⇒ λ(ζ ′, ζN ) = fx(x + Ax(ζ ′ + ζNeN )) > 0,

since∂λ

∂ζN(ζ ′, ζN ) = ∇fx(x + Ax(ζ ′ + ζNen)) · AxeN = ∇fx(x + Axζ) · ∇fx(x)

|∇fx(x)| > 0

in the neighborhood A−1x (U(x)− x) ⊂ A−1

x (U(x)− x) of (0, 0).

Example 5.1 (R. Adams, N. Aronszajn, and K. T. Smith [1]).Consider the open convex (Lipschitzian) set Ω0 = ρ eiθ : 0 < ρ < 1, 0 < θ < π/2and its image Ω = T (Ω0) by the C0,1-diffeomorphism (see Figure 2.3)

T (ρ eiθ) = ρ ei(θ−log ρ), T−1(ρ eiθ) = ρ ei(θ+log ρ).

It is readily seen that, as ρ goes to zero, the image of the two pieces of the boundaryof Ω0 corresponding to θ = 0 and θ = π/2 begin to spiral around the origin. As aresult Ω is not locally the epigraph of a function at the origin.

0 1

0

1

(a) quarter disk Ω0

0 1

0

1

(b) T (Ω0) spirals around 0

Figure 2.3. Domain Ω0 and its image T (Ω0) spiraling around the origin.

Example 5.2.This second example can be found in F. Murat and J. Simon [1], where it isattributed to M. Zerner. Consider the Lipschitzian function λ defined on [0, 1] asfollows: λ(0) = 0, and on each interval [1/3n+1, 1/3n]

λ(s) def=

⎧⎪⎪⎨

⎪⎪⎩

2(

s− 13n+1

),

13n+1 ≤ s ≤ 2

3n+1 ,

−2(

s− 13n

),

23n+1 ≤ s ≤ 1

3n,


where n ranges over all integers n ≥ 0. Associate with λ and a real δ > 0 the set

Ω def= (x1, x2) ∈ R2 : 0 < x1 < 1, |x2 − λ(x1)| < δ x1

(see Figure 2.4). The set Ω is the image of the triangle

Ω0def= (x1, x2) ∈ R2 : 0 < x1 < 1, |x2| < δ x1

through the C0,1-homeomorphism

T (x1, x2)def= (x1, x2 + λ(x1)), T−1(y1, y2) = (y1, y2 − λ(y1)).

Since the triangle Ω0 is Lipschitzian, its image is a set of class C0,1. But Ω is notlocally Lipschitzian in (0, 0) since Ω zigzags like a lightning bolt as it gets closer tothe origin. Thus, however small the neighborhood around (0, 0), a direction eN (0, 0)cannot be found to make the domain locally the epigraph of a function.

0 1

0

1

(a) triangle Ω0

0 1

0

1

(b) T (Ω0) zigzags to 0

Figure 2.4. Domain Ω0 and its image T (Ω0) zigzagging towards the origin.

5.4 Locally Lipschitzian Sets: Some Examples and Properties

The important subfamily of sets that are locally a Lipschitzian epigraph (that is,locally a C0,1 epigraph) enjoys most of the properties of smooth domains. Convexsets and domains that are locally the epigraph of a C1- or smoother function arelocally Lipschitzian. In a bounded path-connected Lipschitzian set, the geodesicdistance between any two points is uniformly bounded. Their boundary has zerovolume and locally finite boundary measure (cf. Theorem 5.7 below). Sobolevspaces defined on Ω have linear extension to RN.

5.4.1 Examples and Continuous Linear Extensions

According to this definition the whole space RN and the closed unit ball with itscenter or a small crack removed are not locally Lipschitzian since in the first case


the boundary is empty and in the second case the conditions cannot be satisfied atthe center or along the crack, which is a part of the boundary. Similarly the set

Ω def=∞⋃

n=1

Ωn, Ωndef=

y ∈ RN :∣∣∣∣y −

12n

∣∣∣∣ <1

2n+2

is not locally Lipschitzian since the conditions of Definition 5.2 (i) are not satisfiedin 0 ∈ ∂Ω. However, the set

Ω def=∞⋃

n=1

Ωn, Ωndef=

y ∈ RN : |y − n| <1

2n+2

is locally Lipschitzian, but not Lipschitzian or equi-Lipschitzian in the sense ofDefinition 5.2.

One of the important properties of a bounded (open) Lipschitzian domain Ωis the existence of a continuous linear extension of functions of the Sobolev spaceHk(Ω) from Ω to functions in Hk(RN) defined on RN, that is,

E : Hk(Ω)→ Hk(RN) and ∀ϕ ∈ Hk(Ω), (Eϕ)∣∣Ω = ϕ (5.28)

(cf. the Calderon extension theorem in R. A. Adams [1, p. 83, Thm. 4.32, p. 91] andJ. Necas [1, Thm. 3.10, p. 80]). This property is also important in the existence ofoptimal domains when it is uniform for a given family. For instance, this will occurfor the family of domains satisfying the uniform cone property of section 6.4.1.

5.4.2 Convex Sets

Theorem 5.6. Any convex subset Ω of RN such that Ω = RN and int Ω = ∅ islocally Lipschitzian, and for each x ∈ ∂Ω the neighborhood Vx and the function ax

can be chosen convex.

Proof. (i) Construction of the function ax. Start with the unit vector eN and thehyperplane H orthogonal to eN . Pick any point x0 in the interior of Ω and chooseε > 0 such that B(x0, 2ε) ⊂ int Ω. Since Ω = RN, ∂Ω = ∅ and we can associatewith each x ∈ ∂Ω the direction d(x) = (x0− x)/|x0− x| and the matrix Ax ∈ O(N)such that d(x) = AxeN . Note that the hyperplane AxH is orthogonal to the vectord(x) = AxeN . By convexity, the point x−

0 = x − (x0 − x) /∈ Ω and the minimumdistance δ from x−

0 to Ω is such that 0 < δ < |x − x0|. As a result there exists ρ,0 < ρ < ε, such that for all ζ ′ ∈ H, |ζ ′| < ρ, the line

Lζ′def= x + Ax(ζ ′ + ζNeN ) : |ζN | ≤ |x− x0|

from the point x0 + Axζ ′ to x−0 + Axζ ′ in the direction AxeN contains a point of

Ω and a point of its complement. Hence there exists ζN , |ζN | ≤ |x − x0|, suchthat y = x + Ax(ζ ′ + ζNeN ) ∈ ∂Ω ∩ Lζ′ minimizes ζN = AxeN · (y − x) over ally ∈ Ω ∩ Lζ′ . If y1 and y2 are two distinct minimizing points such that ζ1N = ζ2N ,


then y1 = x + Ax(ζ ′ + ζ1NeN ) = x + Ax(ζ ′ + ζ2NeN ) = y2. As a result, for allζ ′ ∈ H, |ζ ′| < ρ, the function

ax(ζ ′) def= infy∈Ω∩Lζ′

AxeN · (y − x)

is well-defined and finite and there exists a unique ζN , |ζN | < |x − x0|, such thatax(ζ ′) = ζN and y = x + Ax(ζ ′ + ζNeN ) is the unique minimizer.

Choose the neighborhoods

U(x) def=

y ∈ RN :

|PHA−1x (y − x)| < ρ

|A−1x (y − x) · eN | < |x− x0|

, Vx

def= z ∈ H : |z| < ρ .

Then condition (5.20) of Definition 5.2 is satisfied and it is easy to verify that thefunction ax satisfies conditions (5.21) and (5.22).

(ii) Convexity of the function ax. By construction the neighborhoods Vx andU(x) are convex. The set U(x) ∩ Ω is convex. Then recall from (5.21)–(5.22) that

U(x) ∩ x + Ax(ζ ′ + ζNeN ) : ζ ′ ∈ Vx, ζN ≥ ax(ζ ′) = U(x) ∩ Ω.

Thus the set on the left-hand side is convex. In particular,

∀y1, y2 ∈ Ω ∩ U(x), ∀α ∈ [0, 1], αy1 + (1− α)y2 ∈ Ω ∩ U(x).

In particular, for any two y1 and y2 in ∂Ω ∩ U(x),

yj = x + Ax(ζj ′ + ζjNeN ), ζj

N = ax(ζj ′), j = 1, 2,

and

yα = αy1 + (1− α)y2 = x + Ax[αζ1 + (1− α)ζ2] ∈ Ω ∩ U(x)

⇒ αζ1N + (1− α)ζ2

N ≥ ax

((αζ1 + (1− α)ζ2)′

)

⇒ αax(ζ1′) + (1− α)ax(ζ2′) ≥ ax

(αζ1′ + (1− α)ζ2′)

and ax is convex.(iii) The function ax is Lipschitzian. It is sufficient to show that ax is bounded

in Vx. Then we conclude from I. Ekeland and R. Temam [1, Lem. 3.1, p. 11]that the function is continuous and convex and hence Lipschitz continuous in Vx.By construction the minimizing point y belongs to U(x), which is bounded.

5.4.3 Boundary Measure and Integral for Lipschitzian Sets

The boundary measure on ∂Ω, which was defined in section 3.2 from the localC1-diffeomorphism, can also be defined from the local graph representation of theboundary and extended to the Lipschitzian case.

Assuming that ∂Ω is compact, there is a finite subcover of open neighborhoodsU(x) for ∂Ω that can be represented by a finite family of Lipschitzian graphs. Let


Ujmj=1, Uj

def= U(xj), be a finite open cover of ∂Ω corresponding to some finitesequence xjm

j=1 of points of ∂Ω. Denote by eN , H, Vjmj=1, Vj = Vxj , and aj,

aj = axj , the associated elements of Definition 5.2. Introduce the notation

Γ = ∂Ω, Γj = Γ ∩ Uj , 1 ≤ j ≤ m, (5.29)

and the map hj = hxj and its inverse gj = gxj as defined in (5.26)–(5.27):

hx(ξ) def= x + Ax(ξ′ + [ξN + ax(ξ′)] eN ),

gx(y) def= (PHA−1x (y − x), eN · A−1

x (y − x)− ax(PHA−1x (y − x))).

By Rademacher’s theorem the Lipschitzian function ax is differentiable almost ev-erywhere in Vx and belongs to W 1,∞(Vx

)(cf., for instance, L. C. Evans and

R. F. Gariepy [1]). Since hj is defined from the Lipschitzian function aj , Dhj isdefined almost everywhere in Vx×R, but also HN−1 almost everywhere on Vx×0.Thus the canonical density for C1-domains still makes sense and is given by the sameformula (3.13):

ωj(ζ ′) def= ωxj (ζ′) = | det Dhj(ζ ′, 0)| | ∗(Dhj)−1(ζ ′, 0) eN |. (5.30)

It is easy to verify that for almost all ξ′ ∈ Vj

Dhj(ξ′, ξN ) = Aj

⎡

⎢⎢⎢⎢⎢⎢⎢⎣

1 0 . . . . . . 0

0. . .

......

. . ....

0. . . 0

∂1aj(ξ′)∂2aj(ξ′) . . . ∂N−1aj(ξ′) 1

⎤

⎥⎥⎥⎥⎥⎥⎥⎦

, det Dhj(ξ′, ξN ) = 1,

where ∂iaj is the partial derivative of aj with respect to the ith component ofζ ′ = (ζ1, . . . , ζN−1) ∈ Vj . The matrix Dhj is invertible and its coefficients belongto L∞ and ∗(Aj)−1 = Aj . By direct computation

∗(Dhj)−1(ξ′, ξN ) = Aj

⎡

⎢⎢⎢⎢⎢⎢⎢⎣

1 0 . . . 0 −∂1aj(ξ′)

0. . .

... −∂2aj(ξ′)...

. . . 0...

.... . . −∂N−1aj(ξ′)

0 0 . . . 0 1

⎤

⎥⎥⎥⎥⎥⎥⎥⎦

, (5.31)

and finally

ωj(ζ ′) = | ∗(Dhj)−1(ζ ′, 0) eN |

= |(Aj)−1 ∗(Dhj)−1(ζ ′, 0) eN | =√

1 + |∇aj(ζ ′)|2.(5.32)


Let r1, . . . , rm be a partition of unity for the Uj ’s, that is,⎧⎪⎪⎨

⎪⎪⎩

rj ∈ D(Uj), 0 ≤ rj(x) ≤ 1,m∑

j=1

rj(x) = 1 in a neighborhood U of Γ(5.33)

such that U ⊂ ∪mj=1Uj . For any function f in C0(Γ) the functions fj and fj hj

defined as

fj(y) = f(y)rj(y), y ∈ Γj ,

fj hj(ζ ′, 0) = fj(x + Aj(ξ′ + aj(ξ′) eN ), ζ ′ ∈ Vj ,(5.34)

respectively, belong to C0(Γj) and C0(Vj). The integral of f on Γ is then defined as

∫

Γf dΓ def=

m∑

j=1

∫

Γj

fj dΓj ,

∫

Γj

fj dΓ def=∫

Vj

fj(hj(ζ ′, 0))ωj(ζ ′) dζ ′. (5.35)

Since ωj ∈ L∞(Vj), 1 ≤ j ≤ m, this integral is also well-defined for all f in L1(Γ);that is, the function fj hj ωj belongs to L1(Vj) for all j, 1 ≤ j ≤ m.

As in section 2, the tangent plane to Γj at y = hj(ζ ′, 0) is defined by

Dhj(ζ ′, 0) AjH,

where AjH is the hyperplane orthogonal to AjeN . An outward normal field to Γj

is given by

mx(ζ) = − ∗(Dhj)−1(ζ ′, ζN )eN = Aj

⎡

⎢⎢⎢⎣

∂1aj(ζ ′)...

∂N−1aj(ζ ′)−1

⎤

⎥⎥⎥⎦

and the unit outward normal to Γj at y = hj(ζ ′, 0) ∈ Γj is given by

n(y) = n(hj(ζ ′, 0)) =1√

1 + |∇aj(ζ ′)|2Aj

⎡

⎢⎢⎢⎣

∂1aj(ζ ′)...

∂N−1aj(ζ ′)−1

⎤

⎥⎥⎥⎦. (5.36)

The matrix Aj can be removed by redefining the functions ax on the space AjHorthogonal to AjeN . The formulae also hold for Ck,ℓ-domains, k ≥ 1.

We have seen that sets that are locally a Ck,ℓ epigraph, k ≥ 0, have nice prop-erties. For instance, their boundary has zero “volume.” However, they generallyhave no locally finite “surface measure” (cf. M. C. Delfour, N. Doyon, and J.-P. Zolesio [1, 2] for specific examples) as we shall see in section 6.5. Fortunately,Lipschitzian sets have locally finite “surface measure.”


Theorem 5.7. Let Ω ⊂ RN be locally Lipschitzian. Then

HN−1(U(x) ∩ ∂Ω

)≤√

1 + c2x mN−1

(Vx

), (5.37)

where mN−1 and HN−1 are the respective (N − 1)-dimensional Lebesgue measureand (N − 1)-dimensional Hausdorff measure, and cx > 0 is the Lipschitz constantof ax in Vx. If, in addition, ∂Ω is compact, then

HN−1(∂Ω) <∞. (5.38)

Proof. From Theorem 5.4 and from formula (5.32),

HN−1(U(x) ∩ ∂Ω

)=∫

V (x)

[1 + |∇ax(ζ ′)|2

]1/2dζ ′ ≤

[1 + c2

x

]1/2mN−1

(Vx

).

When the boundary of Ω is compact, then we can find a finite subcover of openneighborhoods U(xj)m

j=1. From the previous estimate, HN−1(∂Ω) is bounded bya finite sum of bounded terms.

5.4.4 Geodesic Distance in a Domain and in Its Boundary

We can now justify property (2.29) in Theorem 2.6.

Theorem 5.8. Let Ω ⊂ RN be bounded, open, and locally Lipschitzian.

(i) If Ω is path-connected, then

∃cΩ, ∀x, y ∈ Ω, distΩ(x, y) ≤ cΩ |x− y|. (5.39)

(ii) If ∂Ω is path-connected, then

∃c∂Ω, ∀x, y ∈ ∂Ω, dist∂Ω(x, y) ≤ c∂Ω |x− y|. (5.40)

Remark 5.6.The boundary of a path-connected, bounded, open, and locally Lipschitzian set isgenerally not path-connected. As an example consider the set x ∈ R2 : 1 < |x| < 2whose boundary is made up of the two disjoint circles of radii 1 and 2.

Remark 5.7.An open subset of RN with compact path-connected boundary is generally not path-connected as can be seen from a domain in R2 made up of two tangent squares.

Proof. For each x ∈ ∂Ω, there exists rx > 0 such that B(x, rx) ⊂ U(x). Since∂Ω is compact, there exist a finite sequence xim

i=1 of points of ∂Ω and a finitesubcover Bim

i=1, Bi = B(xi, Ri), Ri = rxi . Moreover, from part (i) of the proofof Theorem 5.2,

∃R > 0,∀x ∈ ∂Ω, ∃i, 1 ≤ i ≤ m, such that B(x, R) ⊂ Bi.


Always from part (i) of the proof of Theorem 5.2, the neighborhoods U and V canbe chosen as

Udef= B(0, R) and V

def= BH(0, R),

and to each point x ∈ ∂Ω we can associate the following new neighborhoods andfunctions:

V ′x

def= V, A′x

def= Axi , U(x) def= x + A′xU,

ζ ′ 5→ a′x(ζ ′) def= axi(ζ


′i) : V → R .

Since V = BH(0, R) is path-connected and for each ζ ′ ∈ V the map ζN 5→x + Ax[ζ ′ + ζNeN ] is continuous,

Lζ′def= B(x, R) ∩ x + Ax[ζ ′ + ζNeN ] : ζN ∈ R

is a (path-connected) line segment. This is nice, but we need to further reduce thesize of the neighborhoods U ′(x) and V ′

x to make the piece of boundary ∂Ω ∩ U ′(x)and hence Ω∩U ′(x) path-connected. Since the family of functions a′

x(ζ ′) : x ∈ ∂Ωis uniformly bounded and equicontinuous, there exists c > 0 such that

∀x ∈ ∂Ω, ∀ζ ′1, ζ

′2 ∈ V, |a′

x(ζ ′2)− a′

x(ζ ′1)| ≤ c |ζ ′

2 − ζ ′1|.

Choose ρ, 0 < ρ ≤ R/(4c + 4). Then

∀x ∈ ∂Ω, ∀ζ ′ ∈ BH(0, ρ), |a′x(ζ ′)| ≤ R/4 ⇒ |Ax(ζ ′ + ax(ζ ′)eN )| < R

⇒ x + Ax[ζ ′ + ax(ζ ′)eN ] : ζ ′ ∈ BH(0, ρ) ⊂ ∂Ω ∩B(x, R).

Since BH(0, ρ) is path-connected and the map ζ ′ 5→ x+Ax[ζ ′ +ax(ζ ′)eN ] is contin-uous, the image of BH(0, ρ) is path-connected. Redefine V ′ as BH(0, ρ) and U ′ asB(0, R) ∩ y ∈ RN : PH(y) ∈ BH(0, ρ) and take the restrictions of the functionsa′

x to BH(0, ρ). From this construction, for all x ∈ ∂Ω the new sets U(x′)∩ ∂Ω andU(x′) ∩ int Ω are path-connected.

Using the same notation for the finite sequence xi ∈ ∂Ω, let U ′(xi) : 1 ≤ i ≤k be the new finite subcovering of ∂Ω. For each x ∈ int Ω there exists 0 < rx < R/2such that B(x, rx) ⊂ int Ω. Consider the new set ΩR = Ω\∪k

i=1 U ′(xi). Since ΩR iscompact, there exists a finite open covering of ΩR by the sets B(xi, ri) : k + 1 ≤i ≤ m, xi ∈ ΩR, and ri = rxi . By construction B(xi, ri) ⊂ int Ω.

(i) We first show that distΩ(x, y) is uniformly bounded for any two pointsof Ω. Since Ω is covered by a finite number of balls, it is sufficient to show thatthe geodesic distance between any two points x and y in each ball is bounded. Fork + 1 ≤ i ≤ m and x, y ∈ B(xi, ri), distΩ(x, y) = |x− y| < R.

For 1 ≤ i ≤ k, and x, y ∈ U ′(xi), there is a graph representation x = xi +Ai(ζ ′ + ζNeN ) and y = xi + Ai(ξ′ + ξNeN ). Choose as first path the line fromx = xi + Ai(ζ ′ + ζNeN ) to xi + Ai(ζ ′ + ai(ζ ′)eN ), follow the boundary along thesecond path λζ ′+(1−λ)ξ′, ai(λζ ′+(1−λ)ξ′) : λ ∈ [0, 1] from xi+Ai(ζ ′+ai(ζ ′)eN )


to xi+Ai(ξ′+ai(ξ′)eN ), and finally as third path the line from xi+Ai(ξ′+ai(ξ′)eN )to y = xi + Ai(ξ′ + ξNeN ). Its length is bounded by

2R +∫ 1

0

√|ζ ′ − ξ′|2 + |∇ai(λζ ′ + (1− λ)ξ′) · (ζ ′ − ξ′)|2 dλ+ 2R,

which is bounded by 2R +√

1 + c2 2ρ+ 2R.Finally we proceed by contradiction. By definition of the geodesic distance

distΩ(x, y)/|x − y| ≥ 1. If (5.39) is not true, there exist sequences xn and ynin Ω such that distΩ(xn, yn)/|xn − yn| goes to +∞, and hence, by boundedness ofdistΩ(xn, yn), |xn − yn| goes to zero. Since Ω is bounded there exist subsequences,still denoted by xn and yn, and a point x ∈ Ω such that xn → x and yn → x.If x ∈ int Ω, there exists r′ > 0 such that B(x, r′) ⊂ int Ω and N such that forall n > N , xn, yn ∈ B(x, r′), where distΩ(xn, yn) = |xn − yn| → 0 and we get acontradiction.

If x ∈ ∂Ω, PH(B(x, ρ)−x) = BH(0, ρ) = V ′. There exists N such that, for alln > N , xn, yn ∈ B(0, ρ) and there exist (ζ ′

n, ζnN ), and (ξ′n, ξnN ), ζ ′

n, ξ′n ∈ BH(0, ρ),

ξnN ≥ a′x(ξn), and ζnN ≥ a′

x(ζn) such that xn = x + Ax[ξ′n + ξnNeN ] and yn =

x + Ax[ζ ′n + ζnNeN ]. The path p(λ); 0 ≤ λ ≤ 1 defined as

p(λ) = x + Ax

[λζ ′

n + (1− λ)ξ′n

ax (λζ ′n + (1− λ)ξ′

n) + λ [ζnN − ax(ζ ′n)] + (1− λ) [ξnN − ax(ξ′

n)]

]

stays in B(x, R) ∩ Ω. Indeed λζ ′n + (1− λ)ξ′

n ∈ BH(0, ρ),

eN · A−1x (p(λ)− x)

= ax (λζ ′n + (1− λ)ξ′

n) + λ [ζnN − ax(ζ ′n)] + (1− λ) [ξnN − ax(ξ′

n)]≥ ax (λζ ′

n + (1− λ)ξ′n) ,

and p(λ) ∈ B(x, R). This last property follows from the estimate

|p(λ)− x| ≤√

|λζ ′n + (1− λ)ξ′

n|2 + |ax (λζ ′n + (1− λ)ξ′

n) |2

+ |λ [ζnN − ax(ζ ′n)] + (1− λ) [ξnN − ax(ξ′

n)]|

≤√

1 + c2 |λζ ′n + (1− λ)ξ′

n| + λ |ζnN − ax(ζ ′n)| + (1− λ) |ξnN − ax(ξ′

n)|

≤√

1 + c2 ρ+ λ 2ρ+ (1− λ) 2ρ = (2 +√

1 + c2) ρ < R.

Since the path lies in B(x, R) ∩ Ω, its length can be estimated by computing

p′(λ) = Ax

[ζ ′n − ξ′

n

∇ax(λζ ′n + (1− λ)ξ′

n) · (ζ ′n − ξ′

n) + [ζnN − ξnN ] + ax(ξ′n)− ax(ζ ′

n)

]

⇒ |p′(λ)| ≤ |ζn − ξn| + 2c|ζ ′n − ξ′

n| ≤ (1 + 2c) |yn − xn|

⇒ distΩ(xn, yn) ≤∫ 1

0|p′(λ)| dλ ≤ (1 + 2c) |yn − xn| ≤ (1 + 2c) 2ρ < R

and we again get a contradiction.(ii) The case where ∂Ω is path-connected follows by the same constructions

and proof as in part (i).


5.4.5 Nonhomogeneous Neumann and Dirichlet Problems

With the help of the previous definition of boundary measure in section 5.4.3, we cannow make sense of the nonhomogeneous Neumann and Dirichlet problems (for theLaplace equation) in Lipschitzian domains. Assuming that the functions aj belongto W 1,∞(Vj), it can be shown that the classical Stokes divergence theorem holdsfor such domains. Given a bounded smooth domain D in RN and a Lipschitziandomain Ω in D, then

∀ϕ ∈ C1(D,RN),∫

Ωdiv ϕ dx =

∫

Γϕ · n dΓ, (5.41)

where the outward unit normal field n is defined by (5.36) for almost all ∈ Γj (withy = hx(ζ ′, 0), ζ ′ ∈ Vj) as

n(y) = Aj

∗(∂1aj(ζ ′), . . . , ∂N−1aj(ζ ′),−1)√1 + |∇aj(ζ ′)|2

.

The trace of a function g in W 1,1(D) on Γ is defined through a W 1,∞(D)N -extensionN of the normal n as

∀ϕ ∈ D(RN),∫

Γgϕ dΓ def=

∫

Ωdiv (gϕN) dx (5.42)

(cf. S. Agmon, A. Douglis, and L. Nirenberg [1, 2]). The right-hand side iswell-defined in the usual sense so that by the Stokes divergence theorem we obtainthe trace g

∣∣Γ defined on Γ. This trace is uniquely defined, denoted by γΓg, and

γΓ ∈ L(W 1,1(Ω), L1(Γ)

). (5.43)

Let Ω be a Lipschitzian domain in D. Given f ∈ L2(D) and g ∈ H1(D), considerthe following problem:

∃y ∈ H1(Ω) such that ∀ϕ ∈ H1(Ω),∫

Ω∇y ·∇ϕ dx =

∫

Ωfϕ dx +

∫

Γg ϕ dΓ.

(5.44)

If the condition∫

Ωf dx +

∫

Γg dΓ = 0 (5.45)

is satisfied, then by the Lax–Milgram theorem, problem (5.44)–(5.45) has a uniquesolution in H1(Ω)/R: any two solutions of problem (5.44) can differ only by aconstant. For instance, we can associate with the solution y = y(Ω) in H1(Ω)/Rof (5.44) the following objective function:

J(Ω) def= c(Ω, y(Ω)

), c(Ω,ϕ) def=

∫

Ω(|∇ϕ(x)|−G)2 dx (5.46)

for some G in L2(D).

6. Sets Locally Described by a Geometric Property 101

For the nonhomogeneous Dirichlet problem let Ω be a Lipschitzian domainin D and consider the weak form of the problem

∃y ∈ H1(Ω) such that ∀ϕ ∈ H2(Ω) ∩H10 (Ω),

−∫

Ωyϕ dx = −

∫

Γg∂ϕ

∂ndΓ +

∫

Ωf ϕ dx

(5.47)

for data (f, g) in L2(D)×H1(D). If this problem has a solution y, then by Green’stheorem

∫

Ω∇y ·∇ϕ dx−

∫

Γy∂ϕ

∂ndΓ = −

∫

Γg∂ϕ

∂ndΓ +

∫

Ωf ϕ dx (5.48)

and

−y = f ∈ L2(Ω), y|Γ = g ∈ H12 (Γ). (5.49)

Here, for instance, we can associate with y the objective function

J(Ω) def= c(Ω, y(Ω)

), c(Ω,ϕ) def=

12

∫

Ω|ϕ−G|2 dx, G ∈ L2(D). (5.50)

6 Sets Locally Described by a Geometric PropertyA large class of sets Ω can be characterized by geometric segment properties. Thebasic segment property is equivalent to the property that the set is locally a C0

epigraph (cf. section 6.2). For instance, it is sufficient to get the density of Ck(Ω)in the Sobolev space Wm,p(Ω) for any m ≥ 1 and k ≥ m. In this section, weestablish the equivalence of the segment, the uniform segment, and the more recent9uniform fat segment properties with the respective locally C0, the C0, and theequi-C0 epigraph properties (cf. sections 6.2, 6.3, and 6.4). Under the uniformfat segment property the local functions all have the same modulus of continuityspecified by the continuity of the dominating function at the origin (cf. section 6.3).For sets with a compact boundary the three segment properties are equivalent (cf.section 6.1). However, for sets with an unbounded boundary the uniform segmentproperty is generally too meager to make the local epigraphs uniformly boundedand equicontinuous.

Section 6.4 specializes the uniform fat segment property to the uniform cuspand cone properties, which imply that ∂Ω is, respectively, an equi-Holderian andequi-Lipschitzian epigraph.

Section 6.5 discusses the existence of a locally finite boundary measure. Wehave already seen in section 5.4.3 that Lipschitzian sets have a locally finite bound-ary or surface (that is Hausdorff HN−1) measure. So we can make sense of boundaryconditions associated with a partial differential equation on the domain. In gen-eral, this is not true for Holderian domains. To illustrate that point, we constructexamples of Holderian sets for which the Hausdorff dimension of the boundary isstrictly greater than N − 1 and hence HN−1(∂Ω) = +∞.

9This was introduced starting with the uniform cusp property in M. C. Delfour and J.-P. Zo-lesio [37] in 2001 and further refined and generalized in the sequence of papers by M. C. Delfour,N. Doyon, and J.-P. Zolesio [1, 2] in 2005 and M. C. Delfour and J.-P. Zolesio [43] in 2007.


6.1 Definitions and Main ResultsIn this section we use the notation introduced at the beginning of section 5: eN is aunit vector in RN and H = eN⊥. The open segment between two distinct pointsx and y of RN will be denoted by

(x, y) def= x + t(y − x) : ∀t, 0 < t < 1.

Definition 6.1.Let Ω be a subset of RN such that ∂Ω = ∅.

(i) Ω is said to satisfy the segment property if

∀x ∈ ∂Ω, ∃r > 0, ∃λ > 0, ∃Ax ∈ O(N),

such that ∀y ∈ B(x, r) ∩ Ω, (y, y + λAxeN ) ⊂ int Ω.

(ii) Ω is said to satisfy the uniform segment property if

∃r > 0, ∃λ > 0 such that ∀x ∈ ∂Ω, ∃Ax ∈ O(N),

such that

∀y ∈ B(x, r) ∩ Ω, (y, y + λAxeN ) ⊂ int Ω.

(iii) Ω is said to satisfy the uniform fat segment property if there exist r > 0,λ > 0, and an open region O of RN containing the segment (0,λeN ) and not0 such that for all x ∈ ∂Ω,

∃Ax ∈ O(N) such that ∀y ∈ B(x, r) ∩ Ω, y + AxO ⊂ int Ω. (6.1)

Of course, when Ω is an open domain, Ω = int Ω, and we are back to thestandard definition. Yet, as in Definition 3.1, those definitions involve only ∂Ω, Ω,and int Ω and the properties remain true for all sets in the class

[Ω]b =A ⊂ RN : A = Ω and ∂A = ∂Ω

.

A set having the segment property must have an (N−1)-dimensional boundary andcannot simultaneously lie on both sides of any given part of its boundary. In fact,a domain satisfying the segment property is locally a C0 epigraph in the sense ofDefinition 5.1, as we shall see in section 6.2. We first give a few general properties.

Theorem 6.1. Let Ω be a subset of RN such that ∂Ω = ∅.

(i) If

∀x ∈ ∂Ω, ∃λ > 0, ∃Ax ∈ O(N), (x, x + λAxeN ) ∈ int Ω, (6.2)

then int Ω = ∅ and

int Ω = Ω and ∂(int Ω) = ∂Ω.


(ii) Ω satisfies the segment property if and only if !Ω satisfies the segment property.In particular

int Ω = ∅, int Ω = Ω, and ∂(int Ω) = ∂Ω,

int !Ω = ∅, int !Ω = !Ω, and ∂(int !Ω) = ∂Ω.

Proof. (i) Pick any point x ∈ ∂Ω. By definition, for all µ ∈ ]0, 1[ , xµ = x+µλAeN ∈int Ω and int Ω = ∅. As a result there exists a sequence µn > 0→ 0 such that

int Ω ∋ xµn → x ⇒ ∂Ω ⊂ int Ω ⊂ Ω ⇒ Ω ⊂ int Ω ⊂ Ω.

The second identity follows from Lemma 5.1 (i).(ii) Assume that Ω satisfies the segment property. For any x ∈ ∂Ω, there exist

r > 0, λ > 0, and A ∈ O(N) such that for all y ∈ B(x, r)∩Ω, (y, y +λAeN ) ⊂ int Ω.We claim that

∀y ∈ B(x, r) ∩ !Ω, (y, y + λ(−A)eN ) ⊂ int !Ω.

Indeed, if there exist z ∈ B(x, r) ∩ !Ω and α ∈ (0,λ) such that y − αAeN ∈ Ω,then y = (y − αAeN ) + αAeN ∈ int Ω, which leads to a contradiction. In the otherdirection we interchange Ω and !Ω and repeat the proof. Finally, the two sets ofproperties follow from part (i) applied to Ω and !Ω.

We summarize the equivalences between the epigraph and segment propertiesthat will be detailed in Theorems 6.5, 6.6, and 6.7 with the associated constructionsand additional motivation.


(i) (a) Ω is locally a C0 epigraph if and only if Ω has the segment property;(b) Ω is a C0 epigraph if and only if Ω has the uniform segment property;(c) Ω is an equi-C0 epigraph if and only if Ω has the uniform fat segment

property.

(ii) If, in addition, ∂Ω is compact, the six properties are equivalent and there existsa dominating function h ∈ H that satisfies the properties of Theorems 5.1and 5.2.

Proof. (i) From Theorems 6.5, 6.6, and 6.7. (ii) When ∂Ω is compact, from part (i)and the equivalences in Theorem 5.2.

We quote the following density results from R. A. Adams [1, Thm. 3.18,p. 54].

Theorem 6.3. If Ω has the segment property, then the setf |int Ω : ∀f ∈ C∞

0 (RN)

of restrictions of functions of C∞0 (RN) to int Ω is dense in Wm,p(int Ω) for 1 ≤ p <

∞ and m ≥ 1. In particular, Ck(int Ω) is dense in Wm,p(int Ω) for any m ≥ 1 andk ≥ m.


6.2 Equivalence of Geometric Segment and C0 EpigraphProperties

As for sets that are locally a C0 epigraph in Definition 5.1, the three cases ofDefinition 6.1 differ only when ∂Ω is unbounded. We first deal with the first twoand will come back to the third one in section 6.3.

Theorem 6.4. Let ∂Ω = ∅ be compact. Then the segment property and the uniformsegment property of Definition 6.1 are equivalent.

Proof. It is sufficient to show that the segment property implies the uniform segmentproperty. Since ∂Ω is compact there exists a finite open subcover Bim

i=1, Bi =B(xi, rxi), of ∂Ω for some finite sequence xim

i=1 of points of ∂Ω. From part (i) ofthe proof of Theorem 5.2

∃r > 0,∀x ∈ ∂Ω, ∃i, 1 ≤ i ≤ m, such that B(x, r) ⊂ Bi.

Define λ = min1≤i≤m λi > 0. Therefore for each x ∈ ∂Ω, there exists i, 1 ≤ i ≤ m,such that B(x, r) ⊂ Bi and hence, by choosing Ax = Axi ∈ O(N), we get

∀y ∈ Ω ∩Bi, (y, y + λAxeN ) ⊂ (y, y + λiAxieN ) ⊂ int Ω.

By restricting the above property to B(x, r) ⊂ Bi,

∀y ∈ Ω ∩B(x, r), (y, y + λAxeN ) ⊂ (y, y + λiAxeN ) ⊂ int Ω.

Since r and λ are independent of x ∈ ∂Ω, Ω has the uniform segment property.


(i) If Ω satisfies the segment property, then Ω is locally a C0 epigraph. Forx ∈ ∂Ω, let B(x, rx), λx, and Ax be the associated open ball, the height, andthe rotation matrix. Then there exists ρx,

0 < ρx ≤ rxλdef= min rx,λx/2 , (6.3)

which is the largest radius such that

BH(0, ρx) ⊂PH(A−1

x (y − x)) : ∀y ∈ B(x, rxλ) ∩ ∂Ω

.

The neighborhoods of Definition 5.1 can be chosen as

Vxdef= BH(0, ρx) and

U(x) def= B(x, rxλ) ∩y ∈ RN : PH(A−1

x (y − x)) ∈ Vx

,

(6.4)

where10 BH(0, ρx) denotes the open ball of radius ρx in the hyperplane H. Foreach ζ ′ ∈ Vx, there exists a unique yζ′ ∈ ∂Ω∩U(x) such that PHA−1

x (yζ′−x) =ζ ′ and the function

ζ ′ 5→ ax(ζ ′) def= (yζ′ − x) · (AxeN ) : Vx → R10Note that U(x) − x = Ax [B(0, rxλ) ∩ ζ : PHζ ∈ Vx].


is well-defined, bounded,

∀ζ ′ ∈ Vx, |ax(ζ ′)| < rxλ, (6.5)

and uniformly continuous in Vx, that is, ax ∈ C0(Vx).

(ii) Conversely, if Ω is locally a C0 epigraph, both Ω and !Ω satisfy the segmentproperty.

Proof. (i) By assumption for each x ∈ ∂Ω, ∃r > 0, ∃λ > 0, ∃A ∈ O(N) such that

∀y ∈ B(x, r) ∩ Ω, (y, y + λAeN ) ⊂ int Ω.

Consider the set

Wdef=PHA−1(y − x) : ∀y ∈ B(x, rλ) ∩ ∂Ω

, rλ = minr,λ/2.

Therefore

∀ζ ′ ∈W, ∃y ∈ B(x, r) ∩ ∂Ω, PHA−1(y − x) = ζ ′.

If there exist y1, y2 ∈ B(x, rλ)∩∂Ω such that PHA−1(y1−x) = PHA−1(y2−x) = ζ ′,then y2 ∈ (y1, y1 + AeN · (y2 − y1) AeN ) ⊂ (y1, y1 + λAeN ) ∈ int Ω since |y2 − y1| <2rλ < λ and we get a contradiction. Therefore, the map

ζ ′ 5→ a(ζ ′) def= AeN · (y − x) = eN · A−1(y − x) : W → R

is well-defined and a(0) = 0.(W is open and 0 ∈ W ). By definition, 0 ∈ W ⊂ BH(0, rλ). Given any point

ξ′ ∈W , the segment property is satisfied at the point

yξ′def= x + A(ξ′ + a(ξ′)eN ) ∈ ∂Ω ∩B(x, rλ),

and (yξ′ , yξ′ + λAeN ) ⊂ int Ω and (yξ′ , yξ′ − λAeN ) ⊂ int !Ω by Theorem 6.1 (ii):

(yξ′ , yξ′ + λAeN ) ∩B(x, rλ) ⊂ int Ω ∩B(x, rλ),(yξ′ , yξ′ − λAeN ) ∩B(x, rλ) ⊂ int !Ω ∩ ∩B(x, rλ).

Therefore, there exists a neighborhood BH(ξ′, R) of ξ′ such that BH(ξ′, R) ⊂BH(0, rλ) and for all ζ ′ ∈ BH(ξ′, R), Lζ′ ∩ int Ω ∩ B(x, rλ) = ∅, where Lζ′ =x + A(ζ ′ + R eN ) is the line through x + Aζ ′ in the direction AeN .

If ξ′ is not an interior point of W , there exists a sequence ζ ′n ⊂ BH(ξ′, R),

ζ ′n → ξ′, such that for all n, (Lζ′

n∩ B(x, rλ)) ∩ ∂Ω = ∅. Therefore, either Lζ′

n∩

B(x, rλ) ⊂ int Ω or Lζ′n∩ B(x, rλ) ⊂ int !Ω. Since for points ζ ′

n ∈ BH(ξ′, R) wealready know that Lζ′

n∩B(x, rλ)∩ int Ω = ∅, then Lζ′

n∩B(x, rλ) ⊂ int Ω. Choose

αdef=

12

(a(ξ′)−

√r2λ − |ξ′|2

)


and the sequence

zndef= x + A(ζ ′

n + αeN )→ zdef= yξ′ − 1

2

(a(ξ′) +

√r2λ − |ξ′|2

)AeN .

We get the following estimate:

|z − x| <12

(|yξ′ − x| + rλ) < rλ,

|zn − x| < |zn − z| +12

(|yξ′ − x| + rλ) = |ζ ′n − ξ′| +

12

(|yξ′ − x| + rλ) .

There exists N such that, for all n ≥ N ,

|ζ ′n − ξ′| = |zn − z| <

12

(rλ − |yξ′ − x|) ⇒ |zn − x| < rλ

and zn ∈ B(x, rλ)∩Lζ′n⊂ int Ω∩B(x, rλ). The limit point z belongs to Ω∩B(x, rλ).

However, PHA−1(z − x) = ξ′ and

AeN · (z − yξ′) = −12

(a(ξ′) +

√r2λ − |ξ′|2

)< 0,

since, by construction of W and a, |a(ξ′)| <√

r2λ − |ξ′|2. This means that

z ∈ (yξ′ , yξ′ − λAeN ) ∩B(x, rλ) ⊂ int !Ω ∩ ∩B(x, rλ)

and we get a contradiction with the fact that z ∈ Ω ∩ ∩B(x, rλ). This proves thatall points of W are interior points and, hence, that W is open. Since 0 ∈W , thereexists ρ, 0 < ρ ≤ rλ, such that ρ is the largest radius for which BH(0, ρ) ⊂W .

(Choice of the neighborhoods and conditions of Definition 5.1.) Finally theneighborhoods of Definition 5.1 can be chosen as

Vdef= BH(0, ρ) and U def= B(x, rλ) ∩

y : PHA−1(y − x) ∈ BH(0, ρ)

.

By construction, condition (5.20) in Definition 5.2 is satisfied. It is then easy tocheck properties (5.21) and (5.22).

(The function a is bounded and uniformly continuous in V .) Consider a pointξ′ ∈ V and let yξ′ ∈ ∂Ω ∩B(x, rλ) be the associated unique point such that

ξ′ = PHA−1(yξ′ − x) and a(ξ′) = AeN · (yξ′ − x).

By construction, since ξ′ ∈PHA−1(y − x) : y ∈ B(x, rλ) ∩ ∂Ω

, |a(ξ′)| ≤ |yξ′ −

x| < rλ and

ℓdef= lim inf

ζ′→ξ′a(ζ ′) and L

def= lim supζ′→ξ′

a(ζ ′)

are finite. Hence the points

yℓdef= x + A(ζ ′ + ℓeN ) ∈ ∂Ω ∩B(x, rλ),

yLdef= x + x + A(ζ ′ + LeN ) ∈ ∂Ω ∩B(x, rλ)


belong to ∂Ω as limit points of points of ∂Ω. If yℓ = yξ′ , then |yℓ − yξ′ | ≤ |yℓ −x| + |yξ′ − x| < 2rλ ≤ λ and either yξ′ ∈ (yℓ, yℓ + λd) or yℓ ∈ (yξ′ , yξ′ + λd).By the segment property, this means that either yξ′ or yℓ belongs to int Ω, whichcontradicts the fact that both points belong to ∂Ω. Therefore yℓ = yξ′ and by thesame argument yL = yξ′ . Hence

lim infζ′→ξ′

a(ζ ′) = ℓ = (yℓ − x) · AeN = (yξ′ − x) · AeN = a(ξ′),

a(ξ′) = (yξ′ − x) · AeN = (yL − x) · AeN = L = lim supζ′→ξ′

a(ζ ′),

and a is continuous in ξ′ ∈ W . In view of the fact that a is uniformly bounded,then a ∈ C(W ) and a is uniformly continuous in W .

(ii) Pick a point x ∈ ∂Ω. By Definition 5.1, there exist neighborhoods U(x)and V (x), Ax ∈ O(N), and a continuous function ax : Vx → R with the appropriateproperties. Choose rx > 0 such that B(x, 2rx) ⊂ U(x). We want to show that forλx = rx

∀y ∈ B(x, rx) ∩ Ω, (y, y + λxAxeN ) ⊂ int Ω.

From property (5.3) in Definition 5.1


x (y − x) ∈ Vx

⇒ BH(0, 2rx) ⊂ Vx

and hence for all y ∈ B(x, rx) ∩ Ω we have ζ ′ def= PHA−1(y − x) ∈ Vx. Moreover,y ∈ B(x, rx) ∩ Ω ⊂ U(x) ∩ Ω implies that

(y − x) · AxeN ≥ ax(ζ ′).

Consider the segment (y, y + λxAxeN ). For each 0 < t < 1

(y + tλxAxeN − x) · AxeN = (y − x) · AxeN + tλx ≥ ax(ζ ′) + tλx > ax(ζ ′)

⇒ (y, y + λxAxeN ) ⊂ A+,

|y + tλxAxeN − x| ≤ |y − x| + λx < rx + λx < 2rx

⇒ (y, y + λxAxeN ) ⊂ B(x, 2rx) ∩A+ ⊂ U(x) ∩ int Ω⇒ (y, y + λxAxeN ) ⊂ int Ω

and Ω satisfies the segment property.Furthermore, since Ω is locally a C0 epigraph, so is !Ω by changing AxeN into

−AxeN . Hence !Ω also satisfies the segment property.

Theorem 6.6. Let Ω be a subset of RN such that ∂Ω = ∅. Ω has the uniformsegment property if and only if Ω is a C0 epigraph. In particular, the neighbor-hoods U(x) at x ∈ ∂Ω in RN and Vx at 0 in H can be chosen in the followingway: given rλ = minr,λ/2 and the largest radius ρ > 0 such that BH(0, ρ) ⊂PHz : ∀z ∈ B(x, rλ) ∩ ∂Ω, let

Vdef= BH(0, ρ) and U

def= B(0, rλ) ∩ y : PHz ∈ BH(0, ρ)


and, for each x ∈ ∂Ω, define

Vxdef= V and U(x) def= x + AxU.

Moreover, the functions ax : V → R can be chosen uniformly bounded,

∀x ∈ ∂Ω, ∀ζ ′ ∈ V, |ax(ζ ′)| ≤ rλ,

and uniformly continuous in V , that is, ax ∈ C0(V ).

Remark 6.1.It is important to notice that, when ∂Ω is unbounded, the uniform segment propertydoes not generally imply that the family of functions ax : x ∈ ∂Ω can be chosenuniformly bounded and equicontinuous. To restore that property the segment willhave to be fattened, as we shall see in the next section.

Remark 6.2.The modulus of continuity in 0 of the function h of part (ii) determines the modulusof continuity of the family of functions ax : x ∈ ∂Ω.

Proof. (i) By Theorem 6.5, Ω satisfies the segment property if and only if Ω islocally a C0 epigraph. It remains to sharpen this result.

(⇒) Ω satisfies a uniform segment property for some r > 0 and λ > 0, andwe can now repeat the constructions in the proof of part (i) of Theorem 6.5. Letrλ = minr,λ/2. With each x ∈ ∂Ω can be associated

PHA−1

x (y − x) : ∀y ∈ B(x, rλ) ∩ ∂Ω

= PHz : ∀z ∈ B(x, rλ) ∩ ∂Ω

since A−1x B(x, rλ) = B(x, rλ). The set

Wdef= PHz : ∀z ∈ B(x, rλ) ∩ ∂Ω

is independent of x since rλ is now independent of x. Hence the largest radius ρ > 0such that BH(0, ρ) ⊂W is also independent of x. The family of bounded uniformlycontinuous functions ax defined on V = BH(0, ρ) such that ∂Ω is locally the graphfunction of ax in the neighborhood

U(x) = B(x, rλ) ∩y : PHA−1

x (y − x) ∈ BH(0, ρ)

= x + AxU, Udef= B(0, rλ) ∩ y : PHz ∈ BH(0, ρ)

of x. The neighborhoods Vx and U(x) are now independent of x up to a rotationaround the origin. By Definition 5.2 (ii), Ω is a C0 epigraph. Moreover, by Theo-rem 6.5 (i) the functions ax : V → R can be chosen uniformly bounded by rλ (cf.(6.5)), and uniformly continuous in V , that is, ax ∈ C0(V ).

(⇐) The proof is the same as the proof of part (ii) of Theorem 6.5. Theuniform segment property follows from the fact that the neighborhoods U(x) − xof 0 are, up to a rotation around 0, independent of x ∈ ∂Ω. Therefore there existsr > 0 such that, for all x ∈ ∂Ω, B(0, 2r) ⊂ U(x) − x and hence B(x, 2r) ⊂ U(x).As a result we can repeat the proof with r in place of rx and λ = r, which are bothindependent of x.


d = Ax(0, eN ) d = Ax(0, eN )

y y

fat setAxO

fat setAxO

Figure 2.5. Examples of arbitrary and axially symmetrical O around thedirection d = Ax(0, eN ).

6.3 Equivalence of the Uniform Fat Segment and the Equi-C0

Epigraph PropertiesThe uniform fat segment property strengthens the uniform segment property byfattening the segment of Definition 6.1 (ii). Before proving the equivalence withan equi-C0 epigraph, we first show that the open set O of Definition 6.1 (iii) isequivalent to a parametrized axisymmetrical open region of the form

O(h, ρ,λ) def=ζ ′ + ζNeN ∈ RN : ζ ′ ∈ BH(0, ρ), lim suph(|ζ ′)|) < ζN < λ

(6.6)

for some ρ > 0, λ > 0 and a dominating function h that belongs to the space ofdominating functions introduced in (5.11) (see Figure 2.5):

H = h : [0,∞[→ [0,∞[ : h(0) = 0 and h is continuous in 0 . (6.7)

The use of the lim suph rather than h is necessary to make O(h, ρ,λ) open sincewe only assume that h is continuous in 0.

Lemma 6.1. (i) Given ρ > 0, λ > 0, and h ∈ H, the region O(h, ρ,λ) containsthe segment (0,λeN ), does not contain 0, and is open.

(ii) Let λ > 0 be a real number and O be an open subset of RN containing (0,λeN )and not 0. Then there exist ρ > 0 and a continuous function h : [0, ρ]→ [0,∞[which is monotone strictly increasing such that the set

O(h, ρ,λ/2) def= ξ : ξ′ ∈ BH(0, ρ) and h(|ζ ′|) < ξN < λ/2

is open and contains (0,λeN/2) and not 0, and O(h, ρ,λ/2) ⊂ O.

Proof. (i) By definition

(0,λeN ) =ζNeN : 0 = lim

ξ′→0h(|ξ′|) < ζN < λ

⊂ O(h, ρ,λ),

since h(0) = 0 and h is continuous at the origin. Also 0 = 0+0eN /∈ O(h, ρ) since itwould yield the contradiction 0 = limξ′→0 h(|ξ′|) < ζN = 0. To show that O(h, ρ,λ)


is open, it is sufficient to show that for each point ξ = ξ′ + ξNeN ∈ O(h, ρ,λ) thereexists a neighborhood of ξ contained in O(h, ρ,λ). But this is true by definition ofthe lim sup: given (ξ′, ξN ) ∈ O(h, ρ,λ), lim sup h(|ξ′|) < ξN < λ and there exists aneighborhood V (ξ′) ⊂ BH(0, ρ) of ξ′. Hence

ξ ∈ζ ′ + ζNeN ∈ RN : ζ ′ ∈ V (ξ′), lim suph(|ζ ′)|) < ζN < λ

⊂ O(h, ρ,λ),

and O(h, ρ,λ) is open.(ii) For each integer n ≥ 0, let λn = λ/2n+1. Since the segment [λ1eN ,λ0eN ]

is contained in O, there exists a largest radius r0 > 0 such that the cylinder

C0def= ζ : |ζ ′| < r0 and λ1 < ζN < λ0

is contained in O. Again, since the segment [λ2eN ,λ1eN ] is contained in O, thereexists a largest radius 0 < r1 ≤ r0/2 such that the cylinder

C1def= ζ : |ζ ′| < r1 and λ2 < ζN < λ1

is contained in O. Similarly, for n ≥ 2, there exists a largest radius 0 < rn ≤ rn−1/2such that the cylinder

Cndef= ζ : |ζ ′| < r1 and λn < ζN < λn−1

is contained in O. By construction, 2n rn ≤ r0, rn 0, ∪n∈NCn ⊂ O is openand contains (0,λeN/2) and not 0. Since the boundary of that set is piecewiseconstant, we make a few adjustments to make it piecewise linear and continuous.First construct the set

O1def=ζ : λ1 < ζN < λ0 and |ζ ′| < r0

ζN − λ1

λ0 − λ1+ r1

λ0 − ζNλ0 − λ1

.

For n ≥ 2, choose

Ondef=ζ : λn < ζN ≤ λn−1 and |ζ ′| < rn−1

ζN − λn

λn−1 − λn+ rn

λn−1 − ζNλn−1 − λn

.

From the condition 0 < rn ≤ rn−1/2, the following union is open:

O′ def= ∪n≥1On, (0,λeN/2) ⊂ O′ ⊂ O, and 0 /∈ O′.

Set ρ = r0 and define the function h : [0, ρ]→ [0,∞[ on each interval [rn−1, rn[as follows:

h(θ) def= rn−1θ − λn

λn−1 − λn+ rn

λn−1 − θλn−1 − λn

, rn−1 ≤ θ < rn.

It is continuous and monotone strictly increasing since rn−1 < rn and λn−1 > λn.Moreover, it is easy to check that O′ = O(h, ρ,λ/2).


We now complete the equivalences between the epigraph and the segmentproperties of Theorems 6.5 and 6.6.

Theorem 6.7. Let Ω be a subset of RN such that ∂Ω = ∅. Ω is an equi-C0 epigraphif and only if Ω has the uniform fat segment property.

Proof. (⇐) Assume that the uniform fat segment property is verified for r > 0,λ > 0, and the open set O. By Lemma 6.1 (ii), there exist ρ > 0 and a continuousfunction h : [0, ρ]→ [0,∞[ which is monotone strictly increasing, the set

O(h, ρ,λ/2) def= ξ : ξ′ ∈ BH(0, ρ) and h(|ζ ′|) < ξN < λ/2

is open and contains (0,λeN/2) and not 0, and O(h, ρ,λ/2) ⊂ O. So the uni-form fat segment property is verified for r > 0, λ/2 > 0, and the new open setO(h, ρ,λ/2). In particular, Ω satisfies the uniform segment property with λ/2 in-stead of λ. By Theorem 6.6, the neighborhoods at x ∈ ∂Ω can be chosen in thefollowing way: given rλ = minr,λ/4 and the largest radius ρ′ > 0 such thatBH(0, ρ′) ⊂ PHz : ∀z ∈ B(x, rλ) ∩ ∂Ω, let

Vdef= BH(0, ρ′) and U

def= B(0, rλ) ∩ y : PHz ∈ BH(0, ρ′)

and, for each x ∈ ∂Ω, define

Vxdef= V and U(x) def= x + AxU.

By Theorem 6.5 (i) with λ/2 instead of λ, the function ax : V → R is uniformlycontinuous in V and

∀ζ ′ ∈ V, |ax(ζ ′)| < rλ.

Given ζ ′ and ξ′ in BH(0, ρ′) such that ξ′ − ζ ′ ∈ BH(0, ρ), consider the points

yζ′def= x + Ax(ζ ′ + ax(ζ ′)eN ) ∈ B(x, rλ) ∩ ∂Ω,

yξ′def= x + Ax(ξ′ + ax(ξ′)eN ) ∈ B(x, rλ) ∩ ∂Ω.

By subtracting,

yξ′ = yζ′ + Ax(ξ′ − ζ ′ + (ax(ξ′)− ax(ζ ′))eN ).

If ax(ξ′)− ax(ζ ′) > h(|ξ′ − ζ ′|), then

h(|ξ′ − ζ ′|) < ax(ξ′)− ax(ζ ′) < λ/4 + λ/4 = λ/2⇒ yξ′ = yζ′ + Ax(ξ′ − ζ ′ + (ax(ξ′)− ax(ζ ′))eN ) ∈ yζ′ + AxO(h, ρ,λ/2) ⊂ int Ω.

But this contradicts the fact that yξ′ ∈ ∂Ω.Since the argument can be repeated with ξ′ in place of ζ ′, for all x ∈ ∂Ω,

∀ζ ′, ξ′ ∈ BH(0, ρ′) such that |ξ′ − ζ ′| < ρ′, |ax(ξ′)− ax(ζ ′)| ≤ h(|ξ′ − ζ ′|). (6.8)


Since h is continuous in 0, for all ε > 0, there exists δ, 0 < δ < ρ, such that θ < δimplies h(θ) < ε. Hence, for all x ∈ ∂Ω,

∀ζ ′, ξ′ ∈ BH(0, ρ′) such that |ξ′ − ζ ′| < δ, |ax(ξ′)− ax(ζ ′)| ≤ h(|ξ′ − ζ ′|) < ε

and the family ax is equicontinuous with respect to x ∈ ∂Ω.(⇒) We go back to the proof of (i) ⇒ (ii) in Theorem 5.1 that we copy

below for the benefit of the reader in order to keep track of all the parameters.By Definition 5.1 (iii), Ω is an equi-C0 epigraph and the family of functions ax ∈C0(V ) : x ∈ ∂Ω is uniformly bounded and equicontinuous. In particular, U(x) andVx can be chosen in such a way that Vx and A−1

x (U(x) − x) are independent of x:there exist bounded open neighborhoods V of 0 in H and U of 0 in RN such that

∀x ∈ ∂Ω, Vxdef= V and ∃Ax ∈ O(N) such that U(x) def= x + AxU.

Choose r > 0 such that B(0, 3r) ⊂ U . From property (5.3) in Definition 5.1


x (y − x) ∈ V⇒ BH(0, 3r) ⊂ V.

Use the modulus of continuity defined in (5.14):

∀θ ∈ [0, 3r], h(θ) = supζ′∈H|ζ′|=1

supx∈∂Ω

supξ′∈V

ξ′+θζ′∈V

|ax(θζ ′ + ξ′)− ax(ξ′)|. (6.9)

The function h(θ), as the sup of the family θ 5→ |ax(θζ ′ + ξ′)−ax(ξ′)| of continuousfunctions with respect to (ζ ′, ξ′), is lower semicontinuous with respect to θ andbounded in [0, 3r] since the ax’s are uniformly bounded. By construction h(0) = 0and by equicontinuity of the ax, for all ε > 0, there exists δ > 0 such that for allθ ∈ [0, 3r], |θ| < δ,

∀x ∈ ∂Ω, ∀ζ ′ ∈ H, |ζ ′| = 1, |ax(θζ ′ + ξ′)− ax(ξ′)| < ε ⇒ |h(θ)| < ε,

where h is continuous in 0. In view of Remark 5.3, choose ρ = 3r and the newneighborhoods V ′ = BH(0, ρ) and U ′ = ξ ∈ B(0, ρ) : PHξ ∈ V ′ = B(0, ρ) andextend h by the constant h(ρ) to [ρ,∞[ . Therefore h ∈ H and, by definition of h,for all x ∈ ∂Ω and for all ξ′ and ζ ′ in BH(0, ρ) such that |ξ′ − ζ ′| < ρ

|a′x(ξ′)− a′

x(ζ ′)| =∣∣∣∣a

′x

(ζ ′ + |ξ′ − ζ ′| ξ

′ − ζ ′

|ξ′ − ζ ′|

)− a′

x(ζ ′)∣∣∣∣ ≤ h(|ξ′ − ζ ′|)

⇒ ∀x ∈ ∂Ω, ∀ξ′, ζ ′ ∈ BH(0, ρ) |ξ′ − ζ ′| < ρ, |a′x(ξ′)− a′

x(ζ ′)| ≤ h(|ξ′ − ζ ′|).

Now that we have recalled the proof of Theorem 5.1 we complete the proofby showing that the uniform fat segment property of Definition 6.1 is verified withλ = r > 0, r > 0, and

O(h, r, r) def=ζ ′ + ζNeN ∈ RN : ζ ′ ∈ BH(0, r), lim suph(|ζ ′|) < ζN < r

.


By Lemma 6.1 (i), the region O(h, r, r) contains the segment (0, r eN ), does notcontain 0, and is open. Given a point x ∈ ∂Ω, we want to show that

∀y ∈ B(x, r) ∩ Ω, y + AxO(h, r, r) ⊂ int Ω.

For y ∈ B(x, r) ∩ Ω ⊂ U(x) ∩ Ω

ζ ′ def= PHA−1x (y − x) ∈ V and ζN

def= (y − x) · AxeN ≥ ax(ζ ′).

Associate with ξ ∈ O(h, r, r) the point

zdef= y + Axξ = x + Ax(ζ ′ + ξ′ + (ζN + ξN )eN ).

By construction

PHA−1x (z − x) = ζ ′ + ξ′ and AxeN · (z − x) = ζN + ξN ,

ζN + ξN > ax(ζ ′) + lim suph(|ξ′|) ≥ ax(ζ ′) + lim inf h(|ξ′|) ≥ ax(ζ ′) + h(|ξ′|)⇒ ζN + ξN > ax(ζ ′) + h(|ξ′|)

> ax(ζ ′) + ax(ζ ′ + ξ′)− ax(ζ ′) = ax(ζ ′ + ξ′)

and z ∈ A+x . In addition

|z − x| ≤ |z − y| + |y − x| ≤√

2r + r < 3r ⇒ z ∈ U(x) ∩A+x = U(x) ∩ int Ω

and y + AxO(h, r, r) ⊂ int Ω. This proves that Ω satisfies the uniform fat segmentproperty of Definition 6.1 with λ = r > 0, r > 0, and O(h, r, r).

6.4 Uniform Cone/Cusp Properties and Holderian/Lipschitzian Sets

In this section we make the connection between the notions of Holderian and Lip-schitzian sets of Definition 5.2 and the dominating function h generating the setO(h, ρ,λ) of the uniform fat segment property. We first recall some elements ofDefinition 5.2.


(i) Ω is locally Lipschitzian if Ω is locally a C0,1 epigraph.

Ω is locally Holderian if Ω is locally a C0,ℓ epigraph for some ℓ, 0 < ℓ < 1.

(ii) Ω is Lipschitzian if Ω is a C0,1 epigraph.

Ω is Holderian if Ω is a C0,ℓ epigraph for some ℓ, 0 < ℓ < 1.

(iii) Ω is equi-Lipschitzian if Ω is an equi-C0,1 epigraph.

Ω is equi-Holderian if Ω is an equi-C0.ℓ epigraph for some ℓ, 0 < ℓ < 1.


It is natural to associate with Definition 6.2 the following family of dominatingfunctions:

hℓ(θ) = λ (θ/ρ)ℓ , 0 < ℓ ≤ 1,

in H. For ℓ = 1, the region O(h1, ρ,λ) is the open cone of height λ and aperture ωgiven by tanω = ρ/λ, 0 < ω < π/2,

h1(θ) = λθ

ρ=

1tanω

θ.

For 0 < ℓ < 1, O(hℓ, ρ,λ) defines a cuspidal region around λeN of height λ.

6.4.1 Uniform Cone Property and Lipschitzian Sets

Lipschitzian sets have also been equivalently characterized by a purely geometricuniform cone property which seems to have been originally introduced by S. Ag-mon [1]. In this section we establish the equivalence of the two sets of definitionsand express the previous properties and formulae of Definition 5.2 in terms of theparameters of the cone. One of the important properties of a family of open do-mains satisfying the uniform cone property is that the extension operator from thedomains to RN is uniformly continuous.

Notation 6.1.Given λ > 0 and 0 < ω < π/2, denote by C(λ,ω) the open cone

C(λ,ω) def=

y ∈ RN :1

tanω∣∣PH(y)

∣∣ < y · eN < λ

,

where PH is the orthogonal projection onto the hyperplane H = eN⊥ orthogonalto the direction eN (cf. Figure 2.6). Further, associate with an arbitrary point

aperture ω

direction AxeN

height λvertex x

Figure 2.6. The cone x + AxC(λ,ω) in the direction AxeN .


x ∈ RN, an Ax ∈ O(N), a direction AxeN , and the translated and rotated conex + AxC(λ,ω). It is readily seen that

C(λ,ω) = O(h, ρ,λ) with h(θ) = θ/ tanω and ρ = λ tanω.


(i) Ω is said to satisfy the local uniform cone property if

∀x ∈ ∂Ω, ∃λ > 0, ∃ω > 0, ∃r > 0, ∃Ax ∈ O(N),

such that

∀y ∈ B(x, r) ∩ Ω, y + AxC(λ,ω) ⊂ int Ω.

(ii) Ω is said to satisfy the uniform cone property if

∃λ > 0, ∃ω > 0, ∃r > 0, ∀x ∈ ∂Ω, ∃Ax ∈ O(N),

such that

∀y ∈ B(x, r) ∩ Ω, y + AxC(λ,ω) ⊂ int Ω.

Theorem 6.8. Let Ω be a subset of RN such that ∂Ω = ∅. Ω is equi-Lipschitzianif and only if Ω satisfies the uniform cone property.

The proof will be the same as the one of Theorem 6.9 in the next section.

6.4.2 Uniform Cusp Property and Holderian Sets

Definition 6.4.Let Ω be a subset of RN such that ∂Ω = ∅, 0 < ℓ < 1, and hℓ(θ) = λ(θ/ρ)ℓ.

(i) Ω is said to satisfy the cusp property of index ℓ, 0 < ℓ < 1, if

∀x ∈ ∂Ω, ∃h ∈ H, ∃λ > 0, ∃r > 0, ∃Ax ∈ O(N),

such that

∀y ∈ B(x, r) ∩ Ω, y + AxO(hℓ, ρ,λ) ⊂ int Ω.

(ii) Ω is said to satisfy the uniform cusp property of index ℓ, 0 < ℓ < 1, if

∃λ > 0, ∃h ∈ H, ∃r > 0, ∀x ∈ ∂Ω, ∃Ax ∈ O(N),

such that

∀y ∈ B(x, r) ∩ Ω, y + AxO(hℓ, ρ,λ) ⊂ int Ω.

The Lipschitzian case of Definitions 6.2 and 6.3 corresponds to ℓ = 1 andρ = λ tanω in the above definition, but we wanted to keep the two terminologiesdistinct.

Remark 6.3.In some applications, it might be interesting to relax the uniform cusp propertyby permitting the axis of the cuspidal region to bend: this makes the region look


like a horn, and the corresponding property becomes a horn condition or property.Horn-shaped domains have been studied in several contexts in the literature. Inparticular conditions on domains have been introduced in the context of extensionoperators and embedding theorems: the domains of F. John [1]; the (ε, δ)-domainsof P. W. John [1]; and the domains satisfying a flexible horn condition (which isa broader notion than the previous two) of O. V. Besov [1, 2].

Theorem 6.9. Let Ω be a subset of RN such that ∂Ω = ∅ and let 0 < ℓ < 1. Ω isequi-Holderian of index ℓ if and only if Ω has the uniform cusp property of index ℓ.

Proof. By Theorem 6.7, Ω has the uniform fat segment property if and only if Ω isan equi-C0 epigraph. It remains to sharpen the equivalence.

(⇐) In addition, Ω has the uniform cusp property of index ℓ. From inequal-ity (6.8) in the proof of Theorem 6.7 for all x ∈ ∂Ω and

∀ζ ′, ξ′ ∈ BH(0, ρ′) such that |ξ′ − ζ ′| < ρ, |ax(ξ′)− ax(ζ ′)| ≤ hℓ(|ξ′ − ζ ′|)

and since hℓ(θ) = λ(θ/ρ)ℓ, the functions ax are equi-C0,ℓ and Ω is equi-Holderianof index ℓ.

(⇒) In addition, Ω is equi-Holderian of index ℓ. In the proof of Theorem 6.7,we have constructed the modulus of continuity h ∈ H in (6.9):

∀θ ∈ [0, 3r], h(θ) = supζ′∈H|ζ′|=1

supx∈∂Ω

supξ′∈V

ξ′+θζ′∈V

|ax(θζ ′ + ξ′)− ax(ξ′)|.

Since the family ax : x ∈ ∂Ω is equi-C0,ℓ, the modulus h in (6.9) is also C0,ℓ andthere exists c such that |h(θ)| ≤ c |θ|ℓ. As a result for all x ∈ ∂Ω and all ζ ′, ξ′ ∈ Vsuch that |ξ′ − ζ ′| < 3r

|ax(ξ′)− ax(ζ ′)| ≤ h(|ξ′ − ζ ′|) ≤ c|ξ′ − ζ ′|θ.

Let hℓ(θ) = c θℓ. Then O(hℓ, r, r) ⊂ O(h, r, r) and Ω satisfies the uniform fatsegment property for O(hℓ, r, r), but this is precisely the uniform cusp property ofindex ℓ.

6.5 Hausdorff Measure and Dimension of the BoundaryWe have already seen in section 5.4.3 that Lipschitzian sets have a locally finiteboundary measure HN−1(∂Ω). In particular, when ∂Ω is compact, HN−1(∂Ω) isfinite. We now construct examples of Holderian sets of index α, 0 < α < 1, withcompact boundary (verifying the uniform cusp property) for which the Hausdorffdimension of the boundary is strictly greater than N − 1 and hence their boundarymeasure HN−1(∂Ω) = +∞. We first give an upper bound.

Theorem 6.10. Let Ω be a subset of RN with compact boundary. If Ω satisfies theuniform cusp property of Definition 6.4 (ii) associated with the function h(θ) = θα,0 < α < 1, then the Hausdorff dimension of ∂Ω is less than or equal to N − α.


Proof. From Theorem 6.9, Ω is equi-Holderian of index α, that is, an equi-C0,α

epigraph. Let r > 0, ρ > 0, and λ > 0 be the parameters of Theorem 6.6, andlet AxeN , Ax ∈ O(N), be the direction associated with the point x ∈ ∂Ω. Byassumption the functions ax : V → R satisfy the condition

∀ζ ′1, ζ

′2 ∈ V, |ax(ζ ′

2)− ax(ζ ′1)| ≤ c |ζ ′

2 − ζ ′1|α

. (6.10)

Since ∂Ω is compact, there exists a finite number of points xi ∈ ∂Ω : 1 ≤ i ≤ msuch that ∂Ω ⊂ ∪m

i=1U(xi).Given ε, 0 < ε < ρ, let NΩ(ε) be the number of hypercubes of dimension

N and side ε required to cover ∂Ω and let NΩ,i(ε) be the number of hypercubesof dimension N and side ε required to cover ∂Ω ∩ U(xi). We have the followingestimate:

NΩ,i(ε) ≤(rλε

)N−1 c(√

N − 1 ε)α

ε.

Indeed the neighborhood

V = BH(0, ρ) ⊂ BH(0, rλ)

can be covered by [rλ/ε]N−1 (N − 1)-dimensional hypercubes of side ε. On each(N − 1)-dimensional hypercube of side ε the variation between the minimum andthe maximum of the function ax is bounded by

c(√

(N − 1)ε2)α

= c(√

N − 1 ε)α

.

So the number of N -dimensional hypercubes of side ε necessary to cover the hyper-surface above each (N − 1)-dimensional hypercube of side ε is

[ cε

(√N − 1 ε

)α].

Finally

NΩ,i(ε) ≤(rλε

+ 1)N−1 ( c

ε

(√N − 1 ε

)α+ 1)

≤ 1εN−1

1ε1−α (rλ + ε)N−1

(c(√

N − 1)α

+ ε1−α)

≤ 1εN−α (rλ + ε)N−1

(c(√

N − 1)α

+ ε1−α)

.

As a result for all β > N − α

NΩ,i(ε) ≤m∑

i=1

NΩ,i(ε) ≤ m1

εN−α (rλ + ε)N−1(c(√

N − 1)α

+ ε1−α)

⇒ NΩ(ε)εβ ≤ εβ−N+α m (rλ + ε)N−1(c(√

N − 1)α

+ ε1−α)

⇒ ∀β > N − α, Hβ(∂Ω) = 0.

By definition, the Hausdorff dimension of ∂Ω is less than or equal to N − α.


It is possible to construct examples of sets verifying the uniform cusp propertyfor which the Hausdorff dimension of the boundary is strictly greater than N − 1and hence HN−1(∂Ω) = +∞.

Example 6.1.This following two-dimensional example of an open domain with compact boundarysatisfying the uniform cusp condition for the function h(θ) = θα, 0 < α < 1, caneasily be generalized to an N -dimensional example. Consider the open domain

Ω def= (x, y) : −1 < x ≤ 0 and 0 < y < 2∩ (x, y) : 0 < x < 1 and f(x) < y < 2∩ (x, y) : 1 ≤ x < 2 and 0 < y < 2

in R2 (see Figure 2.7), where f : [0, 1]→ R is defined as follows:

f(x) def= dC(x)α, 0 ≤ x ≤ 1,

and C is the Cantor set on the interval [0, 1] and dC(x) is the distance functionfrom the point x to the set C (dC is uniformly Lipschitzian of constant 1). Thisfunction is equal to 0 on C. Any point in [0, 1]\C belongs to one of the intervals oflength 3−k, k ≥ 1, which has been deleted from [0, 1] in the sequential constructionof the Cantor set. Therefore the distance function dC(x) is equal to the distancefunction to the two end points of that interval. In view of this special structure itcan be shown that

∀x, y ∈ [0, 1], |dC(y)α − dC(x)α| ≤ |y − x|α.

Denote by Γ the piece of the boundary ∂Ω specified by the function f = dC . On Γthe uniform cusp condition is verified with ρ = 1/6, λ = (1/6)α, and h(θ) = θα (see

f(x)

x

C(λ, h, ρ, e2)

−1 0 1 20

1 domain Ω

2

y

Figure 2.7. Domain Ω for N = 2, 0 < α < 1, e2 = (0, 1), ρ = 1/6,λ = (1/6)α, h(θ) = θα.


0 1/9 2/9 1/3 2/3 7/9 8/9 1

Figure 2.8. f(x) = dC(x)1/2 constructed on the Cantor set C for 2k + 1 = 3.

Figure 2.8). Clearly the number NΩ(ε) of hypercubes of dimension N and side εrequired to cover ∂Ω is greater than the number NΓ(ε) of hypercubes of dimensionN and side ε required to cover Γ. The construction of the Cantor set is done bysequentially deleting intervals. At step k = 0 the interval (1/3, 2/3) of width 3−1

is removed. At step k a total of 2k intervals of width 3−(k+1) are removed. Thus ifwe pick ε = 3−(k+1), the interval [0, 1] can be covered with exactly 3(k+1) intervals.Here we are interested in finding a lower bound to the total number of squares ofside ε necessary to cover Γ. For this purpose we keep only the 2k intervals removedat step k. Vertically it takes

[(2−13−(k+1))α

3−(k+1)

]≥(2−13−(k+1))α

3−(k+1) − 1.

Then we have for β ≥ 0

NΩ(ε) ≥ NΓ(ε)≥ 2k

((2−13−(k+1))α

3−(k+1) − 1

)

≥ 2k−α3(k+1)(1−α) − 2k =(3(1−α)2

)k2−α3(1−α) − 2k

and

NΩ(ε)(3−k)1+β ≥ 3−k(1+β)((

3(1−α)2)k

2−α3(1−α) − 2k

)

≥(3−(α+β)2

)k2−α3(1−α) −

(2

3(1+β)

)k

.

The second term goes to zero as k goes to infinity. The first term goes to infinity as kgoes to infinity if 3−(α+β)2 > 1, that is, 0 < α+β < ln 2/ ln 3. Under this condition,H1+β(∂Ω) = H1+β(Γ) = +∞ for all 0 < α < ln 2/ ln 3 and all 0 ≤ β < ln 2/ ln 3−α.


Therefore given 0 < α < ln 2/ ln 3

∀β, 0 ≤ β + α < ln 2/ ln 3, H1+β(∂Ω) = +∞

and the Hausdorff dimension of ∂Ω is strictly greater than 1.

Given 0 < α < 1, it is possible to construct an optimal example of a set verify-ing the uniform cusp property for which the Hausdorff dimension of the boundaryis exactly N − α and hence HN−1(∂Ω) = +∞.

Example 6.2 (Optimal example of a set that verifies the uniform cusp propertywith h(θ) = |θ|α, 0 < α < 1, and whose boundary has Hausdorff dimension exactlyequal to N − α.).

For that purpose, we need a generalization of the Cantor set. Denote by C1the Cantor set. Recall that each x, 0 ≤ x ≤ 1, can be written uniquely (if we makea certain convention) as

x =∞∑

j=1

aj(3, x)3j

,

where aj(3, x) can be regarded as the jth digit of x written in basis 3. From thisthe Cantor set is characterized as follows:

x ∈ C1 ⇐⇒ ∀j, aj(3, x) = 1.

Similarly for an arbitrary integer k ≥ 1, each x ∈ [0, 1] can be uniquely written inthe form

x =∞∑

j=1

aj(2k + 1, x)(2k + 1)j

and we can define the set Ck as

x ∈ Ck ⇐⇒ ∀j, aj(2k + 1, x) = k.

In a certain sense, if k1 > k2, Ck1 contains more points than Ck2 . We now use thesesets to construct the family of set Dk as follows:

x ∈ D1 ⇐⇒ 2x ∈ C1

and for k > 1, x ∈ Dk ⇐⇒ 2k+1(x− 2k) ∈ Ck.

Note that if k1 = k2, Dk1

⋂Dk2 = ∅ since the Dk’s contain only points from the

interval [1 − 2k−1, 1 − 2k]. Consider now the set Ddef= ∪∞

k=1Dk and go back toExample 6.1 with the function f replaced by the function f(x) def= dD(x)α. Again itcan be shown that for all x, y ∈ [0, 1], |dD(y)α − dD(x)α| ≤ |y − x|α. Note that onthe interval [1− 2k−1, 1− 2k] we have dD(x)α = dDk(x)α.

Denote by Γ the piece of boundary ∂Ω specified by the function f = dD andby Γk the part of boundary ∂Ω specified by the function f = dD (= dDk) on theinterval [1 − 2k−1, 1 − 2k]. Once again on Γ the uniform cusp property is verified


with ρ = 1/6, λ = (1/6)α, and h(θ) = θα. Clearly the number NΩ(ε) of hypercubesof dimension N and side ε required to cover ∂Ω is greater than the number NΓk(ε)of hypercubes of dimension N and side ε required to cover Γk. The constructionof the set Ck is also done sequentially by deleting intervals. At step j = 0 theinterval ]k/(2k + 1), (k + 1)/(2k + 1)[ of width (2k + 1)−1 is removed. At step ja total of 2j intervals of width (2j + 1)−(j+1) are removed. If we consider theintervals that remain at step j, a total of 2j+1 nonempty disjoint intervals of width(k/(2k + 1))j+1 remain in the set Ck. Each of these intervals contains a gap oflength (k/(2k + 1))j+1 1/(2k + 1) created at step j + 1.

If we construct the set Dk in the same way, at step j a total of 2j nonemptydisjoint intervals of width (k/(2k + 1))j+1 1/2k remain in the set Dk. Each of theseintervals contains a gap of length (k/(2k + 1))j+1 1/(2k(2k + 1)). Pick

ε =12k

(k

2k + 1

)j+1

and look for a lower bound on the number of squares of side ε necessary to cover Γk.For this purpose, consider only the 2j+1 nonempty disjoint intervals remaining atstep j. As they each contain a gap of length

(k

2k + 1

)j+1 12k(2k + 1)

vertically it takes[((

k

2k + 1

)j+1 12k+1(2k + 1)

)α2k

(2k + 1

k

)j+1]

≥((

k

2k + 1

)j+1 12k+1(2k + 1)

)α2k

(2k + 1

k

)j+1

− 1

ε-cubes. Then we have for β ≥ 0

NΩ(ε) ≥ NΓ(ε) ≥ 2j+1

(((k

2k + 1

)j+1 12k+1(2k + 1)

)α2k

(2k + 1

k

)j+1

− 1

)

≥(

2(2k + 1)kα

k(2k + 1)α

)j+1( 2k

2α(k+1)(2k + 1)α

)− 2j+1

=(2(2k + 1)1−αkα−1)j+1

(2k(1−α)−α

(2k + 1)α

)− 2j+1

and hence

ε1+β NΩ(ε)

≥(

12k

(k

2k + 1

)j+1)1+β((

2(2k + 1)1−αkα−1)j+1(

2k(1−α)−α

(2k + 1)α

)− 2j+1

)

≥((

k

2k + 1

)α+β

2

)j+12k(1−α)−α

2k(1+β)(2k + 1)α− 2j+1

(12k

(k

2k + 1

)j+1)1+β

.


The second term goes to zero as j goes to infinity. The first term goes to infinityas j goes to infinity if

∀k,

(k

2k + 1

)α+β

2 > 1, that is ∀k, 0 < α+ β <log 2

log ((2k + 1)/k).

As k can be chosen arbitrarily large, the former inequality reduces to 0 < α+β < 1.Under this condition there exists an integer k for which

H1+β(∂Ω) = H1+β(Γk) = +∞

for all 0 < α < 1 and all 0 ≤ β < 1− α. Therefore, given 0 < α < 1

∀β, 0 ≤ β < 1− α, H1+β(∂Ω) = +∞.

This implies that the Hausdorff dimension of ∂Ω is greater than or equal to 2− α,which is the upper bound we obtained in Theorem 6.10.

Chapter 3

Courant Metrics onImages of a Set

1 IntroductionA natural way to construct a family of variable domains is to consider the images ofa fixed subset of RN by some family of transformations of RN. The structure andthe topology of the images can be specified via the natural algebraic and topologicalstructures of the space of transformations or equivalence classes of transformationsfor which the full power of function analytic methods is available. There are manyways to do that, and specific constructions and choices are again very much problemdependent.

In 1972 A. M. Micheletti [1] introduced what may be one of the firstcomplete metric topologies on a family of domains of class Ck, k ≥ 1, that are theimages of a fixed open domain through a family of Ck-diffeomorphisms of RN. Therethe natural underlying algebraic structure is the group structure of the compositionof transformations with the identity transformation as the neutral element. Heranalysis culminates with the construction of a complete metric on the quotient ofthe group by an appropriate closed subgroup of transformations leaving the fixedsubset unaltered. She called it the Courant metric1 because it is proved in thebook of R. Courant and D. Hilbert [1, p. 420] that the nth eigenvalue of theLaplace operator depends continuously on the domain Ω, where Ω = (I + f)Ω0 isthe image of a fixed domain Ω0 by I + f and f is a smooth mapping. But thereis no notion of a metric in that book. Her constructions naturally extend to otherfamilies of transformations of RN or of fixed holdalls D.

In section 2 we first extend her generic constructions associated with thespace Ck

0 (RN,RN) of mappings from RN into RN to a larger family of Banachspaces of mappings such as Ck(RN,RN), Ck,1(RN,RN), or Bk(RN,RN), and be-yond to Frechet spaces such as B(RN,RN) or C∞

0 (RN,RN) of infinitely continuously

1“Nello studiare la continuita dell’n-esimo autovalore dell’operatore di Laplace - ∆Ω relativoad un aperto limitato Ω con dati di Dirichlet nulli, considerato comme funzione dell’aperto Ω,Courant introduce una nozione di vicinanza tra due domini basata su un diffeomorphismo del tipoI + ψ con ψ ∈ C1(Rm), che transforma l’uno nell’altro.” (cf. A. M. Micheletti [1]).

123

124 Chapter 3. Courant Metrics on Images of a Set

differentiables mappings. We emphasize the geodesic character of the constructionof the metric and its interpretation as trajectories of bounded variation on the group.

The next step in the construction is the choice of the closed subgroup of trans-formations of RN that is very much problem dependent. Originally, it was chosenas the set of transformations that leave the underlying set or pattern unaltered.However, in some applications, it could be unaltered up to a translation, a rotation,or a flip. The underlying set or pattern can be a closed set or an open crack-freeset.2 This includes closed submanifolds of RN. It is shown that, as long as thesubgroup is closed, we get a complete Courant metric on the quotient group. Inthis section we also characterize the tangent space to the group of transformationsof RN that leads to the Courant metric. It is an example of an infinite-dimensionalmanifold (cf. Theorems 2.14 and 2.17 in sections 2.5 and 2.6).

In section 3, we free the constructions from the framework of bounded con-tinuously differentiable transformations to reach the spaces of all homeomorphismsor Ck-diffeomorphisms of RN or an open subset D of RN. Again it is shown thatthey are complete metric spaces. Hence, from section 2, their quotient by a closedsubgroup yields a Courant metric and a complete metric topology. With such largerspaces, it now becomes possible to consider subgroups involving not only transla-tions but also isometries, symmetries, or flips in RN or D.

2 Generic Constructions of MichelettiThe original constructions of A. M. Micheletti [1] were carried out for the Banachspace Ck

0 (RN,RN), k ≥ 0.3 In this section, we generalize them to Banach spaces Θof mappings from RN into RN under fairly general assumptions.

2.1 Space F(Θ) of Transformations of RN

Associate with a real vector space Θ of mappings from RN to RN the followingspace of transformations of RN:

F(Θ) def=I + f : f ∈ Θ, (I + f) bijective and (I + f)−1 − I ∈ Θ

, (2.1)

where x $→ I(x) def= x : RN → RN is the identity mapping.4 It will be shown thatF(Θ) is a group for the composition (F G)(x) def= F (G(x)) of transformations ofRN. Equivalent right-invariant metrics on F(Θ) will be introduced under Assump-tions 2.1 and 2.2 and related to some notion of geodesics in the group F(Θ). Thecompleteness of F(Θ) will require Assumption 2.3 and either

2Cf. Definition 7.1 in Chapter 8.3Ck

0 (RN,RN) denotes the space of k times continuously differentiable functions from RN toRN, whose derivatives vanish at infinity. It is also denoted by Bk

0 (RN,RN) in the literature.4For Θ = Ck

0 (RN,RN), this definition is equivalent to the one of A. M. Micheletti [1]:

F(Ck0 (RN,RN)) def=

F : RN → RN : F − I ∈ Ck

0 (RN,RN) and F −1 ∈ Ck(RN,RN)

.

2. Generic Constructions of Micheletti 125

(i) the assumption that Θ ⊂ C0(RN,RN) and, for all x ∈ RN, the mappingf $→ f(x) : Θ → RN is continuous that makes the group a complete metricspace, or

(ii) Assumption 2.4 that makes the group a complete (metric) topological group.

Finally, the right-invariant Courant metric will be defined as the quotient metric

∀F, G ∈ F(Θ), dG([F ], [H]) def= infG,G∈G

d(F G, H G), [F ] def= F G (2.2)

associated with the quotient group F(Θ)/G of F(Θ) by a closed subgroup G ofF(Θ). The subgroup considered by A. M. Micheletti [1] was

G(Ω0)def= F ∈ F(Θ) : F (Ω0) = Ω0

to quotient out all F whose image of the fixed subset Ω0 of RN is Ω0 and make thequotient group F(Θ)/G(Ω0) isomorphic to the set of images

X (Ω0)def= F (Ω0) : ∀F ∈ F(Θ) (2.3)

of Ω0 by the elements of F(Θ). The abstract results will be applied with Θequal to Banach spaces such as Ck

0 (RN,RN) ⊂ Ck(RN,RN) ⊂ Bk(RN,RN) andCk,1(RN,RN), k ≥ 0, and, through special constructions, to the Frechet spacesC∞

0 (RN,RN) ⊂ B(RN,RN) = ∩k≥0Bk(RN,RN).We proceed by introducing the assumptions step by step.

Assumption 2.1.Θ is a real vector space of mappings from RN into RN and

∀g ∈ Θ, ∀ I + f ∈ F(Θ), g (I + f) ∈ Θ.

Theorem 2.1. Let Θ be a real vector space of mappings from RN into RN. F(Θ) isa group for the composition if and only if Assumption 2.1 is verified. In particular(I + f)−1 − I = −f (I + f)−1 ∈ Θ.

Example 2.1.(1) Θ = B0(RN,RN), space of bounded continuous functions. For g ∈ B0(RN,RN)and I +f ∈ F(B0(RN,RN)), the composition g (I +f) is continuous and ∥g (I +f)∥C0 = ∥g∥C0 <∞.

(2) Θ = C0(RN,RN), space of bounded uniformly continuous functions. Forg ∈ C0(RN,RN) and I + f ∈ F(C0(RN,RN)), ∥g (I + f)∥C0 = ∥g∥C0 < ∞, andsince g and I + f are uniformly continuous, so is the composition g (I + f).

(3) Θ = C00 (RN,RN), subspace of C0(RN,RN). From (2) ∥g (I + f)∥C0 =

∥g∥C0 < ∞ and g (I + f) ∈ C0(RN,RN). It remains to show that g (I + f) ∈C0

0 (RN,RN). Since g ∈ C00 (RN,RN), for all ε > 0, there exists ρ0 > 0 such that

for all x such that |x| > ρ0, |g(x)| < ε. But f also belongs to C00 (RN,RN) and


there exists ρ ≥ 2ρ0 such that for all x such that |x| > ρ, |f(x)| < ρ0. In particular,|x+f(x)| ≥ |x|− |f(x)| > ρ−ρ0 ≥ ρ0 and hence |g(x+f(x))| < ε. In conclusion forall ε > 0, there exists ρ > 0 such that for all x such that |x| > ρ, |g(x + f(x))| < εand g (I + f) ∈ C0

0 (RN,RN).(4) Θ = C0,1(RN,RN). This is the space of bounded Lipschitz continuous

functions on RN that is contained in C0(RN,RN). For f, g ∈ C0,1(RN,RN), ∥f (I + g)∥C0 = ∥f∥C0 <∞. Since g is Lipschitz with Lipschitz constant c(g), I + g isalso Lipschitz with Lipschitz constant 1+ c(g) and the composition is also Lipschitzwith constant c(f) (1 + c(g)).

Proof of Theorem 2.1. I ∈ F(Θ) for f = 0. If I + f ∈ F(Θ), (I + f)−1 is bijective,(I + f)−1 − I ∈ Θ, (F−1)−1 = F = I + f , and (F−1)−1 − I = f ∈ Θ. Hence(I + f)−1 ∈ F(Θ) and

I = (I + f) (I + f)−1 = (I + f)−1 + f (I + f)−1

⇒ (I + f)−1 − I = −f (I + f)−1.

The last property to check is the composition (I + f) (I + g). It is bijective asthe composition of two bijective transformations. Moreover, by linearity of Θ andAssumption 2.1

(I + f) (I + g)− I = g + f (I + g) ∈ Θ. (2.4)

From (2.4) and the invertibility of the composition

[(I + f) (I + g)]−1 − I = −[g + f (I + g)] (I + g)−1 (I + f)−1 ∈ Θ

by using Assumption 2.1 twice. Conversely, if (2.4) is true for all f ∈ Θ andI + g ∈ F(Θ), by linearity of Θ, Assumption 2.1 is verified.

Associate with F ∈ F(Θ) the following function of I and F :

d(I, F ) def= infF=F1···Fn

Fi∈F(Θ)

n∑

i=1

∥Fi − I∥Θ + ∥F−1i − I∥Θ, (2.5)

where the infimum is taken over all finite factorizations of F in F(Θ) of the form

F = F1 · · · Fn, Fi ∈ F(Θ).

In particular d(I, F ) = d(I, F−1). Extend this function to all F and G in F(Θ):

d(F, G) def= d(I,G F−1). (2.6)

By definition, d is right-invariant since for all F , G, and H in F(Θ)

d(F, G) = d(F H, G H).


To show that d is a metric5 on F(Θ) necessitates a second assumption.

Assumption 2.2.(Θ, ∥ · ∥) is a normed vector space of mappings from RN into RN and for eachr > 0 there exists a continuous function c0(r) such that

∀fini=1 ⊂ Θ such that

n∑

i=1

∥fi∥ < α < r; (2.7)

then

∥(I + f1) · · · (I + fn)− I∥ < α c0(r). (2.8)

Theorem 2.2. Under Assumptions 2.1 and 2.2, d is a right-invariant metricon F(Θ).

Example 2.2.(1), (2), (3) Since the norm on the three spaces B0(RN,RN), C0(RN,RN), andC0

0 (RN,RN) is the same and Assumption 2.2 involves only the norm, it is sufficientto check it for B0(RN,RN). Consider fin

i=1 ⊂ B0(RN,RN) such thatn∑

i=1

∥fi∥C0 < α < r. (2.9)

Using the convention (I + fn) (I + fn) = (I + fn) in the summation

(I + f1) · · · (I + fn)− I

= (I + fn)− I +n−1∑

i=1

(I + fi) · · · (I + fn)− (I + fi+1) · · · (I + fn)

= fn +n−1∑

i=1

fi (I + fi+1) · · · (I + fn)

⇒ ∥(I + f1) · · · (I + fn)− I∥C0 ≤n∑

i=1

∥fi∥C0 < α < r ⇒ c0(r) = r.

(4) Θ = C0,1(RN,RN). Consider a sequence fini=1 ⊂ C0,1(RN,RN) such that

n∑

i=1

max ∥fi∥C0 , c(fi) < α < r. (2.10)

5A function d : X × X → R is said to be a metric on X if (cf. J. Dugundji [1])(i) d(F.G) ≥ 0, for all F, G,

(ii) d(F, G) = 0 ⇐⇒ F = G,

(iii) d(F, G) = d(G, F ), for all F, G,

(iv) d(F, H) ≤ d(F, G) + d(G, H), for all F, G, H.A metric d on a group (F , ) is said to be right-invariant if for all H ∈ F , d(F H, GH) = d(F, G).


From the previous example

∥(I + f1) · · · (I + fn)− I∥C0 ≤n∑

i=1

∥fi∥C0 < α.

In addition,

(I + f1) · · · (I + fn)− I = fn +n−1∑

i=1

fi (I + fi+1) · · · (I + fn)

⇒ c((I + f1) · · · (I + fn)− I) ≤ c(fn) +n−1∑

i=1

c(fi) (1 + c(fi+1)) . . . (1 + c(f1))

≤ c(fn) +n−1∑

i=1

c(fi) e∑i+1

k=1c(fk)

≤[

n∑

i=1

c(fi)

]e[∑n

i=1c(fi)] < α eα

⇒ ∥(I + f1) · · · (I + fn)− I∥C0,1 < α maxeα, 1 < α er

and we can choose c0(r) = er.

Proof of Theorem 2.2. Properties (i) and (iii) are verified by definition. For thetriangle inequality (iv), F H−1 = (F G−1) (G H−1) is a finite factorization ofF H−1. Consider arbitrary finite factorizations of F G−1 and G H−1:

F G−1 = (I + α1) · · · (I + αm), I + αi ∈ F(Θ),

G H−1 = (I + β1) · · · (I + βn), I + βj ∈ F(Θ).

This yields a new finite factorization of F H−1. By definition of d,

d(I, F H−1) ≤m∑

i=1

∥αi∥+ ∥αi (I + αi)−1∥+n∑

i=j

∥βj∥+ ∥βj (I + βj)−1∥,

and by taking infima over each factorization,

d(F, H) = d(I, F H−1) ≤ d(I, F G−1) + d(I,G H−1) = d(F, G) + d(G, H)

using the symmetry property (iii). To verify (ii) it is sufficient to show that F =I ⇐⇒ d(I, F ) = 0. Clearly I I is a factorization of F = I and 0 ≤ d(I, F ) ≤ 0.Conversely, if d(I, F ) = 0, for ε, 0 < ε < 1, there exists a finite factorizationF = F1 · · · Fn such that

n∑

i=1

∥Fi − I∥+ ∥F−1i − I∥ < ε < 1.

By Assumption 2.2, ∥F − I∥+ ∥F−1 − I∥ < ε c0(1), which implies that F − I = 0and F−1 − I = 0. This completes the proof.


An important aspect of the previous construction was to build in the triangleinequality to make d a (right-invariant) metric. However, there are other ways tointroduce a topology on F(Θ). The interest of this specific choice will become ap-parent later on. For instance, the original Courant metric of A. M. Micheletti [1]in 1972 was constructed from the following slightly different right-invariant metricon F(Θ): given H ∈ F(Θ), consider finite factorizations H1 · · · Hm, Hi ∈ F(Θ),1 ≤ i ≤ m, of H and finite factorizations G1 · · · Gn, Gj ∈ F(Θ), 1 ≤ j ≤ n, ofH−1, and define

d1(I,H) def= infH=H1···Hm

m≥1

m∑

i=1

∥Hi − I∥+ infH−1=G1···Gn

n≥1

n∑

i=1

∥Gi − I∥, (2.11)

where the infima are taken over all finite factorizations of H and H−1 in F(Θ) and

∀F, G ∈ F(Θ), d1(G, F ) def= d1(I, F G−1). (2.12)

Another example is the topology generated by the right-invariant semimetric6

∀H ∈ F(Θ), d0(I, H) def= ∥H − I∥+ ∥H−1 − I∥,

∀F, G ∈ F(Θ), d0(G, F ) def= d0(I, F G−1).(2.13)

Given r > 0 and F ∈ F(Θ), let Br(F ) = G ∈ F(Θ) : d0(F, G) < r. Let τ0 be theweakest topology on F(Θ) such that the family Br(F ) : 0 ≤ r < ∞ is the basefor τ0. By definition

d1(F, G) ≤ d(F, G) ≤ d0(F, G)

and the injections (F , d1)→ (F , d)→ (F , d0) are continuous.

Theorem 2.3. Under Assumptions 2.1 and 2.2, the topologies generated by d1, d,and d0 are equivalent.

Proof. Since d0, d, and d1 are right-invariant on the group F(Θ), it is sufficient toprove the equivalence for d0 and d1 around I. For all ε > 0 with δ = ε

∀H ∈ F(Θ), d0(I, H) < δ, d1(I,H) < ε,

since d1(I,H) ≤ d0(I,H). Conversely, by Assumption 2.2, for any δ, 0 < δ < 1,

d1(I,H) < δ < 1 ⇒ d0(I −H) = ∥H − I∥+ ∥H−1 − I∥ < 2δ c(1).

So for any ε > 0, pick δ = min1, ε/c(1)/2. Hence

∀ε > 0, ∃δ > 0, ∀H ∈ F(Θ), d1(I,H) < δ, d0(I,H) < ε.

This concludes the proof of the equivalences.6Given a space X, a function d : X × X → R is said to be a semimetric if(i) d(F.G) ≥ 0, for all F, G,

(ii) d(F, G) = 0 ⇐⇒ F = G,

(iii) d(F, G) = d(G, F ), for all F, G.This notion goes back to Frechet and Menger.


F. Murat and J. Simon [1] in 1976 considered as candidates for Θ the spaces

W k+1,c(RN,RN)def=

f ∈W k,∞(RN,RN) : ∀ 0 ≤ |α| ≤ k + 1, ∂αf ∈ C(RN,RN)

and W k+1,∞(RN,RN), k ≥ 0. The first space W k+1,c(RN,RN) is equivalent tothe space Ck+1(RN,RN), k ≥ 0, algebraically and topologically. The secondspace W k+1,∞(RN,RN) coincides with Ck,1(RN,RN), k ≥ 0. For the spacesCk+1(RN,RN) and Ck,1(RN,RN), k ≥ 0, they obtained the following pseudo tri-angle inequality for the semimetric d0:

d0(F1,F3)≤d0(F1, F2) + d0(F2, F3)

+ d0(F1, F2) d0(F2, F3) P [d0(F1, F2) + d0(F2, F3)] , ∀F1, F2, F3,

(2.14)

where P : R+ → R+ is a continuous increasing function. With that additionalproperty, they called d0 a pseudodistance7 and showed that they could construct ametric from it.

Lemma 2.1. Assume that the semimetric d0 satisfies the pseudo triangle inequality(2.14). Then, for all α, 0 < α < 1, there exists a constant ηα > 0 such that thefunction d(α)

0 : F(Θ)× F(Θ)→ R+ defined as

d(α)0 (F1, F2)

def= infd0(F1, F2), ηαα

is a metric on E.

The Micheletti construction with the metric d1 was given in 2001 for thespaces Ck(RN,RN) and Ck,1(RN,RN), k ≥ 0, in M. C. Delfour and J.-P. Zo-lesio [37] so that pseudodistances can be completely bypassed. The metrics d andd1 constructed from the semimetric d0 under the general Assumptions 2.1 and 2.2are both more general and interesting since they are of geodesic type by the use ofinfima over finite factorizations (I + θ1) · · · (I + θn) of I + θ when the semimetricd0 is interpreted as an energy

n∑

i=1

d0(I + θi, I) =n∑

i=1

∥θi∥+ ∥ − θi (I + θi)−1∥ (2.15)

7This terminology is not standard. In the literature a pseudometric or pseudodistance is usuallyreserved for a function d : X × X → R satisfying

(i) d(F.G) ≥ 0, for all F, G,

(ii’) d(F, G) = 0 ⇐= F = G,

(iii) d(F, G) = d(G, F ), for all F, G.

(iv) d(F, H) ≤ d(F, G) + d(G, H), for all F, G, H.Condition (ii) for a metric is weakened to condition (ii’) for a pseudodistance.


over families of piecewise constant trajectories T : [0, 1]→ F(Θ) starting from I attime 0 to I + θ at time 1 in the group F(Θ) by assigning times ti, 0 = t0 < t1 < · · ·< tn < tn+1 = 1, at each jump:

T (t) def=

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

I, 0 ≤ t < t1,

(I + θ1), t1 < t < t2,

(I + θ2) (I + θ1), t2 < t < t3,

. . .

(I + θi) · · · (I + θ1), ti < t < ti+1,

. . .

(I + θn) · · · (I + θ1), tn < t ≤ 1.

(2.16)

The jump in the group F(Θ) at time ti is given by

[T (t)]ti

def= T (t+i ) T (t−i )−1 − I

= [(I + θi) · · · (I + θ1)] [(I + θi−1) · · · (I + θ1)]−1 − I = θi

as an element of Θ that could be viewed as the tangent space to F(Θ) ⊂ I + Θ.The choice of d in a geodesic context is more interesting than the choice of d1

that would involve a double infima in the energy (2.15). At this stage, in view of theequivalence Theorem 2.3, working with the metric d or d1 is completely equivalent.

Assumption 2.3.(Θ, ∥ · ∥) is a normed vector space of mappings from RN into RN and there existsa continuous function c1 such that for all I + f ∈ F(Θ) and g ∈ Θ

∥g (I + f)∥ ≤ ∥g∥ c1(∥f∥). (2.17)

The next assumption is a kind of uniform continuity.

Assumption 2.4.(Θ, ∥ · ∥) is a normed vector space of mappings from RN into RN. For each g ∈ Θ,for all ε > 0, there exists δ > 0 such that

∀γ ∈ Θ such that ∥γ∥ < δ, ∥g (I + γ)− g∥ < ε. (2.18)

Theorem 2.4. Under Assumptions 2.1 to 2.4, F(Θ) is a topological (metric) group.

Example 2.3 (Checking Assumption 2.3).(1), (2), (3) Since the norm on the three spaces B0(RN,RN), C0(RN,RN), andC0

0 (RN,RN) is the same and Assumption 2.2 involves only the norm, it is sufficientto check it for B0(RN,RN). Consider g ∈ B0(RN,RN) and I +f ∈ F(B0(RN,RN)).Since I + f is a bijection

∥g (I + f)∥C0 ≤ ∥g∥C0 (2.19)

and the constant function c1(r) = 1 satisfies Assumption 2.3.


(4) Θ = C0,1(RN,RN). Consider g ∈ C0,1(RN,RN) and I + f ∈ F(C0,1

(RN,RN)). From (1)

∥g (I + f)∥C0 ≤ ∥g∥C0

and since all the functions are Lipschitz

c(g (I + f)) ≤ c(g) (1 + c(f))⇒ ∥g (I + f)∥C0,1 ≤ ∥g∥C0,1 (1 + c(f)) ≤ ∥g∥C0,1 (1 + ∥f∥C0,1)

and the continuous function c1(r) = 1 + r satisfies Assumption 2.3.

Assumption 2.4 is a little trickier. It involves not only norms but also a“uniform continuity” of the function g that is not verified for g ∈ B0(RN,RN) andfor the Lipschitz part c(g) of the norm of C0,1(RN,RN).

Example 2.4 (Checking Assumption 2.4).(1) Θ = B0(RN,RN). In that space there exists some g that is not uniformlycontinuous. For that g there exists ε > 0 such that for all n > 0, there exists xn, yn,|yn−xn| < 1/n such that |g(yn)− g(xn)| ≥ ε. Define the function γn(x) = yn−xn.Therefore, there exists ε > 0 such that for all n with ∥γn∥C0 < 1/n,

∥g (I + γn)− g∥C0 ≥ |g(xn + γn(xn))− g(xn)| = |g(xn + yn − xn)− g(xn)| ≥ ε

and Assumption 2.4 is not verified. However, B0(RN,RN) ⊂ C0(RN,RN) and forall x ∈ RN, the mapping f $→ f(x) : B0(RN,RN)→ R is continuous.

(2) and (3) Θ = C0(RN,RN). By the uniform continuity of g ∈ C0(RN,RN),for all ε > 0 there exists δ > 0 such that

∀x, y ∈ RN, |g(y)− g(x)| < ε/2.

For ∥γ∥C0 < δ

∀x ∈ RN, |(I + γ)(x)− x| = |γ(x)| ≤ ∥γ∥C0 < δ,

∀x ∈ RN, |g(x + γ(x))− g(x)| < ε/2 ⇒ ∥g (I + γ)− g∥C0 ≤ ε/2 < ε

and Assumption 2.4 is verified. In addition, C0(RN,RN) ⊂ C0(RN,RN) and forall x ∈ RN, the mapping f $→ f(x) : C0(RN,RN)→ R is continuous.

(4) Θ = C0,1(RN,RN). Since g ∈ C0,1(RN,RN)

∀x, y ∈ RN, |g(y)− g(x)| < c(g) |y − x|.

Therefore for

∥γ∥C0,1def= max∥γ∥C0 , c(γ) < δ,

∥γ∥C0 < δ and we get

∥g (I + γ)− g∥C0 ≤ c(g) ∥γ∥C0 < c(g) δ.


Unfortunately, for the part c(g (I + γ)) of the C0,1-norm, for all x = y ∈ RN

|g(y + γ(y))− g(y)− (g(x + γ(x))− g(x))||y − x| ≤ c(g) + c(g) c(I + γ),

c(g (I + γ)− g) ≤ c(g) [1 + c(γ)] < c(g) [1 + δ]

and we cannot satisfy Assumption 2.4. However, C0,1(RN,RN) ⊂ C0(RN,RN) andfor all x ∈ RN, the mapping f $→ f(x) : C0,1(RN,RN)→ R is continuous.

Remark 2.1.Since the spaces Ck

0 (RN,RN) ⊂ Ck(RN,RN) ⊂ Bk(RN,RN) have the same normand Assumptions 2.2 and 2.3 involve only the norms, they will always be verifiedfor Ck

0 (RN,RN) and Ck(RN,RN) if they are verified for Bk(RN,RN), k ≥ 0. SolelyAssumption 2.1 will require special attention.

The fact that Assumption 2.4 is not always verified will not prevent us fromproving that the space (F(Θ), d) is complete. The completeness will be proved inTheorem 2.6 under Assumptions 2.1 to 2.3, where Assumption 2.4 will be replacedby the condition Θ ⊂ C0(RN,RN) and, for all x ∈ RN, the mapping f $→ f(x) :Θ → RN is continuous. The following theorem will be proved in Theorems 2.11,2.12, and 2.13 of section 2.5 and Theorems 2.15 and 2.16 of section 2.6.

Theorem 2.5. Let k ≥ 0 be an integer.

(i) Ck0 (RN,RN) and Ck(RN,RN) satisfy Assumptions 2.1 to 2.4, Ck

0 (RN,RN) ⊂Ck(RN,RN) ⊂ C0(RN,RN), and, for all x ∈ RN, the mapping f $→ f(x) :Ck(RN,RN)→ RN is continuous.

(ii) Bk(RN,RN) and Ck,1(RN,RN) verify Assumptions 2.1 to 2.3, Ck,1(RN,RN) ⊂ Bk(RN,RN) ⊂ C0(RN,RN), and, for all x ∈ RN, the mappingf $→ f(x) : Bk(RN,RN)→ RN is continuous.

Proof of Theorem 2.4. From J. L. Kelley [1], it is sufficient to prove that

∀F ∈ F(Θ), ∀H ∈ F(Θ) such that d(I,H)→ 0, d(I, F−1 H F )→ 0.

Rewrite F−1 H F as follows:

F−1 H F − I = F−1 [I + H − I] F − F−1 F

=[F−1 (I + γ)− F−1] F

=[(F−1 − I) (I + γ)− (F−1 − I)

] F + γ F,

γdef= H − I and g

def= (F−1 − I) (I + γ)− (F−1 − I).

By Assumption 2.3,

∥γ F∥ ≤ ∥γ∥ c1(∥F − I∥), ∥g F∥ ≤ ∥g∥ c1(∥F − I∥);


by Assumption 2.4, for all ε > 0, there exists δ, 0 < δ < min1, ε/(4c0(1) c1(∥F −I∥)), such that

∥γ∥ = ∥H − I∥ < δ c0(1) ⇒∥g∥ =∥(F−1 − I) (I + γ)− (F−1 − I)∥

< ε/(4 c1(∥F − I∥)).

Combining the above properties

∀H such that d(I, H) < δ, ∥H − I∥ < δ c0(1)

⇒ ∥F−1 H F − I∥ ≤ ∥γ∥ c1(∥F − I∥) + ε/2 ≤ δ c0(1) c1(∥F − I∥) + ε/4 ≤ ε/2.

By Assumption 2.2, for all H such that d(I,H) < δ < 1, we have ∥H − I∥ < δ c0(1)and ∥H−1 − I∥ < δ c0(1). Finally, for all ε > 0, there exists δ > 0 such thatd(I,H) < δ implies ∥F−1 H F − I∥ ≤ ε/2. By a similar argument, for all ε > 0,there exists δ′ > 0 such that d(I, H) < δ′ implies ∥F−1 H−1 F − I∥ ≤ ε/2.Therefore, for all ε > 0 there exists δ0 = minδ, δ′ > 0 such that for all H,d(I,H) < δ0,

d(I, F−1 H F ) ≤ ∥F−1 H F − I∥+ ∥F−1 H−1 F − I∥ < ε

and F(Θ) is a topological group for the metric d.

Theorem 2.6. (i) Let Assumptions 2.1 to 2.3 be verified for a Banach space Θcontained in C0(RN,RN) such that, for all x ∈ RN, the mapping f $→ f(x) :Θ→ RN is continuous. Then (F(Θ), d) is a complete metric space.

(ii) Let Assumptions 2.1 to 2.4 be verified for a Banach space Θ. Then F(Θ) isa complete topological (metric) group.

Remark 2.2.F(C0,1(RN,RN)) is a complete metric space since the assumptions of Theorem 2.6are verified, but it is not a topological group since Assumption 2.4 is not verified.

Proof of Theorem 2.6. (i) Let Fn be a Cauchy sequence in F(Θ).(a) Boundedness of Fn−I and F−1

n −I in Θ. For each ε, 0 < ε < 1, thereexists N such that, for all m, n > N , d(Fm, Fn) < ε. By the triangle inequality

|d(I, Fm)− d(I, Fn)| ≤ d(Fm, Fn) < ε,

d(I, Fn is Cauchy, and, a posteriori, is bounded by some constant L. For each n,there exists a factorization Fn = (I + fn

1 ) · · · (I + fnν(n)) such that

0 ≤ν(n)∑

i=1

∥fni ∥+ ∥fn

i (I + fn1 )−1∥ − d(I, Fn) < ε < 1

⇒ν(n)∑

i=1

∥fni ∥+ ∥fn

i (I + fni )−1∥ < L + ε < L + 1.


By Assumption 2.2

∥Fn − I∥+ ∥F−1n − I∥ < 2(L + ε) c0(L + 1)

and the sequences Fn − I and F−1n − I are bounded in Θ.

(b) Convergence of Fn − I and F−1n − I in Θ. For any m, n

Fn − Fm = (Fn F−1m − I) Fm

and by Assumption 2.3

∥Fm − Fn∥ = ∥(Fn F−1m − I) Fm∥ ≤ ∥Fn F−1

m − I∥ c1(∥Fm − I∥)

⇒ ∥Fm − Fn∥ ≤ c+1 ∥Fn F−1

m − I∥, c+1

def= supm

c1(∥Fm − I∥) <∞

since Fm − I is bounded. For all ε, 0 < ε < c0(1) c+1 , there exists N such that

∀n, m, > N, d(I, Fn F−1m ) = d(Fm, Fn) < ε/(c0(1) c+

1 ) < 1.

By Assumption 2.2

∥Fn F−1m − I∥ < c0(1)ε/(c0(1) c+

1 ) = ε/c+1

⇒ ∀n, m, > N, ∥Fm − I − (Fn − I)∥ ≤ c+1 ∥Fn F−1

m − I∥ < ε

and Fn − I is a Cauchy sequence in Θ that converges to some F − I ∈ Θ sinceΘ is a Banach space. Similarly, we get that F−1

n − I is a Cauchy sequence in Θthat converges to some G− I ∈ Θ.

(c) F is bijective, G = F−1, and F ∈ F(Θ). For all n, Fn, F−1n , F , and G

are continuous since Θ ⊂ C0(RN,RN). By continuity of f $→ f(x) : Θ → RN,f $→ x + f(x) : Θ → RN is continuous and [F−1

n (x) − x| + x → [G(x) − x] + x.By the same argument for Fn, x = Fn(F−1

n (x)) → F (G(x)) = (F G)(x) andfor all x ∈ RN, (F G)(x) = x. Similarly, starting from F−1

n Fn = I, we get(G F )(x) = x. Hence, G F = I = F G. So F is bijective, F − I, F−1 − I ∈ Θ,and F ∈ F(Θ).

(d) Convergence of Fn → F in F(Θ). To check that Fn → F in F(Θ) recallthat by Assumption 2.3

d(Fn, F ) = d(I, F F−1n ) ≤∥I − F F−1

n ∥+ ∥I − Fn F−1∥= ∥(Fn − F ) F−1

n ∥+ ∥(F − Fn) F−1∥≤ c1(F−1

n − I)∥Fn − F∥+ c1(F−1 − I)∥F − Fn∥

≤(max

nc1(F−1

n − I) + c1(F−1 − I))∥F − Fn∥,

which goes to zero as Fn − I goes to F − I in Θ.(ii) The proof (ii) differs from the proof of (i) only in part (c). Consider

G F − I = G F −G Fn + (G− F−1n ) Fn.


Since Fn − I is bounded in Θ and F−1n → G, the term (G− F−1

n ) Fn goes to 0as n→∞ by Assumption 2.3. The second terms can be rewritten as

G F −G Fn = F − Fn + (G− I) F − (G− I) Fn = F − Fn + g F − g Fn,

where g = G− I ∈ Θ. The term F − Fn goes to zero.As for the second terms

g F − g Fn =g [I + (F − Fn) F−1

n ]− g Fn.

Let gn = g [I + (F − Fn) F−1n ]− g and let γn = (F − Fn) F−1

n .By boundedness of Fn − I and F−1

n − I in Θ and Assumption 2.3,

∥γn∥ = ∥(F − Fn) F−1n ∥ ≤ ∥F − Fn∥ c1 (∥F−1

n − I∥) ≤ c+1 ∥F − Fn∥,

∥gn Fn∥ ≤ ∥gn∥ c1 (∥Fn − I∥) ≤ c+1 ∥gn∥,

c+1

def= supn

c1 (∥Fn − I∥), c1 (∥F−1n − I∥) <∞.

By Assumption 2.4, for all ε > 0 there exists δ > 0 such that

∥γn∥ < δ c+1 ⇒ ∥gn∥ = ∥g (I + γn)− g∥ < ε/c+

1 .

By combining for all ε > 0, there exists δ > 0 such that

∥F − Fn∥ < δ ⇒ ∥γn∥ ≤ c+1 ∥F − Fn∥ < c+

1 δ

⇒ ∥gn Fn∥ < c+1 ε/c+

1 = ε

⇒ ∥F − Fn∥ < δ ⇒ ∥g F − g Fn∥ < ε.

As a result ∥G F − I∥ is bounded by an expression that goes to zero as n → ∞and G F = I. By an analogous argument, we can prove that F G− I = 0, thatis, F−1 = G. So F is bijective, F − I, F−1 − I ∈ Θ. Hence F ∈ F(Θ).

2.2 Diffeomorphisms for B(RN, RN) and C∞0 (RN, RN)

Consider as other candidates for Θ the spaces

B(RN,RN) def= ∩∞k=0Bk(RN,RN) = ∩∞

k=0Ck(RN,RN), (2.20)

C∞0 (RN,RN) def= ∩∞

k=0Ck0 (RN,RN) (2.21)

and the resulting groups F(B(RN,RN))I + f : f ∈ B(RN,RN), (I + f) bijective and (I + f)−1 − I ∈ B(RN,RN)

and F(C∞0 (RN,RN))

I + f : f ∈ C∞

0 (RN,RN), (I + f) bijective and (I + f)−1 − I ∈ C∞0 (RN,RN)


that satisfy Assumption 2.1 but not the other assumptions since B(RN,RN) andC∞

0 (RN,RN) are Frechet but not Banach spaces. To get around this, observe that

F(B(RN,RN)) = ∩∞k=0F(Bk(RN,RN)) = ∩∞

k=0F(Ck(RN,RN)),

F(C∞0 (RN,RN)) = ∩∞

k=0F(Ck0 (RN,RN)),

where (F(Ck(RN,RN)), dk) and (F(Ck0 (RN,RN)), dk) are topological (metric)

groups with the same metrics dk and the following monotony property:

dk(F, G) ≤ dk+1(F, G)

resulting from the monotony of the norms

∥f∥Ck = max0≤i≤k

∥f∥Ci ≤ max0≤i≤k+1

∥f∥Ci = ∥f∥Ck+1 .

Theorem 2.7. The spaces F(B(RN,RN)) and F(C∞0 (RN,RN)) are complete topo-

logical (metric) groups for the distance

d∞(F, G) def=∞∑

k=0

12k

dk(F, G)1 + dk(F, G)

, (2.22)

where dk is the metric associated with Ck(RN,RN) and Ck0 (RN,RN).8

This theorem is a consequence of the following general lemma.

Lemma 2.2. Assume that (Fk, dk), k ≥ 0, is a family of groups each complete withrespect to their metric dk and that

∀k ≥ 0, Fk+1 ⊂ Fk and dk(F, G) ≤ dk+1(F, G), ∀F, G ∈ ∩∞k=0Fk. (2.23)

Then the intersection F∞ = ∩∞k=0Fk is a group that is complete with respect to the

metric d∞ defined in (2.22). If, in addition, each (Fk, dk) is a topological (metric)group, then F∞ is a topological (metric) group.

Proof. By definition, F∞ is clearly a group.(i) d∞ is a metric on F∞. d∞(F, G) = 0 implies that dk(F, G) = 0 for all k

and F = G and conversely. Also d∞(F, G) = d∞(G, F ). The function x/(1 + x) ismonotone strictly increasing with x ≥ 0. Therefore, dk(F, H) ≤ dk(F, G)+dk(G, H)implies

dk(F, H)1 + dk(F, H)

≤ dk(F, G)1 + dk(F, G) + dk(G, H)

+dk(G, H)

1 + dk(F, G) + dk(G, H)

≤ dk(F, G)1 + dk(F, G)

+dk(G, H)

1 + dk(G, H),

∞∑

k=0

12k

dk(F, H)1 + dk(F, H)

≤∞∑

k=0

12k

dk(F, G)1 + dk(F, G)

+∞∑

k=0

12k

dk(G, H)1 + dk(G, H)

and d∞(F, H) ≤ d∞(F, G) + d∞(G, H).8In view of (2.20), it is important to note that F(B(RN,RN)) is a topological group by using

the family (F(Ck(RN,RN)), dk) in Lemma 2.2 and not the family (F(Bk(RN,RN)), dk).


(ii) (F∞.d∞) is complete. Let Fn be a Cauchy sequence in F∞; that is, forall ε, 0 < ε < 1/2, there exists N0 such that for n, m > N0, d∞(Fn, Fm) < ε. Inparticular

d0(Fn, Fm)1 + d0(Fn, Fm)

≤∞∑

k=0

12k

dk(Fn, Fm)1 + dk(Fn, Fm)

< ε ⇒ d0(Fn, Fm) < ε/(1− ε) < 2ε

and Fn is a Cauchy sequence in F0 that converges to some F in F0. Choosea subsequence of Fn such that there exists N1 ≥ N0 such that for n,m > N1,d∞(Fn, Fm) < ε/22. Then

12

d1(Fn, Fm)1 + d1(Fn, Fm)

≤∞∑

k=0

12k

dk(Fn, Fm)1 + dk(Fn, Fm)

<122 ε ⇒ d1(Fn, Fm) < ε/(1− ε) < 2ε

and Fn is a Cauchy sequence in F1 that converges to some G1 in F1. By continu-ous injection of F1 into F0, G1 = F ∈ F0∩F1. Proceeding in that way, we get thatthere exists F ∈ F∞ = ∩∞

k=0Fk such that Fn → F in Fk for all k ≥ 0. Therefore,F∞ is complete.

2.3 Closed Subgroups GGiven an arbitrary nonempty subset Ω0 of RN, consider the family

X (Ω0)def=F (Ω0) ⊂ RN : ∀F ∈ F(Θ)

(2.24)

of images of Ω0 by the elements of F(Θ). By introducing the subgroup

G(Ω0)def= F ∈ F(Θ) : F (Ω0) = Ω0 , (2.25)

we obtain a bijection

[F ] $→ F (Ω0) : F(Θ)/G(Ω0)→ X (Ω0) (2.26)

between the set of images X (Ω0) and the quotient space F(Θ)/G(Ω0). Using thisbijection, the topological structure of X (Ω0) will be identified with the topologicalstructure of the quotient space.

Lemma 2.3. Let Assumptions 2.1 to 2.3 hold with Θ ⊂ C0(RN,RN) and, for allx ∈ RN, let the mapping f $→ f(x) : Θ → RN be continuous.9 If the nonemptysubset Ω0 of RN is closed or if Ω0 = int Ω0,10 the family

G(Ω0)def= F ∈ F(Θ) : F (Ω0) = Ω0 (2.27)

is a closed subgroup of F(Θ).9This means that F(Θ) ⊂ I + Θ ⊂ Hom(RN,RN).

10Such a set will be referred to as a crack-free set in Definition 7.1 (ii) of Chapter 8. Indeed,by definition Ω is crack-free if !Ω = !Ω. If, in addition, Ω is open, then !Ω = !Ω = !Ω andΩ = !!Ω = int Ω.


Proof. G(Ω0) is clearly a subgroup of F(Θ). It is sufficient to show that for anysequence Fn in G(Ω0) that converges to F in F(Θ), then F ∈ G(Ω0). Theother properties are straightforward. If Fn converges to F in F(Θ), then byAssumption 2.3 we have

∥Fn F−1 − I∥+ ∥F F−1n − I∥ → 0.

From ∥F F−1n − I∥ → 0, for each x ∈ Ω0 the sequence F (F−1

n (x)) ⊂ F (Ω0)converges to x ∈ Ω0 and hence Ω0 ⊂ F (Ω0). From the convergence ∥FnF−1−I∥ →0 for each x ∈ F (Ω0), F−1(x) ∈ Ω0 and the sequence Fn(F−1(x)) ⊂ Ω0 convergesto x ∈ F (Ω0) and hence F (Ω0) ⊂ Ω0. Therefore, Ω0 ⊂ F (Ω0) ⊂ Ω0 and Ω0 =F (Ω0). Since F is a homeomorphism, for all A, F (A) = F (A) and F (!A) = !F (A).As a result int F (A) = F (intA). In view of the identity Ω0 = F (Ω0),

F (Ω0) = F (Ω0) = Ω0 and int Ω0 = int F (Ω0) = F (int Ω0).

From this, if Ω0 is closed, Ω0 = Ω0 and F (Ω0) = Ω0. If Ω0 = int Ω0, we also getF (Ω0) = Ω0. In both cases F ∈ G(Ω0).

Remark 2.3.A. M. Micheletti [1] assumes that Ω0 is a bounded connected open domain ofclass C3 in order to make all the images F (Ω0) bounded connected open domainsof class C3 with Θ = C3

0 (RN,RN). Open domains of class Ck, k ≥ 1, are lo-cally Ck epigraphs and domains that are locally C0 epigraphs are crack-free11 byTheorem 5.4 (ii) of Chapter 2.

Remark 2.4.F. Murat and J. Simon [1] were mainly interested in families of images of locallyLipschitzian (epigraph) domains. Recall that F(C1(RN,RN)) transports locallyLipschitzian (epigraph) domains onto locally Lipschitzian (epigraph) domains (cf.A. Bendali [1] and A. Djadane [1]), but that F(C0,1(RN,RN)) does not, asshown in Examples 5.1 and 5.2 of section 5.3 in Chapter 2.

Of special interest are the closed submanifolds of RN of codimension greaterthan or equal to 1. For instance, we can choose Ω0 = Sn, 2 ≤ n < N , the n-sphereof radius 1 centered in 0. We get a family of closed curves in R3 for n = 2 and ofclosed surfaces in R3 for n = 3. Similarly, by choosing Ω0 = Bn

1 (0), the closed unitball of dimension n, we get a family of curves for n = 1 and a family of surfaces forn = 2. The choice of [0, 1]n yields a family of surfaces with corners for n = 2.

Explicit expressions can be obtained for normals, curvatures, and density mea-sures as in Chapter 2.

Example 2.5 (Closed surfaces in R3).For Ω0 = S2 in R3, f ∈ C1(R3,R3), and F = I + f , the tangent field at the pointξ ∈ S2 is TξS2 = ξ⊥ and the unit normal is ξ. At a point x = F (ξ) of the image

11Cf. Definition 7.1 (ii) of Chapter 8.


F (S2), the tangent space and the unit normal are given by

TF (ξ)F (S2) = DF (ξ)ξ⊥ and n(F (x)) =∗DF (ξ)−1ξ

| ∗DF (ξ)−1ξ| .

Example 2.6 (Curves in RN).Let eN be a unit vector in RN, let H = eN⊥, and let

Ω0def= ζNeN : ζN ∈ [0, 1] ⊂ RN .

For f ∈ C1(RN,RN), and F = I + f , the tangent field at the point ξ ∈ Ω0 isTξΩ0 = R eN and the normal space (TξΩ0)⊥ is H. At a point x = F (ξ) of the curveF (Ω0), the tangent and normal spaces are given by

TF (ξ)F (Ω0) = RDF (ξ)eN and TF (ξ)F (Ω0)⊥ = DF (ξ)eN⊥.

In practice, one might want to use other subgroups, such as all the translationsby some vector a ∈ RN:

Fa(x) def= a + x, x ∈ RN, ⇒ Fa − I = a ∈ Θ (2.28)

⇒ F−1a (x)− a + x = F−a(x) ⇒ F−1

a − I = −a ∈ Θ (2.29)

for all Θ that contain the constant functions such as Bk(RN,RN), Ck(RN,RN),and Ck,1(RN,RN), but not Ck

0 (RN,RN). Therefore

Gτdef=Fa : ∀a ∈ RN (2.30)

is a subgroup of F(Θ). In that case, it is also a closed subgroup of F(Θ):

Fa Fb(x) = a + b + x = Fa+b(x), F0 = I

and (Gτ , ) is isomorphic to the one-dimensional closed abelian group (RN,+). Thismeans that all the topologies on Gτ are equivalent and (Gτ , ) is closed in F(Θ).

2.4 Courant Metric on the Quotient Group F(Θ)/GWhen F(Θ) is a topological right-invariant metric group and G is a closed subgroupof F(Θ), the quotient metric

dG(F G, H G) def= infG,G∈G

d(F G, H G) (2.31)

induces a complete metric topology on the quotient group F(Θ)/G.12 The quotientmetric dG(Ω0) with Θ = Ck

0 (RN,RN) was referred to as the Courant metric by12Cf., for instance, D. Montgomery and L. Zippin [1 sect. 1.22, p. 34, sect. 1.23, p. 36])(i) If G is a topological group whose open sets at e have a countable basis, then G is metrizable

and, moreover, there exists a metric which is right-invariant.

(ii) If G is a closed subgroup of a metric group F , then F/G is metrizable.


A. M. Micheletti [1]. In that case F(Θ) is a topological (metric) group. However,we have seen that for Θ = C0,1(RN,RN) and B0(RN,RN), F(Θ) is not a topologicalgroup. Yet, the completeness of (F(Θ), d) and the right-invariance of d are sufficientto recover the result for an arbitrary closed subgroup G of F . It is important to havethe weakest possible assumptions on G to secure the widest range of applications.

Theorem 2.8. Assume that (F , d) is a group with right-invariant metric d thatis complete for the topology generated by d. For any closed subgroup G of F , thefunction dG : F × F → R,

dG(F G, H G) def= infG∈G

d(F, H G), (2.32)

is a right-invariant metric on F(Θ)/G and the space (F/G, dG) is complete. Thetopology induced by dG coincides with the quotient topology of F/G.

Proof. (i) F/G is clearly a group with unit element G under the composition law[F G] [H G] def= [F H] G.

(ii) (dG is a metric). Since d is right-invariant, the definition (2.31) is equiva-lent to


d(F, H G). (2.33)

Since d is symmetrical, so is dG . For the triangle inequality let g1, g2 ∈ G:

dG(F G, H G) ≤ d(F, H g1 g2) ≤ d(F, G g2) + d(G g2, H g1 g2)≤ d(F, G g2) + d(G, H g1)

by right-invariance. The triangle inequality for dG follows by taking the infima overg1, g2 ∈ G. Since F/G is a group, it remains to show that dG(I G, H G) = 0 if andonly if H ∈ G. Clearly dG(IG, IG) ≤ d(I, I) = 0. Conversely, let dG(IG, HG) =0. For all n > 0, there exists gn ∈ G such that d(g−1

n , H) = d(I,H gn) < 1/n.Therefore g−1

n → H in F(Θ). But since G is closed limn→∞ g−1n ∈ G and H ∈ G.

In particular, the canonical mapping F → π(F ) = F G : F(Θ) → F(Θ)/G iscontinuous since dG(I G, F G) ≤ d(I, F ) for all F ∈ F(Θ).

(iii) (Completeness). Consider a Cauchy sequence Fn G in F/G. It issufficient to show that there exists a subsequence which converges to some F G ∈F(Θ)/G. Construct the subsequence Fν G such that

dG(Fν G, Fν+1 G) < 1/(2ν).

Then there exists a sequence Hν in F such that (i) Hν ∈ Fν G and (ii)d(Hν , Hν+1) < 1/(2ν). We proceed by induction. By definition of dG , if dG(F1 G, F2G) < 1/2, there exist H1 ∈ F1G and H2 ∈ F2G such that d(H1, H2) < 1/2.Similarly, if Hν ∈ Fν G, it follows that there exists Hν+1 with the required prop-erty. Because dG(Fν G, Fν+1 G) < 1/2ν , there exist G1 and G2 in G such thatd(Fν G1, Fν+1 G2) < 1/2ν . If G3 is such that Fν G1 = Hν G3, then we can


choose Hν+1 = Fν+1 G2 G−13 since

d(Hν , Fν+1 G2 G−13 ) = d(Hν G3, Fν+1 G2) = d(Fν G1, Fν+1 G2).

It is easy to check that Hν is a Cauchy sequence in F . Indeed

∀i < j, d(Hi, Hj) ≤j−1∑

n=i

d(Hn, Hn+1) ≤12i

+ · · · +12j≤ 1

2i−1 .

By completeness of F , Hν converges to some H in F . From (ii) the canonicalmap π : F → F/G is continuous. So the sequence π(Hν) = Fν G converges toπ(H) = H G in F/G.

(iv) Associate with G the equivalence relation FR∼ H if F H−1 ∈ G. By

definition, the quotient topology on F/R is the finest topology T on F/R forwhich the canonical map F $→ π(F ) : F → F/R is continuous. But, by definition,F/R = F/G and, from part (ii), π is also continuous for the right-invariant quotientmetric. Therefore, the identity map i : (F/R, T ) → (F/G, dG) is continuous. Thetopology T in [I] is defined by the fundamental system of neighborhoods π(F ) :d(F, I) < ε. But dG(π(F ),π(I)) ≤ d(F, I) < ε and i(π(F ) : d(F, I) < ε) ⊂π(F ) : dG([F ], [I]) < ε that is a neighborhood of [I] in the (F/G, dG) topology.Therefore i−1 is continuous and the two topologies coincide.

We now have all the ingredients to conclude.

Theorem 2.9. Let Θ be any of the spaces Bk(RN,RN), Ck(RN,RN), Ck0 (RN,RN),

Ck,1(RN,RN), k ≥ 0, or C∞(RN,RN), C∞0 (RN,RN).

(i) The group (F(Θ), d) is a complete right-invariant metric space. For Θ equalto Ck(RN,RN) or Ck

0 (RN,RN), 0 ≤ k ≤ ∞, (F(Θ), d) is also a topologicalgroup.

(ii) For any closed subgroup G of F(Θ), the function dG : F(Θ)× F(Θ)→ R,


d(F, H G), (2.34)

is a right-invariant metric on F(Θ)/G and the space (F(Θ)/G, dG) is complete.The topology induced by dG coincides with the quotient topology of F(Θ)/G.

(iii) Let Ω0, ∅ = Ω0 ⊂ RN, be closed or let Ω0 = int Ω0. Then

G(Ω0)def= F ∈ F(Θ) : F (Ω0) = Ω0 (2.35)

is a closed subgroup of F(Θ).

Finally, it is interesting to recall that A. M. Micheletti [1] used the quotientmetric to prove the following theorem.


Theorem 2.10. Fix k = 3, the Courant metric of F(C30 (RN,RN))/G(Ω0), and a

bounded open connected domain Ω0 of class C3 in RN. Let X (Ω0) be the family ofimages of Ω0 by elements of F(C3

0 (RN,RN))/G(Ω0).The subset of all bounded open domains Ω in X (Ω0) such that the spectrum

of the Laplace operator −∆ on Ω with homogeneous Dirichlet conditions on theboundary ∂Ω does not have all its eigenvalues simple is of the first category.13

This theorem says that, up to an arbitrarily small perturbation of the domain,the eigenvalues of the Laplace operator of a C3-domain can be made simple.

2.5 Assumptions for Bk(RN, RN), Ck(RN, RN),and Ck

0 (RN, RN)2.5.1 Checking the Assumptions

By definition, for all integers k ≥ 0

Ck0 (RN,RN)) ⊂ Ck(RN,RN) ⊂ Bk(RN,RN) ⊂ B0(RN,RN) ⊂ C0(RN,RN)

∀x ∈ RN, f $→ f(x) : B0(RN,RN)→ RN

with continuous injections between the spaces and continuity of the evaluation map.The spaces are all Banach spaces endowed with the same norm:14

∥f∥Ckdef= max

0≤|α|≤k∥∂αf∥C .

As a consequence, the evaluation map f $→ f(x) is continuous for all the spaces.The following results are quoted directly from A. M. Micheletti [1] and

apply to all spaces Θ ⊂ Ck(RN,RN). Denote by

S(x)[y1][y2], . . . , [yk]

at the point x ∈ RN a k-linear form with arguments y1, . . . , yk.

Lemma 2.4. Assume that F and G are two mappings from RN to RN such that Fis k-times differentiable in an open neighborhood of x and G is k-times differentiablein an open neighborhood of F (x). Then the kth derivative of G F in x is the sumof a finite number of k-linear applications on RN of the form

(h1, h2, . . . , hk) $→ G(ℓ)(F (x))[F (λ1)(x)[h1][h1] . . . [hλ1 ]

]

. . .[F (λℓ)(x)[hk−λℓ+1] . . . [hk]

],

(2.36)

where ℓ = 1, . . . , k and λ1 + · · · + λℓ = k.13A set B is said to be nowhere dense if its closure has no interior or, alternatively, if !B is

dense. A set is said to be of the first category if it is the countable union of nowhere dense sets(cf. J. Dugundji [1, Def. 10.4, p. 250]).

14Here RN is endowed with the norm |x| =∑N

i=1 x2i

1/2.


Proof. We proceed by induction on k. The result is trivially true for k = 1. Assum-ing that it is true for k − 1, we prove that it is also true for k. This is an obviousconsequence of the observation that for a mapping from RN into L((RN)k−1,RN)of the form

x $→ G(ℓ)(F (x))[F (λ1)(x)] . . . [F (λℓ)(x)],

where ℓ = 1, . . . , k − 1 and λ1 + · · · + λℓ = k − 1, the differential in x is

G(ℓ+1)(F (x))[F (1)][F (λ1)(x)] . . . [F (λℓ)(x)]

+ G(ℓ)(F (x))[F (λ1+1)(x)] . . . [F (λℓ)(x)]

+ · · · + G(ℓ)(F (x))[F (λ1)(x)] . . . [F (λℓ+1)(x)], (2.37)

and the result is true for k.

Lemma 2.5. Given g and f in Ck(RN,RN), let ψ = g (I + f). Then for eachx ∈ RN

|ψ(x)| = |g(x + f(x))|,|ψ(1)(x)| ≤ |g(1)(x + f(x))| [1 + |f (1)(x)|],

|ψ(i)(x)| ≤ |g(1)(x + f(x))| |f (i)(x)|

+i∑

j=2

|g(j)(x + f(x))| aj(|f (1)(x)|, . . . , |f (i−1)(x)|)

for i = 2, . . . , k, where aj is a polynomial with positive coefficients.

Proof. This is an obvious consequence of Lemma 2.4.

Theorem 2.11. Assumptions 2.1 and 2.3 are verified for the spaces Ck0 (RN,RN),

Ck(RN,RN), and Bk(RN,RN), k ≥ 0.

Proof. (i) Assumption 2.3. From Lemma 2.5 it readily follows that f (I + g) ∈Bk(RN,RN) for all f ∈ Bk(RN,RN) and I + g in F(Bk(RN,RN)). Moreover,

∥ψ∥C = ∥f∥C ,

∥ψ(1)∥C ≤ ∥f (1)∥C (1 + ∥g(1)∥C),

∥ψ(i)∥C ≤∥f (1)∥C ∥g(i)∥C +i∑

j=2

∥f (j)∥C aj(∥g(1)∥C , . . . , ∥g(i−1)∥C)

for i = 2, . . . , k, where aj is a polynomial. Hence

∥ψ∥Ck = maxi=0,...,k

∥ψi∥C ≤ ∥f∥Ck c1(∥g∥Ck)

for some polynomial function c1 that depends only on k.


(ii) Assumption 2.1. The case of Bk(RN,RN) is a consequence of (i). ForCk(RN,RN) and each i, ψ(i) is uniformly continuous as the finite sum and product ofcompositions of uniformly continuous functions by Lemma 2.4. As for Ck

0 (RN,RN),for all I +g ∈ F(Ck

0 (RN,RN)) and f ∈ Ck0 (RN,RN), h = f (I +g) ∈ Ck(RN,RN)

from the previous case. It remains to show that h ∈ Ck0 (RN,RN). This amounts

to showing that |h(x)| and |h(i)(x)| go to zero as |x| → ∞, i = 1, . . . , k. We againproceed by induction on k. Since |g(x)|→ 0 as |x|→∞, |x+ g(x)| ≥ |x|− |g(x)|→∞ as |x| → ∞ and |h(x)| = |f(x + g(x))| → 0 as |x| → ∞ and Assumption 2.1 istrue for k = 0. Always, from Lemma 2.5 and the identity h(x) = f(x + g(x)),

|h(1)(x)| ≤ |f (1)(x)| [1 + |g(1)(x)|]→ 0

as |x| → 0 and Assumption 2.1 is true for k = 1. For k ≥ 2 and 2 ≤ i ≤ k, again,from Lemma 2.5

|h(i)(x)| ≤ |f (1)(x + g(x))| ∥g(i)∥C

+i∑

j=2

|f (j)(x + g(x))| aj(∥g(1)∥C , . . . , ∥g(i−1)∥C)

for i = 2, . . . , k. Since for each j, |f (j)(x)|→ 0 as |x|→∞, by the same argument,|f (j)(x + g(x))|→ 0 as |x|→∞ and hence the whole right-hand side goes to 0.

Theorem 2.12. Assumption 2.2 is verified for the spaces Ck0 (RN,RN), Ck(RN,RN),

and Bk(RN,RN), k ≥ 0.

It is a consequence of the following lemma of A. M. Micheletti [1].

Lemma 2.6. Given integers r ≥ 0 and s > 0 there exists a constant c(r, s) > 0with the following property: if the sequence f1, . . . , fn in Cr(RN,RN) is such that

n∑

i=1

∥fi∥Cr < α, 0 < α < s, (2.38)

then for the map F = (I + fn) · · · (I + f1),

∥F − I∥Cr ≤ α c(r, s). (2.39)

Proof. We again proceed by induction on r. For simplicity define

Fidef= (I + fi) · · · (I + f1), Fn = F.

By the definition of F we have

F − I = f1 + f2 (I + f1) + · · · + fn (I + fn−1) · · · (I + f1)= f1 + f2 F1 + · · · + fn Fn−1.

(2.40)


Then ∥F − I∥C0 ≤∑n

i=1 ∥fi∥C0 ≤ α and c(0, s) = 1 for all integers s > 0. From(2.40) and Lemma 2.5 we further get

supx

|(F − I)(1)(x)| ≤ ∥f1∥C1 + ∥f2∥C1 [1 + ∥f1∥C1 ]

+ · · · + ∥fn∥C1 [1 + ∥fn−1∥C1 ] . . . [1 + ∥f1∥C1 ]

≤n∑

i=1

∥fi∥C1

n∏

i=1

(1 + ∥fi∥C1) ≤(

n∑

i=1

∥fi∥C1

)e(∑n

i=1∥fi∥C1) ≤ α eα ≤ α es

(2.41)

and ∥F − I∥C1 ≤ α eα. Hence c(1, s) = es for all integers s > 0. We now show thatif the result is true for r − 1, it is true for r. We have to evaluate |(Fn − I)(r)(x)|.It is obvious that Fn − I = (I + fn) Fn−1 − I = Fn−1 − I + fn Fn−1 and hence

(Fn − I)(r)(x) = (Fn−1 − I)(r)(x) + (fn Fn−1)(r)(x). (2.42)

For r ≥ 2 from Lemma 2.5 we have

|(fn Fn−1)(r)(x)| ≤ |f (1)n (Fn−1(x))| |(Fn−1 − I)(r)(x)|

+r∑

j=2

|f (j)n (Fn−1(x))| aj(|(Fn−1 − I)(1)(x)|, . . . , |(Fn−1 − I)(r−1)(x)|).

By the induction assumption |(Fn−1 − I)(i)(x)| ≤ ∥(Fn−1 − I)∥Cr−1 ≤ α c(r− 1, s),i = 1, . . . , r−1, and the fact that aj is a polynomial dependent on r for j = 2, . . . , r,there exists a constant L(r, s) such that

|(fn Fn−1)(r)(x)|≤ |f (1)

n (Fn−1(x))| |(Fn−1 − I)(r)(x)| + L(r, s)r∑

j=2

|f (j)n (Fn−1(x))|

≤ ∥fn∥Cr |(Fn−1 − I)(r)(x)| + (r − 1)L(r, s)∥fn∥Cr .

(2.43)

Define M(r, s) = (r − 1)L(r, s). From (2.42) and (2.43), we get

|(Fn − I)(r)(x)| ≤ [1 + ∥fn∥Cr ] |(Fn−1 − I)(r)(x)| + M(r, s)∥fn∥Cr .

After repeating this procedure n− 1 times we have

|(Fn − I)(r)(x)| ≤ [1 + ∥fn∥Cr ] . . . [1 + ∥f2∥Cr ]∥f1∥Cr

+ [1 + ∥fn∥Cr ] . . . [1 + ∥f2∥Cr ]M(r, s)∥f2∥Cr + · · · + M(r, s)∥fn∥Cr

≤ maxM(r, s), 1 eαα.

Henceforth c(r, s) = max M(r, s), 1es.

Theorem 2.13. Assumption 2.4 is verified for Ck(RN,RN) and Ck0 (RN,RN).

It is a consequence of the following slightly modified versions of lemmasof A. M. Micheletti [1].


Lemma 2.7. Given f ∈ Ck(RN,RN) and γ ∈ C0(RN,RN), for all ε > 0, thereexists δ > 0 such that

∀γ, ∥γ∥C0 < δ, sup0≤i≤k

∥f (i) (I + γ)− f (i)∥C0 < ε. (2.44)

Proof. Since f and its derivatives f (i) all belong to C0(RN,RN), they are uniformlycontinuous functions. Hence, for each ε > 0 there exists δ > 0 such that

∀x, y ∈ RN, |x− y| < δ, sup0≤i≤k

|f (i)(y)− f (i)(x)| < ε.

In particular for γ ∈ C0(RN,RN) such that ∥γ∥C0 < δ

|x + γ(x)− x| = |γ(x)| ≤ ∥γ∥C0 < δ ⇒ sup0≤i≤k

|f (i)(x + γ(x))− f (i)(x)| < ε

and we get the result.

Lemma 2.8. For each g ∈ Ck(RN,RN), for all ε > 0, there exists δ > 0 such that

∀γ ∈ Ck(RN,RN) such that ∥γ∥Ck < δ, ∥g (I + γ)− g∥Ck < ε. (2.45)

Proof. Again proceed by induction on k. For k = 0 the result is a consequence ofLemma 2.7. Then check that if it is true for k− 1 it is true for k. From Lemma 2.4the kth derivative of g(I+γ)−g is the sum of a finite number of k-linear mappingsof the form

g(r)(x + γ(x))[(I + γ)(λ1)(x)] . . . [(I + γ)(λr)(x)]− g(r)(x)[I(λ1)(x)] . . . [I(λr)(x)]

=

g(r)(x + γ(x))− g(r)(x)

[(I + γ)(λ1)(x)] . . . [(I + γ)(λr)(x)]

+ g(r)(x)[(I + γ)(λ1)(x)] . . . [(I + γ)(λr)(x)]− g(r)(x)[I(λ1)(x)] . . . [I(λr)(x)],

where r = 1, . . . , k and λ1 + · · · + λr = k. Since

|(I + γ)(λ1)(x)| |(I + γ)(λ2)(x)| . . . |(I + γ)(λr)(x)| ≤ (∥γ∥Ck + 1)r,

the norm of the mapping can be bounded above by the following expression:

supx

|g(r)(x + γ(x))− g(r)(x)| (∥γ∥Ck + 1)r + ∥γ∥Ck p(∥γ∥Ck , ∥g∥Ck), (2.46)

where p is a polynomial. From this upper bound and Lemma 2.7 the result of thelemma is true for k.

2.5.2 Perturbations of the Identity and Tangent Space

By construction

F(Θ) ⊂ I + Θ


and the tangent space in each point of the affine space I + Θ is Θ. In general,I + Θ is not a subset of F(Θ), but, for some spaces of transformations Θ, there isa sufficiently small ball B around 0 in Θ such that the so-called perturbations ofthe identity , I + B, are contained in F(Θ). In particular, for θ ∈ B, the familyof transformations Tt = I + tθ ∈ F(Θ), 0 ≤ t ≤ 1, is a C1-path in F(Θ) and thetangent space to each point of F(Θ) can be shown to be exactly Θ.

Theorem 2.14. Let Θ be equal to Ck0 (RN,RN), Ck(RN,RN), or Bk(RN,RN),

k ≥ 1.

(i) The map f $→ I + f : B(0, 1) ⊂ Θ→ F(Θ) is continuous.

(ii) For all F ∈ F(Θ), the tangent space TF F(Θ) to F(Θ) is Θ.

Remark 2.5.The generic framework of Micheletti is similar to an infinite-dimensional Riemann-ian manifold.

Proof. (i) For k ≥ 1 and f sufficiently small, F is bijective and has a unique inverse.Given y ∈ RN, consider the mapping

S(x) def= y − f(x), x ∈ RN .

Then for any x1 and x2,

S(x2)− S(x1) = −[f(x2)− f(x1)]

⇒ |S(x2)− S(x1)| ≤ |f(x2)− f(x1)| ≤ ∥f (1)∥C0 |x2 − x1|.

For ∥f (1)∥C1 < 1, S is a contraction and, for each y, there exists a unique x suchthat y − f(x) = S(x) = x, [I + f ](x) = y. Therefore I + f is bijective.

But in order to show that F ∈ F(Bk(RN,RN)) we also need to prove that[I + f ]−1 ∈ Bk(RN,RN)). The Jacobian matrix of F (x) is equal to I + f (1)(x) and

∀x ∈ RN, |F (1)(x)− I| ≤ ∥F (1) − I∥C0 = ∥f (1)∥C0 .

If there exists x ∈ RN such that F (1)(x) is not invertible, then for some 0 = y ∈ RN,F (1)(x)y = 0 and

|y| = |F (1)(x)y − y| ≤ |F (1)(x)− I||y| ≤ ∥f (1)∥C0 |y| ⇒ 1 ≤ ∥f (1)∥C0 < 1,

which is a contradiction. Therefore, for all x ∈ RN, F (1)(x) is invertible. As a resultthe conditions of the implicit function theorem are met and from L. Schwartz[3, Vol. 1, Thm. 29, p. 294, Thm. 31, p. 299] we have that F−1 = [I + f ]−1 ∈Ck(RN,RN).

It remains to prove that F−1 = [I+f ]−1 ∈ Bk(RN,RN) and the continuity. Weagain proceed by induction on k. Set y = F (x). By definition g(y) = F−1(y)− y =x − F (x) = −f(x) and ∥g∥C0 = ∥f∥C0 . Always, from Lemma 2.5 and the identityg(y) = −f(x) = −f(F−1(y)) = −f(y + g(y)),

|g(1)(y)| ≤ |f (1)(x)| [1 + |g(1)(y)|] ⇒ |g(1)(y)| [1− |f (1)(x)|] ≤ |f (1)(x)|.


Since |f (1)(x)| ≤ ∥f∥C0 < 1

0 ≤ |g(1)(y)| ≤ |f (1)(x)|1− ∥f∥C0

⇒ ∥g(1)∥C0 ≤ |f (1)(x)|1− ∥f∥C0

≤ ∥f (1)∥C0

1− ∥f∥C0

. (2.47)

For k ≥ 2, again from Lemma 2.5, we have

0 ≤ |g(k)(y)| ≤∑k

j=2 |f (j)(x)| aj(|g(1)(y)|, . . . , |g(k−1)(y)|)1− |f (1)(x)|

≤∑k

j=2 |f (j)(x)| aj(|g(1)(y)|, . . . , |g(k−1)(y)|)1− ∥f (1)∥C0

.

(2.48)

So there exists a constant ck such that

∥F−1 − I∥Ck ≤ ck ∥F − I∥Ck = ck ∥f∥Ck .

Therefore, for k ≥ 1,

∀f ∈ Bk(RN,RN) such that ∥f (1)∥C0 < 1, F = I + f ∈ F(Bk(RN,RN)).

Note that the condition is on ∥f (1)∥C0 and not ∥f∥Ck , but by definition ∥f (1)∥C0 ≤∥f∥C1 ≤ ∥f∥Ck and the condition ∥f∥Ck < 1 yields the same result. In particular,the map f $→ I + f : B(0, 1) ⊂ Bk(RN,RN)→ F (Bk(RN,RN)) is continuous. Allthis can now be specialized to the spaces Ck

0 (RN,RN) and Ck(RN,RN) by usingthe identity (I + f)−1 − I = −f (I + f)−1.

(ii) Since F(Θ) is contained in the affine space I + Θ, any tangent elementto F(Θ) will be contained in Θ. From part (i) for k ≥ 1 and for all θ ∈ Θ thereexists τ > 0 such that for all t, 0 ≤ t ≤ τ , I + tΘ ∈ F(Θ). For any F ∈ F(Θ),t $→ (I + tθ) F : [0, τ ]→ F(Θ) is a continuous path in F(Θ) and the limit of

[(I + tθ) F ] F−1 − I

t=

(I + tθ)− I

t= θ

will be an element of the tangent space TF F(Θ) in F to F(Θ). Therefore for allF ∈ F(Θ), TF F(Θ) = Θ.

2.6 Assumptions for Ck,1(RN, RN) and Ck,10 (RN, RN)

2.6.1 Checking the Assumptions

By definition, for all integers k ≥ 0

Ck,1(RN,RN) def=

f ∈ Ck(RN,RN) :

∀α, 0 ≤ |α| ≤ k, ∃c > 0, ∀x, y ∈ Ω|∂αf(y)− ∂αf(x)| ≤ c |x− y|


endowed with the same norm:15

∥f∥Ck,1def= max

max

0≤|α|≤k∥∂αf∥C , ck(f)

= max ∥f∥Ck , ck(f) , (2.49)

c(f) def= supy =x

|f(y)− f(x)||y − x| and ∀k ≥ 1, ck(f) def=

∑

|α|=k

c(∂αf), (2.50)

and the convention c0(f) = c(f). Ck,10 (RN,RN) is the space of functions f ∈

Ck,1(RN,RN) such that f and all its derivatives go to zero at infinity. They are allBanach spaces. By definition

Ck,10 (RN,RN) ⊂ Ck,1(RN,RN) ⊂ B0(RN,RN) ⊂ C0(RN,RN),

∀x ∈ RN, f $→ f(x) : B0(RN,RN)→ RN

with continuous injections between the spaces and continuity of the evaluation map.As a consequence, the evaluation map f $→ f(x) is continuous for all the spaces.We deal only with the case Ck,1(RN,RN). The case Ck,1

0 (RN,RN) is obtained fromthe case Ck,1(RN,RN) as in the previous section.

Theorem 2.15. Assumptions 2.1 and 2.3 are verified for the spaces Ck,10 (RN,RN)

and Ck,1(RN,RN), k ≥ 0.

Proof. The composition and multiplication of bounded and Lipschitz functions isbounded and Lipschitz. For f ∈ C0,1(RN,RN) and I + g ∈ F(C0,1(RN,RN)),∥f (I + g)∥C0 = ∥f∥C0 < ∞. Since g is Lipschitz with Lipschitz constant c(g),I + g is also Lipschitz with Lipschitz constant 1 + c(g) and the composition is alsoLipschitz with constant c(f (I + g)) ≤ c(f) (1 + c(g)). Hence

∥f (I + g)∥C0,1 ≤ ∥f∥C0,1 (1 + c(g)) ≤ ∥f∥C0,1 (1 + ∥g∥C0,1)

and the continuous function c1(r) = 1 + r satisfies Assumption 2.3. For k = 1,D(f (I + g)) = Df (I + g) (I + Dg) and

∥D(f (I + g))∥C ≤ ∥Df∥C (1 + ∥Dg∥C),c(D(f (I + g))) ≤ c(Df) (1 + c(g)) (1 + ∥Dg∥C) + ∥Df∥C (1 + c(Dg))⇒ ∥D(f (I + g))∥C0,1 ≤ ∥Df∥C0,1 (1 + ∥Dg∥C0,1) (1 + c(g))

≤ ∥Df∥C0,1 (1 + ∥Dg∥C0,1) (1 + ∥g∥C0,1)⇒ ∥f (I + g)∥C1,1 ≤ ∥f∥C1,1 (1 + ∥Dg∥C0,1) (1 + ∥g∥C0,1)

≤ ∥f∥C1,1 (1 + ∥g∥C1,1)2.

The general case is obtained by induction over k.

Theorem 2.16. Assumption 2.2 is verified for Ck,10 (RN,RN) and Ck,1(RN,RN),

k ≥ 0.

15Here RN is endowed with the norm |x| =∑N

i=1 x2i

1/2.


Proof. Consider a sequence fini=1 ⊂ C0,1(RN,RN) such that

n∑

i=1

max∥fi∥C0 , c(fi) < α < r. (2.51)

From Theorem 2.12

∥(I + f1) · · · (I + fn)− I∥C0 ≤n∑

i=1

∥fi∥C0 < α.

In addition,

(I + f1) · · · (I + fn)− I = fn +n−1∑

i=1

fi (I + fi+1) · · · (I + fn)

⇒ c((I + f1) · · · (I + fn)− I) ≤ c(fn) +n−1∑

i=1

c(fi) (1 + c(fi+1)) . . . (1 + c(f1))

≤ c(fn) +n−1∑

i=1

c(fi) e∑i+1

k=1c(fk)

≤[

n∑

i=1

c(fi)

]e[∑n

i=1c(fi)] < α eα

⇒ ∥(I + f1) · · · (I + fn)− I∥C0,1 < α maxeα, 1 < α er

and we can choose c0(r) = er for k = 0. We use the same technique for k ≥ 1 butwith heavier computations.

2.6.2 Perturbations of the Identity and Tangent Space

By construction

F(Θ) ⊂ I + Θ

and the tangent space in each point of the affine space I + Θ is Θ. In general,I + Θ is not a subset of F(Θ), but, for some spaces of transformations Θ, there isa sufficiently small ball B around 0 in Θ such that the so-called perturbations ofthe identity , I + B, are contained in F(Θ). In particular, for θ ∈ B, the familyof transformations Tt = I + tθ ∈ F(Θ), 0 ≤ t ≤ 1, is a C1-path in F(Θ) and thetangent space to each point of F(Θ) can be shown to be exactly Θ.

Theorem 2.17. Let Θ be equal to Ck,10 (RN,RN) or Ck,1(RN,RN), k ≥ 0.

(i) The map f $→ I + f : B(0, 1) ⊂ Θ→ F(Θ) is continuous.

(ii) For all F ∈ F(Θ), the tangent space TF F(Θ) to F(Θ) is Θ.

Remark 2.6.The generic framework of Micheletti is similar to an infinite-dimensional Riemann-ian manifold.


Proof. (i) Consider perturbations of the identity of the form F = I + f , f ∈C0,1(RN,RN). Consider for any y ∈ RN the map

x $→ S(x) def= y − f(x) : RN → RN .

For all x1 and x2

S(x2)− S(x1) = −[f(x2)− f(x1)],|S(x2)− S(x1)| = |f(x2)− f(x1)| ≤ c(f) |x2 − x1|,

and f is contracting for c(f) < 1. Hence for each y ∈ RN there exists a uniquex ∈ RN such that

S(x) = x ⇒ y − f(x) = x ⇒ F (x) = y

and F is bijective. It remains to prove that gdef= F−1 − I ∈ C0,1(RN,RN) to

conclude that I + f ∈ F(C0,1(RN,RN)).For any y ∈ RN and x = F−1(y)

F−1(y)− y = x− F (x),

∥g∥C0 = ∥F−1 − I∥C0 = supy∈RN

|F−1(y)− y| = supy∈RN

|(I − F )(F−1(y))|

= supx∈RN

|(I − F )(x)| = ∥F − I∥C0 <∞

since F : RN → RN is bijective and ∥g∥C0 = ∥f∥C0 .For all y1 and y2 and xi = F−1(yi),

|g(y2)− g(y1)| = |F−1(y2)− y2 − (F−1(y1)− y1)|≤ |x2 − F (x2)− (x1 − F (x1)| = |f(x2)− f(x1)|≤ c(f)|x2 − x1|≤ c(f)|F−1(y2)− F−1(y1)| = c(f)|g(y2)− g(y1) + (y2 − y1)|

⇒ |g(y2)− g(y1)| ≤c(f)

1− c(f)|y2 − y1| ⇒ c(g) ≤ c(f)

1− c(f)<∞.

Therefore

∀f ∈ C0,1(RN,RN) such that c(f) < 1, F = I + f ∈ F(C0,1(RN,RN)).

Moreover since c(f) ≤ ∥f∥C0,1 , the condition ∥f∥C0,1 < 1 yields c(f) < 1 and I+f ∈F(C0,1(RN,RN)). Furthermore, the map f $→ I + f : B(0, 1) ⊂ C0,1(RN,RN) →F(C0,1(RN,RN)) is continuous. We use the same technique for k ≥ 1 plus thearguments of Theorem 2.14 (i).

(ii) Same proof as Theorem 2.14 (ii).

3. Generalization to All Homeomorphisms and Ck-Diffeomorphisms 153

3 Generalization to All Homeomorphismsand Ck-Diffeomorphisms

With a variation of the generic constructions associated with the Banach space Θ ofsection 2, it is possible to construct a metric on the whole space of homeomorphismsHom(RN,RN) or on the whole space Diffk(RN,RN) of Ck-diffeomorphisms of RN,k ≥ 1, or of an open subset D of RN.

First, recall that, for an open subset D of RN, the topology of the vector spaceCk(D,RN) can be specified by the family of seminorms

∀K compact ⊂ D, nK(F ) def= ∥F∥Ck(K). (3.1)

This topology is equivalent to the one generated by the monotone increasing sub-family Kii≥1 of compact sets

Kidef=

x ∈ D : d!D(x) ≥ 1i

and |x| ≤ i

, i ≥ 1, (3.2)

where d!D(x) = infy∈!D |y − x|. This leads to the construction of the metric

δk(F, G) def=∞∑

i=1

12i

nKi(F −G)1 + nKi(F −G) (3.3)

that generates the same topology as the initial family of seminorms and makesCk(D,RN) a Frechet space with metric δk.

Given an open subset D of RN, consider the group

Homk(D)def=F ∈ Ck(D,RN) : F : D → D is bijective and F−1 ∈ Ck(D,RN)

(3.4)

of transformations of D. For k = 0, Hom0(D) = Hom(D) and for k ≥ 1, Homk(D) =Diffk(D), where the condition F−1 ∈ Ck(D,RN) is redundant. One possible choiceof a topology on Homk(D) is the topology induced by Ck(D,RN). This is theso-called weak topology on Homk(D) (cf., for instance, M. Hirsch [1, Chap. 2]).However, Homk(D) is not necessarily complete for that topology.

In order to get completeness, we adapt the constructions of A. M. Miche-letti [1]. Associate with F ∈ Homk(D) and each compact subset K ⊂ D thefollowing function of I and F :

qK(I, F ) def= infF=F1···FnFℓ∈Homk(D)

n∑

ℓ=1

nK(I, Fℓ) + nK(I, F−1ℓ ), (3.5)

where the infimum is taken over all finite factorizations F = F1 · · · Fn, Fi ∈Homk(D), of F in Homk(D). By construction, qK(I, F−1) = qK(I, F ). Extend thisdefinition to all pairs F and G in Homk(D):

qK(F, G) def= qK(I,G F−1). (3.6)


The function qK is a pseudometric16 and the family qK : K compact ⊂ D definesa topology on Homk(D). This topology is metrizable for the metric

dk(F, G) def=∞∑

i=1

12i

qKi(F, G)1 + qKi(F, G)

, (3.7)

constructed from the monotone increasing subfamily Kii≥1 of compact sets de-fined in (3.2). By definition, dk(F, G) = dk(G, F ) and dk is symmetrical. It isright-invariant since for all F , G, and H in Homk(D))

∀F, G, and H ∈ Homk(D), dk(F, G) = dk(F H, G H),

∀F ∈ Homk(D), dk(I, F ) = dk(I, F−1).

Theorem 3.1. Homk(D), k ≥ 0, is a group under composition, dk is a right-invariant metric on Homk(D), and (Homk(D), dk) is a complete metric space.

We need Lemmas 2.4 and 2.5 and the analogues of Lemma 2.6 and Theo-rem 2.11 for the spaces Ck(K,RN), K a compact subset of D, whose elements arebounded and uniformly continuous.

Lemma 3.1. Given integers r ≥ 0 and s > 0 there exists a constant c(r, s) > 0with the following property: for any compact K ⊂ D and a sequence F1, . . . , Fn inCr(K,RN) is such that

n∑

ℓ=1

∥Fℓ − I∥Cr(K) < α, 0 < α < s; (3.8)

then for the map F = Fn · · · F1,

∥F − I∥Cr(K) ≤ α c(r, s). (3.9)

Theorem 3.2. Given k ≥ 0, there exists a continuous function c1 such that for allF, G ∈ Homk(D) and any compact K ⊂ D

∥F G∥Ck(K) ≤ ∥F∥Ck(K) c1(∥G∥Ck(K)). (3.10)

16A map d : X × X → R on a set X is called a pseudometric (or ecart, or gauge) whenever

(i) d(x, y) ≥ 0, for all x and y,

(ii) x = y ⇒ d(x, y) = 0,

(iii) d(x, y) = d(y, x),

(iv) d(x, z) ≤ d(x, y) + d(y, z)

(J. Dugundji [1, p. 198]). This notion is analogous to the one of seminorm for topological vectorspaces, but for a group the terminology semimetric (cf. footnote 7) is used for a metric withoutthe triangle inequality.


Proof. From Lemma 2.5 it readily follows that ψ = F G ∈ Homk(D). Moreover,

∥ψ∥C0(K) = ∥F∥C0(K),

∥ψ(1)∥C0(K) ≤ ∥F (1)∥C0(K) ∥G(1)∥C0(K),

∥ψ(i)∥C0(K) ≤∥F (1)∥C0(K) ∥G(i)∥C0(K)

+i∑

j=2

∥F (j)∥C0(K) aj(∥G(1)∥C0(K), . . . , ∥G(i−1)∥C0(K))

for i = 2, . . . , k, where aj is a polynomial. Hence

∥ψ∥Ck(K) = maxi=0,...,k

∥ψi∥C(K) ≤ ∥F∥Ck(K) c1(∥G∥Ck(K))

for some polynomial function c1 that depends only on k.

Proof of Theorem 3.1. (i) Homk(D) is a group. The composition of Ck-diffeomorphisms is again a Ck-diffeomorphism. By definition, each element has aninverse and the neutral element is I.

(ii) dk is a metric. dk(F, G) ≥ 0, dk(F, G) = dk(G, F ). For the triangleinequality F H−1 = (F G−1) (G H−1) is a finite factorization of F H−1.Consider arbitrary finite factorizations of F G−1 and G H−1:

F G−1 = H1 · · · Hm, Hi ∈ Ck(D, D),

G H−1 = L1 · · · Ln, Lj ∈ Ck(D, D).

This yields a new finite factorization of F H−1. By definition of qKi ,

qKi(I, F H−1) ≤m∑

ℓ=1

nKi(I, Hℓ) + nKi(I,H−1ℓ ) +

n∑

ℓ′=1

nKi(I, Lℓ′) + nKi(I, L−1ℓ′ )

and by taking infima over each factorization,

qKi(F, H) = qKi(I, F H−1) ≤qKi(I, F G−1) + qKi(I,G H−1)= qKi(F, G) + qKi(G, H).

By definition of dk, since Ki is a monotonically increasing sequence of compactsets and qKi is a monotonically increasing sequence of pseudometrics, we thenget the triangle inequality

dk(F, H) ≤ dk(F, G) + dk(G, H).

Since Homk(D) is a group and dk is right-invariant, it remains to show thatdk(I, F ) = 0 if and only if F = I. Clearly if F = I, qKi(I, F ) = 0 since the qKi ’sare pseudometrics and dk(I, F ) = 0. Conversely, if

dk(I, F ) def=∞∑

i=1

12i

qKi(I, F )1 + qKi(I, F )

= 0,


then for all Ki, qKi(I, F ) = 0. For each Ki and all ε, 0 < ε < 1, there exists a finitefactorization F = F1 · · · Fn such that

n∑

ℓ=1

nKi(Fℓ − I) + nKi(F−1ℓ − I) < ε < 1

⇒ nKi(F − I) ≤∑

ℓ

nKi(Fℓ · · · Fn − Fℓ+1 · · · Fn).

From Lemma 3.1

∀ε > 0, nKi(F − I) < ε c(k, 1)

and (F − I)|Ki = 0. Since this is true for all i, F − I = 0 on D and dk is a metric.(iii) Completeness. Let Fn be a Cauchy sequence in Homk(D).(a) Boundedness of Fn and F−1

n in Ck(D,RN). For all ε, 0 < ε < 1, thereexists N such that, for all m, n > N , dk(Fm, Fn) < ε. By the triangle inequality

|dk(I, Fm)− dk(I, Fn)| ≤ dk(Fm, Fn) < ε.

For all i, qKi(I, Fn is Cauchy and, a posteriori, bounded by some constant L. Foreach n, there exists a factorization Fn = F 1

n · · · F ν(n)n such that

0 ≤ν(n)∑

i=1

∥F in − I∥Ck(Ki) + ∥(F i

n)−1 − I∥Ck(Ki) − dk(I, Fn) < ε < 1

⇒ν(n)∑

i=1

∥F in − I∥Ck(Ki) + ∥(F i

n)−1 − I∥Ck(Ki) < L + ε < L + 1.

By Lemma 3.1

∥Fn − I∥Ck(Ki) + ∥F−1n − I∥Ck(Ki) < 2(L + ε) c(k, L + 1)

and the sequences Fn − I and F−1n − I and, a fortiori, Fn and F−1

n arebounded in Ck(Ki,RN).

(b) Convergence of Fn and F−1n in Ck(D,RN). For any m, n

Fn − Fm = (Fn F−1m − I) Fm

and by Theorem 3.2

∥Fm − Fn∥Ck(Ki) = ∥(Fn F−1m − I) Fm∥Ck(Ki)

≤ ∥Fn F−1m − I∥Ck(Ki) c1(∥Fm∥Ck(Ki))

⇒ ∥Fm − Fn∥Ck(Ki) ≤ c+1 ∥Fn F−1

m − I∥Ck(Ki),

c+1

def= supm

c1(∥Fm∥Ck(Ki)) <∞


since Fm is bounded in Ck(Ki,RN). For all ε, 0 < ε < c(k, 1) c+1 (where c(k, 1)

is the constant of Lemma 3.1),

∃N, ∀n, m > N, dk(I, Fn F−1m ) = dk(Fm, Fn) < ε/(c(k, 1) c+

1 ) < 1.

By Lemma 3.1

∥Fn F−1m − I∥Ck(Ki) < c(k, 1)ε/(c(k, 1) c+

1 ) = ε/c+1

⇒ ∀n, m, > N, ∥Fm − Fn)∥Ck(Ki) ≤ c+1 ∥Fn F−1

m − I∥Ck(Ki) < ε

and, since this is true for all Ki, Fn is a Cauchy sequence in Ck(D,RN) thatconverges to some F ∈ Ck(D,RN) since Ck(D,RN) is a Frechet space for thefamily of seminorms qKi. Similarly, we get that F−1

n is a Cauchy sequence inCk(D,RN) that converges to some G ∈ Ck(D,RN).

(c) F is bijective, G = F−1, and F ∈ Homk(D). For all n, Fn, F−1n , F

and G are continuous since Ck(D,RN) ⊂ C0(D,RN). By continuity of F $→F (x) : Ck(D,RN) → RN, F−1

n (x) → G(x). By the same argument for Fn,x = Fn(F−1

n (x))→ F (G(x)) = (F G)(x) and for all x ∈ RN, (F G)(x) = x. Sim-ilarly, starting from F−1

n Fn = I, we get (GF )(x) = x. Hence, GF = I = F G.So F is bijective, F, F−1 ∈ Ck(D,RN), and F ∈ Homk(D).

(d) Convergence of Fn → F in Homk(D). To check that Fn → F in Homk(D)recall that by Theorem 3.2 for each Ki

qKi(Fn, F ) = qKi(I, F F−1n )

≤ ∥I − F F−1n ∥Ck(Ki) + ∥I − Fn F−1∥Ck(Ki)

= ∥(Fn − F ) F−1n ∥Ck(Ki) + ∥(F − Fn) F−1∥Ck(Ki)

≤ c1(∥F−1n ∥Ck(Ki))∥Fn − F∥Ck(Ki) + c1(∥F−1∥Ck(Ki))∥F − Fn∥Ck(Ki)

≤(max

nc1(∥F−1

n ∥Ck(Ki)) + c1(∥F−1∥Ck(Ki)))∥F − Fn∥Ck(Ki)

that goes to zero as Fn goes to F in Ck(D,RN). This completes the proof.

Remark 3.1.In section 2 it was possible to consider in F(Θ) a subgroup G of translations since

Fa(x) = a + x ⇒ (Fa − I)(x) = a ∈ Θ

for spaces Θ that contain constants. Yet, since all of them were spaces of boundedfunctions in RN, it was not possible to include rotations or flips. But this canbe done now in Homk(RN), and we can quotient out not only by the subgroup oftranslations but also by the subgroups of isometries or rotation that are importantin image processing.

Chapter 4

TransformationsGenerated by Velocities

1 IntroductionIn Chapter 3 we have constructed quotient groups of transformations F(Θ)/G andtheir associated complete Courant metrics. Such spaces are neither linear nor con-vex. In this chapter, we specialize the results of Chapter 3 to spaces of transforma-tions that are generated at time t = 1 by the flow of a velocity field over a generictime interval [0, 1] with values in the tangent space Θ. The main motivation is tointroduce a notion of semiderivatives in the direction θ ∈ Θ on such groups as wellas a tractable criterion for continuity via C1 or continuous paths in the quotientgroup endowed with the Courant metric. This point of view was adopted by J.-P. Zolesio [2, 7] as early as 1973 and considerably expanded in his these d’etat in1979. One of his motivations was to solve a shape differential equation of the typeA V (t) + G(Ωt(V )) = 0, t > 0, where G is the shape gradient of a functional and Aa duality operator.1 At that time most people were using a simple perturbation ofthe identity to compute shape derivatives. The first comprehensive book systemat-ically promoting the velocity method was published in 1992 by J. Soko!lowski andJ.-P. Zolesio [9]. Structural theorems for the Eulerian shape derivative of smoothdomains were first given in 1979 by J.-P. Zolesio [7] and generalized to nonsmoothdomains in 1992 by M. C. Delfour and J.-P. Zolesio [14]. The velocity pointof view was also adopted in 1994 by R. Azencott [1], in 1995 by A. Trouve [2],and in 1998 by A. Trouve [3] and L. Younes [2] to construct complete metricsand geodesic paths in spaces of diffeomorphisms generated by a velocity field witha broad spectrum of applications to imaging.2 The reader is referred to the forth-coming book of L. Younes [6] for a comprehensive exposition of this work and

1Cf. J.-P. Zolesio [7] and the recent book by M. Moubachir and J.-P. Zolesio [1].2The first author would like to thank Robert Azencott, who pointed out his work and the

contributions of his team during a visit in Houston early in 2010. Special thanks also to AlainTrouve for an enlightening discussion shortly after in Paris.

159

160 Chapter 4. Transformations Generated by Velocities

to related papers such as the ones of P. W. Michor and D. Mumford [1, 2, 3]and L. Younes, P. W. Michor, I. Shah, and D. Mumford [1].

In view of the above motivations, this chapter begins with section 2, whichspecializes the results of Chapter 3 to transformations generated by velocity fields. Italso explores the connections between the constructions of Azencott and Michelettithat implicitly use a notion of geodesic path with discontinuities.

Section 3 motivates and adapts the definitions of Gateaux and Hadamardsemiderivatives in topological vector spaces (cf. Chapter 9) to shape functionalsdefined on shape spaces. The analogue of the Gateaux semiderivative for sets isobtained by the method of perturbation of the identity operator, while the analogueof the Hadamard semiderivative comes from the velocity (speed) method. The firstnotion does not extend to submanifolds and does not incorporate the chain rule forthe semiderivative of the composition of functions (for instance, to get the semi-derivative with respect to the parameters of an a priori parametrized geometry),while Hadamard does. In Chapter 9, flows of velocity fields will be adopted asthe natural framework for defining shape semiderivatives. In section 3.3 the veloc-ity and transformation viewpoints will be emphasized through a series of examplesof commonly used families of transformations of sets. They include Ck-domains,Cartesian graphs, polar coordinates, and level sets.

The following two sections give technical results that will be used to charac-terize continuity and semidifferentiability along local paths without restricting theanalysis to the subgroup GΘ of F(Θ) of section 2. In section 4 we establish theequivalence between deformations obtained from a family of C1-paths and defor-mations obtained from the flow of a velocity field. Section 4.1 gives the equivalenceunder relatively general conditions. Section 4.2 shows that Lipschitzian perturba-tions of the identity operator can be generated by the flow of a nonautonomousvelocity field. In section 4.3 the conditions of section 4.2 are sharpened for thespecial families of velocity fields in Ck

0 (RN,RN), Ck(RN,RN), and Ck,1(RN,RN).The constrained case where the family of domains are subsets of a fixed holdall isstudied in section 5. In both sections 4 and 5 we show that, under appropriate con-ditions, starting from a family of transformations is locally equivalent to startingfrom a family of velocity fields. This key result bridges the two points of view.

Section 6 builds on the results of section 4 to establish that the continuity ofa shape function with respect to the Courant metric on F(Θ)/G is equivalent toits continuity along the flows of all velocity fields V in C0([0, τ ]; Θ) for Θ equal toCk+1

0 (RN,RN), Ck+1(RN,RN), and Ck,1(RN,RN), k ≥ 0. This result is of bothintrinsic and practical interest since it is generally easier to check the continuityalong paths than directly with respect to the Courant metric. Finally, from thediscussion in section 3, the technical results of section 4 will again be used in Chap-ter 9 to construct local C1-paths generated by velocity fields V in C0([0, τ ]; Θ) todefine the shape semiderivatives.

2. Metrics on Transformations Generated by Velocities 161

2 Metrics on Transformations Generatedby Velocities

2.1 Subgroup GΘ of Transformations Generated by VelocitiesFrom Chapter 3 recall the definition of the group

F(Θ) def=I + θ : θ ∈ Θ, (I + θ)−1 ∃, and (I + θ)−1 − I ∈ Θ

for the Banach space Θ = C0,1(RN,RN) with norm

∥θ∥Θdef= max

sup

x∈RN|θ(x)|, c(θ)

, c(θ) def= sup

y =x

|θ(y)− θ(x)||y − x| , (2.1)

and the associated complete metric d on F(Θ). This section specializes to a sub-group of transformations I + θ of F(Θ) that are specified via a vector field V on ageneric interval [0, 1] with the following properties:

(i) for all x ∈ RN, the function t &→ V (t, x) : [0, 1] → RN belongs to L1(0, 1;RN) and

∫ 1

0∥V (t)∥C dt =

∫ 1

0sup

x∈RN|V (t, x)| dt <∞; (2.2)

(ii) for almost all t ∈ [0, 1], the function x &→ V (t, x) : RN → RN is Lipschitz-ian and

∫ 1

0c(V (t)) dt =

∫ 1

0supx=y

|V (t, y)− V (t, x)||y − x| dt <∞. (2.3)

In view of the assumptions on V , it is natural to introduce the norm

∥V ∥L1(0,1;Θ)def=∫ 1

0∥V (t)∥Θ dt. (2.4)

Associate with V ∈ L1(0, 1; Θ) the flow

dTt(V )dt

= V (t) Tt(V ), T0(V ) = I. (2.5)

In view of the assumptions on V , for each X ∈ RN, the differential equation

x′(t;X) = V (t, x(t;X)), x(0;X) = X, (2.6)

has a unique solution in W 1,1(0, 1;RN) ⊂ C([0, 1],RN), the mapping X &→ x(·;X) :RN → C([0, 1],RN) is continuous, and for all t ∈ [0, 1] the mapping X &→ Tt(V )(X) def= x(t;X) : RN → RN is bijective. Define

GΘdef=T1(V ) : ∀V ∈ L1([0, 1]; Θ)

. (2.7)

flow

W 1,1(0, 1; RN) ⊂ C([0, 1], RN)

the mapping X → x(·; X) : RN → C([0, 1], RN) is continuous

Θ = C0,1(RN, RN)


Theorem 2.1. (i) The set GΘ with the composition ,

∀V1, V2 ∈ L1(0, 1; Θ), T1(V1) T1(V2), (2.8)

is a subgroup of F(Θ) and for all V ∈ L1(0, 1; Θ),

sup0≤t≤1

∥Tt(V )− I∥Θ = sup0≤t≤1

∥θV (t)∥Θ ≤ ∥V ∥L1 exp2(1+∥V ∥L1 ), (2.9)

where θV (t) def= Tt(V )− I and L1 stands for L1(0, 1; Θ). In particular, for allt, Tt(V ) ∈ F(Θ) and t &→ Tt(V ) : [0, 1]→ F(Θ) is a continuous path.

(ii) For all V ∈ L1(0, 1; Θ), (T1(V ))−1 = T1(V −) for V −(t, x) = −V (1− t, x),

−θV (I + θV )−1 = (T1(V ))−1 − I = T1(V −)− I = θV − , (2.10)

∥V −∥L1(0,1;Θ) = ∥V ∥L1(0,1;Θ), (2.11)

and

sup0≤t≤1

∥Tt(V )−1 − I∥Θ = sup0≤t≤1

∥θ−V (t)∥Θ ≤ ∥V ∥L1 exp2(1+∥V ∥L1 ) . (2.12)

(iii) For all V ∈ L1(0, 1; Θ),

∥T1(V )− I∥Θ + ∥(T1(V ))−1 − I∥Θ ≤ 2 ∥V ∥L1(0,1;Θ) exp2 (1+∥V ∥L1(0,1;Θ)) (2.13)

⇒ d(T1(V ), I) ≤ 2 ∥V ∥L1(0,1;Θ) exp2 (1+∥V ∥L1(0,1;Θ)) . (2.14)

(iv) For all V, W ∈ L1(0, 1; Θ),

max

sup0≤t≤1

∥θW (t)− θV (t)∥Θ, sup0≤t≤1

∥θW −(t)− θV −(t)∥Θ

≤∥V ∥L1 exp(1+∥V ∥L1 )(1 + ∥W∥L1 exp2(1+∥W∥L1 )

)∥W (s)− V (s)∥L1 ,

(2.15)

where L1 stands for L1(0, 1; Θ).

Remark 2.1.Note that the condition (I−θ)−1 exists and (I−θ)−1 ∈ Θ in the definition of F(Θ)is automatically verified for the transformations I +θV generated by a velocity fieldV ∈ L1(0, 1; Θ).

Proof. (i) Choosing V = 0, the identity I is in GΘ. To show that GΘ is closedunder the composition, it is sufficient to construct a W ∈ L1(0, 1; Θ) such thatT1(W ) = T1(V1) T1(V2). Given τ ∈ (0, 1) define the concatenation

(V1 ∗ V2)(t, x) def=

⎧⎪⎪⎨

⎪⎪⎩

1τ

V2

(t

τ, x

), 0 ≤ t ≤ τ,

11− τ V1

(t− τ1− τ , x

), τ < t ≤ 1.

(2.16)


It is easy to check that W = V1 ∗ V2 ∈ L1(0, 1; Θ) as the concatenation of theL1-functions V1 and V2 so that the equation

dTt(W )dt

= W (t) Tt(W ), T0(W ) = I

has a unique solution and T1(W ) ∈ GΘ. On the interval [0, τ ]

Tt(W )(X)−X =∫ t

0V2

( s

τ, Ts(W )(X)

)d

s

τ=∫ t/τ

0V2 (s′, Ts′τ (W )(X)) ds′

⇒ Tt′τ (W )(X)−X =∫ t′

0V2 (s′, Ts′τ (W )(X)) ds′, 0 ≤ t′ ≤ 1

⇒ Tt′τ (W )(X) = Tt′(V2)(X) and Tτ (W ) = T1(V2).

On the interval [τ, 1]

Tt(W )(X)− T1(V2) =∫ t

τV1

(s− τ1− τ , Ts(W )(X)

)ds− τ1− τ

=∫ t−τ

1−τ

0V2(s′, Tτ+s′(1−τ)(W )(X)

)ds′

⇒ Tτ+t′(1−τ)t(W )(X)− T1(V2) =∫ t′

0V2(s′, Tτ+s′(1−τ)(W )(X)

)ds′, 0 ≤ t′ ≤ 1,

Tτ+t′(1−τ)(W )(X) = Tt′(V1) T1(V2) and T1(W )(X) = T1(V1) T1(V2).

To show that each T1(V ) has an inverse in GΘ, consider the new function

y(t;X) def= [T1−t(V ) (T1(V ))−1](X) = x(1− t; (T1(V ))−1(X)), 0 ≤ t ≤ 1,

for which y(0;X) = [T1(V ) (T1(V ))−1](X) = X and y(1;X) = (T1(V ))−1(X).Its equation is given by

dy

dt(t;X) = −V (1− t, y(t;X)), y(0;X) = X,

and the associated vector field in V −(t, x) def= −V (1 − t, x) that clearly belongs toL1(0, 1; Θ). Therefore GΘ is a group of homeomorphisms of RN, that is, GΘ ⊂Hom(RN).

To show that GΘ ⊂ F(Θ), it is necessary to show that for each V , θ =T1(V ) − I ∈ Θ. Given X ∈ RN, consider the functions x(t) = Tt(V )(X) and


θ(t;X) = x(t)−X. By definition

θ(t;X) = x(t)−X =∫ t

0V (s, X + θ(s;X)) ds,

|θ(t;X)| ≤∫ t

0|V (s, X + θ(s;X))− V (s, X)| ds +

∫ t

0|V (s, X)| ds

≤∫ t

0c(V (s)) |θ(s;X)| ds +

∫ t

0|V (s, X)| ds,

supX

|θ(t;X)| ≤∫ t

0c(V (s)) sup

X|θ(s;X)| ds +

∫ t

0supX

|V (s, X)| ds.

Given 0 < α < 1 let gα(t) = exp(∫ t0 c(V (s)) ds/α). Then it is easy to show that

supt∈[0,1]

supX |θ(t;X)|gα(t)

≤ 11− α

∫ 1

0supX

|V (s, X)| ds

⇒ supt∈[0,1]

∥θ(t)∥C ≤gα(1)1− α

∫ 1

0∥V (s)∥C ds.

To prove that θ is Lipschitzian, associate with X, Y ∈ RN, X = Y , the functionsx(t) = Tt(V )(X), y(t) = Tt(V )(Y ), θ(t;X) = x(t)−X, and θ(t;Y ) = y(t)− Y . Bydefinition

θ(t;X)− θ(t;Y ) =∫ t

0V (s, Y + θ(s;Y ))− V (s, X + θ(s;X)) ds,

|θ(t;X)− θ(t;Y )| ≤∫ t

0c(V (s)) |θ(s;Y ))− θ(s;X)| ds +

∫ t

0c(V (s)) |Y −X| ds

⇒ supX =Y

|θ(t;X)− θ(t;Y )||Y −X| ≤

∫ t

0c(V (s)) sup

X =Y

|θ(s;Y ))− θ(s;X)||Y −X| ds +

∫ t

0c(V (s)) ds

⇒ supt∈[0,1]

c(θ(t)) ≤ gα(1)1− α

∫ 1

0c(V (s)) ds.

Combining the last inequality with the one of the first part, for all α, 0 < α < 1,

supt∈[0,1]

∥θ(t)∥Θ ≤gα(1)1− α

∫ 1

0∥V (t)∥Θ dt.

For V = 0, θ = 0; for V = 0, choose α =∫ 10 ∥V (t)∥Θdt/(

∫ 10 ∥V (t)∥Θdt + 1). This

yields (2.9). Since (I + θ)−1 = (T1(V ))−1 = T1(V −) is of the same form for thevelocity field V −, (I + θ)−1 − I = (T1(V ))−1 − I = T1(V −)− I ∈ Θ. Therefore GΘis a subgroup of F(Θ).

(ii) From the construction of the inverse in part (i) and the fact that (T1(V ))−1−I = −θV (I + θV )−1.

(iii) From inequality (2.9) applied to θV and θV − and identity (2.11).


(iv) Let θV (t) = Tt(V )− I and let θW (t) = Tt(W )− I:

θW (t)− θV (t) =∫ t

0W (s) Ts(W )− V (s) Ts(V ) ds

=∫ t

0(W (s)− V (s)) Ts(W ) ds

+∫ t

0V (s) Ts(W )− V (s) Ts(V ) ds,

∥θW (t)− θV (t)∥C ≤∫ t

0∥W (s)− V (s)∥C ds

+∫ t

0c(V (s)) ∥θW (s)− θV (s)∥C ds

⇒ sup0≤t≤1

∥θW (t)− θV (t)∥C ≤gα(1)1− α

∫ 1

0∥W (s)− V (s)∥C ds

for 0 < α < 1 and gα(t) = exp∫ t0 c(V (s) ds/α. Similarly,

θW (t)− θV (t) =∫ t

0(W (s)− V (s)) Ts(W ) ds

+∫ t

0V (s) Ts(W )− V (s) Ts(V ) ds,

c(θW (t)− θV (t)) ≤∫ t

0c(W (s)− V (s)) (1 + c(θW (s))) ds

+∫ t

0c(V (s)) c(θW (s)− θV (s)) ds

⇒ sup0≤t≤1

c(θW (t)− θV (t)) ≤ gα(1)1− α

∫ 1

0c(W (s)− V (s)) (1 + c(θW (s))) ds.

From part (i)

sup0≤s≤1

c(θW (s)) ≤ ∥W∥L1(0,1;Θ) exp2(1+∥W∥L1(0,1;Θ))

and finally

sup0≤t≤1

∥θW (t)− θV (t)∥Θ

≤ gα(1)1− α (1 + ∥W∥L1(0,1;Θ) exp2(1+∥W∥L1(0,1;Θ))) ∥W (s)− V (s)∥L1(0,1;Θ) ds.

By choosing α = ∥V ∥L1(0,1;Θ)/(1 + ∥V ∥L1(0,1;Θ))

sup0≤t≤1

∥θW (t)− θV (t)∥Θ

≤ ∥V ∥L1 exp(1+∥V ∥L1 ) (1 + ∥W∥L1(0,1;Θ) exp2(1+∥W∥L1 )) ∥W (s)− V (s)∥L1 ds.


2.2 Complete Metrics on GΘ and GeodesicsSince GΘ is a subgroup of the complete group F(Θ), its closure GΘ with respect tothe metric d of F(Θ) is a closed subgroup whose elements can be “approximated”by elements constructed from the flow of a velocity field. Is (GΘ, d) complete? Cana velocity field be associated with a limit element? It is very unlikely that (GΘ, d)can be complete unless we strengthen the metric d to make the associated Cauchysequence of velocities Vn converge in L1(0, 1; Θ).

We have seen that for each V ∈ L1(0, 1; Θ), we have a continuous path t &→Tt(V ) in GΘ. Since the part of the metric on I +θV is defined as an infimum over allfinite factorizations (I +θV1) · · · (I +θVn) of I +θV and (I +θV −

1) · · · (I +θV −

n)

of (I + θV )−1,

T1(V ) = T1(V1) T1(V2) · · · T1(Vn),

d(I, T1(V )) = infT1(V )=

T1(V1)···T1(Vn)

n∑

i=1

∥θVi∥Θ + ∥θV −i∥Θ,

is there a velocity V ∗ ∈ L1(0, 1; Θ) that achieves that infimum? Do we have ageodesic path between I and T1(V ) that would be achieved by V ∗ ∈ L1(0, 1; Θ)?

Given a velocity field we have constructed a continuous path t &→ Tt(V ) :[0, 1] → F(Θ). At each t, there exists δ > 0 sufficiently small such that the patht &→ Tt+s(V ) T−1

t (V ) : [0, δ]→ F(Θ) is differentiable in t in the group F(Θ):

Tt+s(V ) T−1t (V )− I

s=

1s

∫ t+s

tV (r) Tr(V ) dr T−1

t (V )

→ V (t) Tt(V ) T−1t (V ) = V (t)

(2.17)

⇒ dFTt

dt= V (t) a.e. in Θ. (2.18)

With this definition a jump from I + θ1 to [I + θ2] [I + θ1] at time 1/2 becomes

T1/2+s(V ) T−11/2(V )− I = [I + θ2] [I + θ1] [I + θ1]−1 − I = I + θ2 − I = θ2,

that is, a Dirac delta function θ2 in the tangent space Θ to F(Θ) at 1/2. Now thenorm of the velocity V = θ2 δ1/2 in the space of measures becomes

∥V ∥M1((0,1);Θ) =∫ 1

0∥θ2∥Θδ1/2 dt = ∥θ2∥Θ.

If we have n such jumps,

∥V ∥M1((0,1);Θ) =∑

1=1,...,n

∥θi∥Θ.

The metric of Micheletti is the infimum of this norm over all finite factorizations.A consequence of this analysis is that a factorization of an element I +θ of the

form (I+θ1)(I+θ2)· · ·(I+θn) is in fact a path t &→ Tt : [0, 1]→ F(Θ) of bounded


variations in F(Θ) between I and I+θ by assigning times 0 < t1 < t2 < · · · < tn < 1to each jump in the group F(Θ):

Tt(V ) def=

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

I, 0 ≤ t < t1,

I + θ1, t1 ≤ t < t2,

(I + θ1) (I + θ2), t2 ≤ t < t3,

. . .

(I + θ1) (I + θ2) · · · (I + θn), tn ≤ t ≤ 1,

V (t) def=n∑

i=1

θi δ(ti) ⇒ dFTt

dt= V (t), ∥V ∥M1((0,1);Θ) =

∑

1=1,...,n

∥θi∥Θ,

where the time-derivative is taken in the group sense (2.17) and V is a boundedmeasure.

With this interpretation in mind, for V ∈ L1(0, 1; Θ) the metric of Michelettishould reduce to

d(I, T1(V )) = infv∈L1(0,1;Θ)T1(v)=T1(V )

∫ 1

0∥v(t)∥Θ dt = inf

v∈L1(0,1;Θ)T1(v)=T1(V )

∫ 1

0

∥∥∥∥dFTt(v)

dt

∥∥∥∥Θ

dt,

where the derivative is in the group F(Θ). As a result we could talk of a geodesicpath in F(Θ) between I and T1(V ). If this could be justified, the next questionis whether GΘ is complete in (F(Θ), d) or complete with respect to the metricconstructed from the function

dG(I, T1(V )) def= infv∈L1(0,1;Θ)T1(v)=T1(V )

∫ 1

0∥v(t)∥Θ dt, (2.19)

dG(T1(V ), T1(W )) def= d(T1(V ) T1(W )−1, I). (2.20)

By definition

dG(T1(V )−1, I) = dG(T1(V ), I), dG(T1(V ), T1(W )) = dG(T1(W ), T1(V ))

and dG is right-invariant. The infimum is necessary since the velocity taking I toT1(V ) is not unique.

Theorem 2.2. Let Θ be equal to Ck,1(RN,RN), Ck+1(RN,RN), Ck+10 (RN,RN),

or Bk+1(RN,RN), k ≥ 0. Then dG is a right-invariant metric on GΘ.

Remark 2.2.The theorem is also true for the right-invariant metric

d′G(T1(V ), I) def= d(T1(W ), T1(V )) + inf

W∈L1(0,1;Θ)T1(W )=T1(V )

∥W∥L1(0,1;Θ), (2.21)

d′G(T1(V ), T1(W )) def= d′

G(T1(V ) T1(W )−1, I). (2.22)


Proof. We give only the proof for C0,1(RN,RN). It remains to show that thetriangle inequality holds and that dG(T1(W ), T1(V )) = 0 if and only if T1(W ) =T1(V ). Given U, V, W in L1(0, 1; Θ), T1(W )T1(U)−1 = T1(W )T1(V )−1 T1(V )T1(U)−1. For all v ∈ L1(0, 1; Θ) such that T1(v) = T1(W ) T1(V )−1 and for allu ∈ L1(0, 1; Θ) such that T1(v) = T1(V ) T1(U)−1, the concatenation v ∗ u is suchthat T1(v ∗ u) = T1(W ) T1(U)−1. By definition

dG(T1(W ) T1(U)−1, I) ≤∥v ∗ u∥L1 = ∥v∥L1 + ∥u∥L1 .

By taking infima with respect to v and u we get the triangle inequality

dG(T1(W ) T1(U)−1, I) ≤ dG(T1(W ) T1(V )−1, I) + dG(T1(V ) T1(U)−1, I).

If T1(V ) = I, then infw∈L1(0,1;Θ)T1(w)=I

∥w∥L1(0,1;Θ) = 0 and dG(T1(V ), I) = 0. Con-

versely, if dG(T1(V ), I) = 0, then inf v∈L1(0,1;Θ)T1(v)=T1(V )

∥v∥L1(0,1;Θ) = 0 and there exists a

sequence vn ∈ L1(0, 1; Θ) such that vn → 0 in L1(0, 1; Θ) and T1(vn) = T1(V ). Bycontinuity of T1(vn) − I with respect to vn from Theorem 2.1 (iv), T1(V ) − I =T1(vn)− I → T1(0)− I = 0 and T1(V ) = I.

Getting the completeness with the metric dG and even d′G is not obvious. For

a Cauchy sequence T1(Vn), the L1-convergence of the velocities Vn to some Vwould yield the convergence of T1(Vn) to T1(V ) and even the convergence of thegeodesics, but the metrics do not seem sufficiently strong to do that. To appreciatethis point, we first establish an inequality that really follows from the triangle in-equality for dG. Associate with W a w such that T1(w) = T1(W ), with V a v suchthat T1(v) = T1(V ), and let u be such that T1(u) = T1(W )T1(V )−1. Then the con-catenation w ∗ v− is such that T1(v ∗u) = T1(W ). From this we get the inequalities

infT1(w)=T1(W )

∥w∥L1 ≤ ∥v ∗ u∥L1 = ∥v∥L1 + ∥u∥L1

and by taking infima we get the inequality

infT1(w)=T1(W )

∥w∥L1 ≤ infT1(u)=T1(W )T1(V )−1

∥u∥L1 + infT1(v)=T1(V )

∥v∥L1

⇒∣∣∣∣ infT1(w)=T1(W )

∥w∥L1 − infT1(v)=T1(V )

∥v∥L1

∣∣∣∣ ≤ infT1(u)=T1(W )T1(V )−1

∥u∥L1 .

Pick any Cauchy sequence T1(Vn). For any ε > 0, there exists N such thatfor all n, m > N

∣∣∣∣ infT1(w)=T1(Vm)

∥w∥L1 − infT1(v)=T1(Vn)

∥v∥L1

∣∣∣∣ ≤ infT1(u)=T1(Vm)T1(Vn)−1

∥u∥L1 < ε.

Therefore, there exists a sequence vn such that T1(vn) = T1(Vn) and

0 ≤ ∥vn∥L1 − infT1(v)=T1(Vn)

∥v∥L1 < ε

⇒ |∥vm∥L1 − ∥vn∥L1 | < 3ε.

The sequence ∥vn∥L1 is bounded in L1(0, 1; Θ), but this does not seem sufficientto get the convergence of the vn’s in L1(0, 1; Θ).


One could think of many ways to strengthen the metric. For instance, onecould introduce the minimization of the whole W 1,1(0, 1; Θ)-norm of the path t &→Tt(v) with respect to v from which the minimizing velocity v(t) = ∂Tt/∂t T−1

t

could be recovered, but the triangle inequality would be lost.

2.3 Constructions of Azencott and TrouveIn 1994 R. Azencott [1] defined the following program. Given a smooth manifoldM , consider the time continuous curve T = (Tt)0≤t≤1 in Aut(M) solutions of

∂Tt

∂t= V (t) Tt, T0 = I, (2.23)

for a continuous time family V (t) of vector fields in a Hilbert space H ⊂ Aut(M)and define

d(I,φ) = infV ∈C([0,1];H)

T1(V )=φ

l(φ) with l(φ) def=∫ 1

0|V (t)|I dt. (2.24)

A fully rigorous construction of this distance, in the context of infinite-dimensionalLie groups, was performed by A. Trouve [2] in 1995 and [3] in 1998. He showsthat the subset of φ ∈ Aut(M) for which d(I,φ) can be defined is a subgroup Aof invertible mappings. He extends this distance between two arbitrary mappingsφ and ψ by right-invariance: d(φ,ψ) def= d(I,ψ φ−1). Hence, in this extendedframework, the new problem should be to find in A

φ = argmin12

∫

M|f − f φ|2dx +

12d(Id,φ)2,

where M is a smooth manifold, the function f : M → R is a high dimensionalrepresentation template, and f is a new observed image belonging to this family (cf.A. Trouve [3]). This approach was further developed by the group of Azencott(cf., for instance, L. Younes [1]).

Quoting from the introduction of A. Trouve [3] in 1998:

. . . Hence, for a large deformation φ, one can consider φ as the con-catenation of small deformations φui . More precisely, if φ = Φn, wherethe Φk are recursively defined by Φ0 = Id and φk+1 = φuk+1 Φk, thefamily Φ = (Φk)0≤k≤n defines a polygonal line in Aut(M) whose lengthl(Φ) can be approximated by l(φ) =

∑nk=1 |uk|e. At a naive level, one

could define the distance d(Id,φ) as the infimum of l(Φn) for all familyΦ such that Φn = φ. For a more rigorous setting, one should considerthe time continuous curve Φ = (Φt)0≤t≤1 in Aut(M). . . .

In fact what he describes as a naive approach is the very precise construction ofA. M. Micheletti [1] in 1972. He is certainly not the first author who hasoverlooked that paper in Italian, where the central completeness result that is ofinterest to us was just a lemma in her analysis of the continuity of the first eigenvalueof the Laplacian. We brought it up only in the 2001 edition of this book.

construction of A. M. Micheletti [1] in 1972.


By introducing a Hilbert space H ⊂ Θ = C0,1(RN,RN) and minimizing overL2(0, 1;H) rather than L1(0, 1; Θ), the bounded sequence vn lives in L2(0, 1;H),where there exist a v ∈ L2(0, 1;H) and a weakly converging subsequence to v. Byusing the complete metric of Micheletti, we had the existence of a F ∈ F(Θ) suchthat T1(vn)→ F , but now we have a candidate v for which F = T1(v). This is verymuch analogous to the Hilbert uniqueness method of J.-L. Lions [1] in the contextof the controllability of partial differential equations (cf. also R. Glowinski andJ.-L. Lions [1, 2]).

Remark 2.3.It is important to understand that finding a complete metric of Courant type forthe various spaces of diffeomorphisms is a much less restrictive problem than con-structing a metric from geodesics. The existence of a geodesic is not required forthe Courant metrics.

3 Semiderivatives via Transformations Generatedby Velocities

3.1 Shape FunctionAll spaces F(Θ) and F(Θ)/G(Ω0) in Chapter 3 associated with the family of sets

X (Ω0) = F (Ω0) : ∀F ∈ F(Θ) (3.1)

are nonlinear and nonconvex and the elements of F(Θ)/G(Ω0) are equivalenceclasses of transformations. Defining a differential with respect to such spaces issimilar to defining a differential on an infinite-dimensional manifold. Fortunately,the tangent space is invariant and equal to Θ in every point of F(Θ). This consid-erably simplifies the analysis.

Definition 3.1.Given a nonempty subset D of RN, consider the set P(D) = Ω : Ω ⊂ D of subsetsof D. The set D will be referred to as the underlying holdall or universe. A shapefunctional is a map

J : A→ E (3.2)from some admissible family A of sets in P(D) into a topological space E.

For instance, A could be the set X (Ω). D can represent some physical ormechanical constraint, a submanifold of RN, or some mathematical constraint. Inmost cases, it can be chosen large and as smooth as necessary for the analysis. Inthe unconstrained case, D is equal to RN.

3.2 Gateaux and Hadamard SemiderivativesConsider the solution (flow) of the differential equation

dx

dt(t, X) = V (t, x(t, X)), t ≥ 0, x(t, X) = X, (3.3)

Definition 3.1. Given a nonempty subset D of RN, consider the set P(D) = Ω : Ω ⊂ D of subsets of D. The set D will be referred to as the underlying holdall or universe. A shape functional is a map J:A→E (3.2) from some admissible family A of sets in P(D) into a topological space E.

3. Semiderivatives via Transformations Generated by Velocities 171

V

X

Ω Ωt = Tt(Ω)

x(t)

Figure 4.1. Transport of Ω by the velocity field V .

for velocity fields V (t)(x) def= V (t, x) (cf. Figure 4.1). It generates the family oftransformations Tt : RN → RN : t ≥ 0 defined as follows:

X &→ Tt(X) def= x(t;X) : RN → RN . (3.4)

Given an initial set Ω ⊂ RN, associate with each t > 0 the new set

Ωtdef= Tt(Ω) = Tt(X) : ∀X ∈ Ω . (3.5)

This perturbation of the initial set is the basis of the velocity (speed) method .The choice of the terminology “velocity” to describe this method is accurate

but may become ambiguous in problems where the variables involved are themselves“physical velocities”: this situation is commonly encountered in continuum mechan-ics. In such cases it may be useful to distinguish between the “artificial velocity”and the “physical velocity.” This is at the origin of the terminology speed method,which has often been used in the literature. The latter terminology is convenient,but is not as accurate as the velocity method. We shall keep both terminologies anduse the one that is most suitable in the context of the problem at hand.

It is important to work with the weakest notion of semiderivative that pre-serves the basic elements of the classical differential calculus such as the chain rulefor the differentiation of the composition of functions. It should be able to handlethe norm of a function or functions defined as an upper or lower envelope of a familyof differentiable functions. Since the spaces of geometries are generally nonlinearand nonconvex, we also need a notion of semiderivative that yields differentials inthe tangent space as for manifolds. We are looking for a very general notion ofsemidifferential but not more! In that context the most suitable candidate is theHadamard semiderivative3 that will be introduced later. Important nondifferentia-ble functions are Hadamard semidifferentiable and the chain rule is verified.

3In his 1937 paper, M. Frechet [1] extends to function spaces the Hadamard derivative andpromotes it as more general than his own Frechet derivative since it does not require the space tohave a metric in infinite-dimensional spaces. In the same paper, he also introduces a relaxationof the Hadamard derivative that is almost a semiderivative. A precise definition of the Hadamardsemiderivative explicitly appears in J.-P. Penot [1] in 1978 and in A. Bastiani [1] in 1964 as awell-known notion.


We proceed step by step. A first candidate for the shape semiderivative of Jat Ω in the direction V would be


J(Ωt(V ))− J(Ω)t

(when the limit exists). (3.6)

This very weak notion depends on the history of V for t > 0 and will be too weak formost of our purposes. For instance, its composition with another function would notverify the chain rule. To make it compatible with the usual notion of semiderivativein a direction θ of the tangent space Θ to F(Θ) in F (Ω), it must be strengthenedas follows.

Definition 3.2.Given θ ∈ Θ

dJ(Ω; θ) def= limV ∈C0([0,τ ];Θ)

V (0)=θt0


, (3.7)

where the limit depends only on θ and is independent of the choice of V for t > 0.

Equivalently, the limit dJ(Ω; θ) is independent of the way we approach Ω for afixed direction θ. This is precisely the adaptation of the Hadamard semiderivative.4

Definitions based on a perturbation of the identity

Tt(X) def= x(t) = X + t θ(X), t ≥ 0, (3.8)

can be recast in the velocity framework by choosing the velocity field V (t) = θT−1t ,

that is,

∀x ∈ RN, ∀t ≥ 0, V (t, x) def= θ(T−1t (x)) = θ (I + tθ)−1(x), (3.9)

which requires the existence of the inverse for sufficiently small t. With that choicefor each X, x(t) = Tt(X) is the solution of the differential equation

dx

dt(t) = V

(t, x(t)

), x(0) = X. (3.10)

Under appropriate continuity and differentiability assumptions,

∀x ∈ RN, V (0)(x) def= V (0, x) = θ(x), (3.11)

∀x ∈ RN, V (0)(x) def=∂V

∂t(t, x)

∣∣∣∣t=0

= − [Dθ(x)] θ(x), (3.12)

where Dθ(x) is the Jacobian matrix of θ at the point x. In compact notation,

V (0) = θ and V (0) = − [Dθ] θ. (3.13)

This last computation shows that at time 0, the points of the domain Ω are si-multaneously affected by the velocity field V (0) = θ and the acceleration field

4Cf. footnote 2 in section 2 of Chapter 9 for the definitions and properties and the comparisonof the various notions of derivatives and semiderivatives in Banach spaces.

shape semiderivative

Definition 3.2. Given θ ∈ Θ dJ(Ω;θ) = V ∈C0([0,τ];Θ) t V (0)=θ (3.7) def J(Ωt(V )) − J(Ω) lim , t0 where the limit depends only on θ and is independent of the choice of V for t > 0.


V (0) = −[Dθ]θ. The same result will be obtained without acceleration term by re-placing I + tθ by the transformation Tt generated by the solutions of the differentialequation

dx

dt(t) = θ(x(t)), x(0) = X, Tt(X) def= x(t),

for which V (0) = θ and V (0) = 0. Under suitable assumptions the two methodswill produce the same first-order semiderivative.

The definition of a shape semiderivative

dJ(Ω; θ) def= limt0

J([I + tθ]Ω)− J(Ω)t

(3.14)

by perturbation of the identity is similar to a Gateaux semiderivative that willnot verify the chain rule. Moreover, second-order semiderivatives will differ by anacceleration term that will appear in the expression obtained by the method ofperturbation of the identity.

3.3 Examples of Families of Transformations of DomainsIn section 3.2 we have formally introduced a notion of semiderivative of a shapefunctional via transformations generated by flows of velocity fields and perturbationsof the identity operator as special cases of the velocity method.

Before proceeding with a more abstract treatment, we present several examplesof definitions of shape semiderivatives that can be found in the literature. Weconsider special classes of domains (C∞, Ck, Lipschitzian), Cartesian graphs, polarcoordinates, and level sets that provide classical examples of parametrized and/orconstrained deformations. In each case we construct the associated underlyingfamily (not necessarily unique) of transformations Tt : 0 ≤ t ≤ τ.

3.3.1 C∞-Domains

Let Ω be an open domain of class C∞ in RN. Recall from section 5 of Chapter 2that in any point x ∈ Γ the unit outward normal is given by

∀y ∈ Γxdef= U(x) ∩ Γ, n(y) =

mx(y)|mx(y)| , (3.15)

where

mx(y) = − ∗(Dhx)−1(h−1x (y)) eN in U(x) (3.16)

⇒ n = −∗(Dhx)−1eN

| ∗(Dhx)−1eN | h−1x . (3.17)

When Γ is compact, it is possible to find a finite sequence of points xj : 1 ≤ j ≤ Jin Γ such that

Γ ⊂ U def=J⋃

j=1

Uj , Ujdef= U(xj).

Under suitable assumptions the two methods will produce the same first-order semiderivative.

second-order semiderivatives will differ by an acceleration term that will appear in the expression obtained by the method of perturbation of the identity.


As in the definition of the boundary integral, associate with Uj a partition ofunity rj:

rj ∈ D(Uj), 0 ≤ rj ≤ 1,J∑

j=1

rj = 1 in U0

for some neighborhood U0 of Γ such that

Γ ⊂ U0 ⊂ U0 ⊂ U .

For the C∞-domain Ω the normal satisfies

n =J∑

j=1

rj n =J∑

j=1

rjmj

|mj |∈ C∞(Γ;RN)

since

∀j, mj

|mj | hj ∈ C∞(B;RN).

Given any ρ ∈ C∞(Γ) and t ≥ 0, consider the following perturbation Γt of Γ alongthe normal field n:

Γtdef= x ∈ RN : x = X + tρ(X)n(X), ∀X ∈ Γ. (3.18)

We claim that for τ sufficiently small and all t, 0 ≤ t ≤ τ , the set Γt is the boundaryof a C∞-domain Ωt by constructing a transformation Tt of RN which maps Ω ontoΩt and Γ onto Γt. First construct an extension N ∈ D(RN,RN) of the normal fieldn on Γ. Define

mdef=

J∑

j=1

rjmj ∈ D(U ,RN). (3.19)

By constructionm ∈ C∞(Γ;RN), (3.20)

m = 0 on Γ, and there exists a neighborhood U1 of Γ contained in U0 where m = 0since m is at least C1.

Now construct a function r0 in D(U1), 0 ≤ r0(x) ≤ 1, and a neighborhood Vof Γ such that

r0 = 1 in V and Γ ⊂ V ⊂ V ⊂ U1.

Define the vector field

∀x ∈ RN, N(x) def= r0(x)m(x)|m(x)| . (3.21)

Hence N belongs to D(RN,RN) since supp N ⊂ V is compact. Moreover

N =m

|m| in V ⇒ N(x) = n(x) on Γ = Γ ∩ V.

For each j, ρ hj ∈ C∞(B0) and the extensions

ρhj (ζ) = ρh

j (ζ ′, ζN ) def= ρ(hj(ζ ′, 0)), ∀ζ ∈ B, ρjdef= ρh

j gj ,


belong, respectively, to C∞(B) and C∞(Uj). Then

ρdef=

J∑

j=1

rj ρj ∈ D(RN)

is an extension of ρ from Γ to RN with compact support since supp rj ⊂ Uj ⊂ U .Define the following transformation of RN:

Tt(X) def= X + t ρ(X)N(X), t ≥ 0. (3.22)

By construction, ρN is uniformly Lipschitzian in RN and, by Theorem 4.2 in sec-tion 4.2, there exists 0 < τ such that Tt is bijective and bicontinuous from RN ontoitself. As a result from J. Dugundji [1] for 0 ≤ t ≤ τ

Ωt = Tt(Ω) = [I + tρN ](Ω),∂Ωt = Tt(∂Ω) = [I + tρN ](∂Ω) = [I + tρn](∂Ω) = Γt.

Since the domain Ωt is specified by its boundary Γt, it depends only on ρ and noton its extension ρ. The special transformation Tt introduced here is of class C∞,that is, Tt ∈ C∞(RN,RN), and 1

t [Tt − I] is proportional to the normal field n on Γ,but it is not proportional to the normal nt on Γt for t > 0. In other words, at t = 0the deformation is along n, but at t > 0 the deformation is generally not along nt.

If J(Ω) is a real-valued shape functional defined on C∞-domains in D, thesemiderivative (if it exists) is defined as follows: for all ρ ∈ C∞(Γ)

dnJ(Ω; ρ) def= limt0

J((I + tρN)Ω

)− J(Ω)

t. (3.23)

It turns out that this limit depends only on ρ and not on its extension ρ.

3.3.2 Ck-Domains

When Ω is a domain of class Ck with boundary Γ, the normal field n belongsto Ck−1(Γ,RN). Therefore, choosing deformations along the normal would yieldtransformations Tt mapping Ck-domains Ω onto Ck−1-domains Ωt = Tt(Ω). Theobvious way to deal with Ck-domains is to relax the constraint that the perturbationρN be carried by the normal. Choose vector fields θ in Dk(RN,RN) and considerthe family of transformations

Tt = I + tθ, Ωt = Tt(Ω), t ≥ 0. (3.24)

This is a generalization of the family of transformations (3.22) in section 3.3.1 fromρN to θ. For k ≥ 1, θ is again uniformly Lipschitzian in RN and, by Theorem 4.2in section 4.2, there exists τ > 0 such that Tt is bijective and bicontinuous from RN

onto itself. Thus for 0 ≤ t ≤ τ ,

Ωt = Tt(Ω) = [I + tθ](Ω) and ∂Ωt = Tt(∂Ω) = [I + tθ](∂Ω).

When Ω is a domain of class Ck with boundary Γ, the normal field n belongs to Ck−1(Γ,RN).


A more restrictive approach to get around the lack of sufficient smoothness of thenormal n to Γ would be to introduce a transverse field p on Γ such that

p ∈ Ck(Γ;RN), ∀x ∈ Γ, p(x) · n(x) > 0. (3.25)

Given p and ρ ∈ Ck(Γ) define for t ≥ 0

Γtdef=x ∈ RN : x = X + tρ(X)p(X), ∀X ∈ Γ

. (3.26)

Choosing Ck-extensions ρ and p of ρ and p we can go back to the case where

θ = ρp ∈ Dk(RN,RN). (3.27)

For any θ ∈ Dk(RN,RN) the semiderivative is defined as

dkJ(Ω; θ) def= limt0

J((I + tθ)(Ω)

)− J(Ω)

t. (3.28)

3.3.3 Cartesian Graphs

In many applications it is convenient to work with domains Ω which are the hy-pograph of some positive function γ in Cartesian coordinates. Such domains aretypically of the form

Ω def=

(x′, xN ) ∈ RN : x′ ∈ U ⊂ RN−1 and 0 < xN < γ(x′)

, (3.29)

where U is a connected open set in RN−1 and γ ∈ C0(U ;R+) is a positive function.Many free boundary and contact problems are formulated over such domains. Usu-ally the domains Ω (and hence the functions γ) will be constrained. The function γcan be specified on U\U or not. In some examples the derivative of γ could also bespecified, that is, ∂γ/∂ν = g on ∂U , where U is smooth and ν is the outward unitnormal field along ∂U .

When γ (resp., ∂γ/∂ν) is specified along ∂U , the directions of deformation µare chosen in C0(U ;R+) such that

µ = 0 (resp., ∂µ/∂ν = 0) on ∂U. (3.30)

For small t ≥ 0 and each such µ, define the perturbed domain

Ωt =(x′, xN ) ∈ RN : 0 < xN < γ(x′) + tµ(x′)

(3.31)

and the family of transformations

(x′, xN ) &→ Tt(x′, xN ) =(

x′, xN +xN

γ(x′)tµ(x′)

): Ω→ Ωt, (3.32)

which can be extended to a neighborhood D of Ω containing the perturbed domainsΩt, 0 ≤ t ≤ t1, for some small t1 > 0. In general, D will be such that D = U × [0, L]for some L > 0.

This construction is also appropriate for domains that are Lipschitzian or ofclass Ck, 1 ≤ k ≤ ∞, that is, when γ is a Lipschitzian or a Ck-function. Again the


transformation Tt is equal to I + tθ, where

θ(x′, xN ) def=(

0, xNµ(x′)γ(x′)

). (3.33)

But of course this θ is not the only choice for that Ωt = Tt(Ω). For instance, letλ : R+ → R+ be any smooth increasing function such that λ(0) = 0 and λ(1) = 1.Then we could consider the transformations

Tt(x′, xN ) def=(

x′, xN + tλ

(xN

γ(x′)

)µ(x′)

). (3.34)

This example illustrates the general principle that the transformation Tt of D suchthat Ωt = Tt(Ω) is not unique.

This means that, at least for smooth domains, only the trace Tt|Γt on Γt isimportant, while the displacement of the inner points does not contribute to thedefinition of Ωt. Nevertheless this statement is to be interpreted with caution. Herewe implicitly assume that the objective function J(Ω) and the associated constraintsare only a function of the shape of Ω. However, in some problems involving singu-larities at inner points of Ω (e.g., when the solution y(Ω) of the state equation has asingularity or when some constraints on the domain are active), the situation mightrequire a finer analysis. One such example is the internal displacement of the inte-rior nodes of a triangularization τh when the solution y(Ω) of a partial differentialequation is approximated by a piecewise polynomial solution over the triangular-ized domain Ωh in the finite element method. Such a displacement does not changethe shape of Ωh, but it does change the solution yh of the problem. When thedisplacement of the interior nodes is a priori parametrized by the boundary nodesthe solution yh will depend only on the position of the boundary nodes, but theinterior nodes will contribute through the choice of the specified parametrization.

3.3.4 Polar Coordinates and Star-Shaped Domains

In some examples domains are star-shaped with respect to a point. Since a domaincan always be translated, there is no loss of generality in assuming that this pointis the origin. Then such domains Ω can be parametrized as follows:

Ω def=x ∈ RN : x = ρζ, ζ ∈ SN−1, 0 ≤ ρ < f(ζ)

, (3.35)

where SN−1 is the unit sphere in RN,

SN−1def=x ∈ RN : |x| = 1

, (3.36)

and f : SN−1 → R+ is a positive continuous mapping from SN−1 such that

mdef= min f(ζ) : ζ ∈ SN−1 > 0. (3.37)

Given any g ∈ C0(SN−1) and a sufficiently small t ≥ 0 the perturbed domains aredefined as

Ωtdef=x ∈ RN : x = ρζ, ζ ∈ SN−1, 0 ≤ ρ < f(ζ) + tg(ζ)

. (3.38)


For example, choose t, 0 ≤ t ≤ t1, for some

t1 =m

∥g∥C0(SN−1)> 0 (3.39)

and define the transformation Tt as

Tt(X) = 0, if X = 0,

Tt(X) =[ρ+ t ρ

f(ζ)g(ζ)]ζ, if X = ρζ = 0.

(3.40)

As in the previous example, Tt is not unique and for any continuous increasingfunction λ : R+ → R+ such that λ(0) = 0 and λ(t) = 1 the transformation

Tt(X) = 0, if X = 0,

Tt(X) =[ρ+ tλ

(ρ

f(ζ)

)g(ζ)

]ζ, if X = ρζ = 0,

(3.41)

yields the same domain Ωt.

3.3.5 Level Sets

In sections 3.3.1 to 3.3.4, the perturbed domain Ωt always appears in the formΩt = Tt(Ω), where Tt is a bijective transformation of RN and Tt is of the form I+tθ.In some free boundary problems (e.g., plasma physics, propagation of fronts) the freeboundary Γ is a level curve of a smooth function u defined over an open domain D.Assume that D is bounded open with smooth boundary ∂D. Let u ∈ C2(D) be apositive function on D such that

u ≥ 0 in D, u = 0 on ∂D,

∃ a unique xu ∈ D such that ∀x ∈ D − xu , |∇u(x)| > 0.(3.42)

If m = maxu(x) : x ∈ D

, then for each t in [0, m[ the level set

Γt = u−1(t) (3.43)

is a C2-submanifold of RN in D, which is the boundary of the open set

Ωt = x ∈ D : u(x) > t . (3.44)

By definition Ω0 = D, for all t1 > t2, Ωt1 ⊂ Ωt2 , and the domains Ωt converge inthe Hausdorff complementary topology5 to the point xu. The outward unit normalfield on Γt is given by

x ∈ Γt, nt(x) = −|∇u(x)|−1∇u(x). (3.45)

This suggests introducing the velocity field

∀x ∈ D − xu , V (x) def= |∇u(x)|−2∇u(x), (3.46)

which is continuous everywhere but at x = xu. If V were continuous everywhere,then for each X, the path x(t;X) generated by the differential equation

dx

dt(t) = V

(x(t)

), x(0) = X (3.47)

5Cf. section 2 of Chapter 6.


would have the property that

u(x(t)

)= u(X) + t (3.48)

since formally d

dtu(x(t)

)= ∇u

(x(t)

)· dx

dt(t) = 1. (3.49)

This means that the map X &→ Tt(X) = x(t;X) constructed from (3.47) would mapthe level set

Γ0 =X ∈ D : u(X) = 0

(3.50)

onto the level setΓt =

x ∈ D : u(x) = t

(3.51)

and eventually Ω0 onto Ωt. Unfortunately, it is easy to see that this last propertyfails on the function u(x) = 1− x2 defined on the unit disk.

To get around this difficulty, introduce for some arbitrarily small ε, 0 < ε <m/2, an infinitely differentiable function ρε : RN → [0, 1] such that

ρε(x) =

0, if |∇u(x)| < ε,

1, if |∇u(x)| > 2ε,(3.52)

and the velocityVε(x) = ρε(x)V (x), x ∈ D. (3.53)

As above, define the transformation

X &→ T εt (X) = x(t;X), (3.54)

where x(t;X) is the solution of the differential equation

dx

dt(t) = Vε

(x(t)

), t ≥ 0, x(0) = X ∈ D. (3.55)

For 0 ≤ t < m− 2ε, Tt maps Γ0 onto Γt; for 0 ≤ s < m− ε such that s + t < m− ε,Tt maps Γs onto Γt+s. However, for s > m − ε, Tt is the identity operator. As aresult, for 0 ≤ t < m− 2ε,

Tt(Ω0) = Ωt and Tt(Γ0) = Γt. (3.56)

Of course ε > 0 is arbitrary and we can make the construction for t’s arbitrary closeto m. This is an example that can be handled by the velocity (speed) method andnot by a perturbation of the identity. Here the domains Ωt are implicitly constrainedto stay within the larger domain D. We shall see in section 5 how to introduce andcharacterize such a constraint.

Another example of description by level sets is provided by the oriented dis-tance function bΩ for some open domain Ω of class C2 with compact boundary Γ(cf. Chapter 7). We shall see that there exists h > 0 and a neighborhood

Uh(Γ) =x ∈ RN : |bΩ(x)| < h

such that bΩ ∈ C2(Uh(Γ)). Then for 0 ≤ t < h the flow corresponding to thevelocity field V = ∇bΩ maps Ω and its boundary Γ onto

Tt(Ω) = Ωt =x ∈ RN : bΩ(x)| < t

, Tt(Γ) = Γt =

x ∈ RN : bΩ(x)| = t

.


4 Unconstrained Families of DomainsHere we study equivalences between the velocity method (cf. J.-P. Zolesio [12,8]) and methods using a family of transformations. In section 4.1, we give somegeneral conditions to construct a family of transformations of RN from a veloc-ity field. Conversely, we show how to construct a velocity field from a family oftransformations of RN. In section 4.2, this construction is applied to Lipschitzianperturbations of the identity. In section 4.3, the various equivalences of section 4.1are specialized to velocities in Ck+1

0 (RN,RN), Ck+1(RN,RN), and Ck,1(RN,RN),k ≥ 0.

4.1 Equivalence between Velocities and Transformations

Let the real number τ > 0 and the map V : [0, τ ] × RN → RN be given. If t isinterpreted as an artificial time, the map V can be viewed as the (time-dependent)velocity field V (t) : 0 ≤ t ≤ τ defined on RN:

x &→ V (t)(x) def= V (t, x) : RN &→ RN . (4.1)

Assume that there exists τ = τ(V ) > 0 such that

(V)∀x ∈ RN, V (·, x) ∈ C

([0, τ ];RN),

∃c > 0, ∀x, y ∈ RN, ∥V (·, y)− V (·, x)∥C([0,τ ];RN) ≤ c|y − x|,(4.2)

where V (·, x) is the function t &→ V (t, x). Note that V is continuous on [0, τ ]×RN.Hence it is uniformly continuous on [0, τ ] × D for any bounded open subset D ofRN and

V (·) ∈ C([0, τ ];C(D;RN)). (4.3)

Associate with V the solution x(t;V ) of the vector ordinary differential equation

dx

dt(t) = V

(t, x(t)

), t ∈ [0, τ ], x(0) = X ∈ RN (4.4)

and define the transformations

X &→ Tt(V )(X) def= xV (t;X) : RN → RN (4.5)

and the maps (whenever the inverse of Tt exists)

(t, X) &→ TV (t, X)def= Tt(V )(X) : [0, τ ]×RN → RN, (4.6)

(t, x) &→ T−1V (t, x) def= T−1

t (V )(x) : [0, τ ]×RN → RN . (4.7)

Notation 4.1.In what follows we shall drop the V in TV (t, X), T−1

V (t, x), and Tt(V ) whenever noconfusion arises.

4. Unconstrained Families of Domains 181

Theorem 4.1.

(i) Under assumptions (V) the map T specified by (4.4)–(4.6) has the followingproperties:

(T1)∀X ∈ RN, T (·, X) ∈ C1([0, τ ];RN) and ∃c > 0,

∀X, Y ∈ RN, ∥T (·, Y )− T (·, X)∥C1([0,τ ];RN) ≤ c|Y −X|,(T2) ∀t ∈ [0, τ ], X &→ Tt(X) = T (t, X) : RN → RN is bijective,

(T3)∀x ∈ RN, T−1(·, x) ∈ C

([0, τ ];RN) and ∃c > 0,

∀x, y ∈ RN, ∥T−1(·, y)− T−1(·, x)∥C([0,τ ];RN) ≤ c|y − x|.

(4.8)

(ii) Given a real number τ > 0 and a map T : [0, τ ] × RN → RN satisfyingassumptions (T1) to (T3), the map

(t, x) &→ V (t, x) def=∂T

∂t

(t, T−1

t (x))

: [0, τ ]×RN → RN (4.9)

satisfies conditions (V), where T−1t is the inverse of X &→ Tt(X) = T (t, X).

If, in addition, T (0, ·) = I, then T (·, X) is the solution of (4.4) for that V .

(iii) Given a real number τ > 0 and a map T : [0, τ ] × RN → RN satisfyingassumptions (T1) and (T2) and T (0, ·) = I, then there exists τ ′ > 0 such thatthe conclusions of part (ii) hold on [0, τ ′].

A more general version of this theorem for constrained domains (Theorem 5.1)will be given and proved in section 5.1.

Proof. (i) Conditions (T1) follow by standard arguments.Condition (T2). Associate with X in RN the function

y(s) = Tt−s(X), 0 ≤ s ≤ t.

Thendy

ds(s) = −V (t− s, y(s)), 0 ≤ s ≤ t, y(0) = Tt(X). (4.10)

For each x ∈ RN, the differential equation

dy

ds(s) = −V (t− s, y(s)), 0 ≤ s ≤ t, y(0) = x ∈ RN (4.11)

has a unique solution in C1([0, t];RN). The solutions of (4.11) define the map

x &→ St(x) def= y(t) : RN → RN

such that

∃c > 0, ∀t ∈ [0, τ ], ∀x, y ∈ RN, |St(y)− St(x)| ≤ c|y − x|. (4.12)


In view of (4.10) and (4.11)

St(Tt(X)) = y(t) = Tt−t(X) = X ⇒ St Tt = I on RN .

To obtain the other identity, consider the function

z(r) = y(t− r;x),

where y(·, x) is the solution of (4.11) for some arbitrary x in RN. By definition

dz

dr(r) = V (r, z(r)), z(0) = y(t, x),

and necessarily

x = y(0;x) = z(t) = Tt(y(t;x)) = Tt(St(x))

⇒ Tt St = I on RN ⇒ St = T−1t : RN → RN .

Conditions (T3). The uniform Lipschitz continuity in (T3) follows from (4.12),and we need only show that

∀x ∈ RN, T−1(·, x) ∈ C([0, τ ];RN).

Given t in [0, τ ] pick an arbitrary sequence tn, tn → t. Then for each x ∈ RN

there exists X ∈ RN such that

Tt(X) = x and Ttn(X)→ Tt(X) = x

from the first condition (T1). But

T−1tn

(x)− T−1t (x) = T−1

tn(Tt(X))− T−1

t (Tt(X))

= T−1tn

(Tt(X))− T−1tn

(Ttn(X)) .

By the uniform Lipschitz continuity of T−1t

|T−1tn

(x)− T−1t (x)| = |T−1

tn(Tt(X))− T−1

tn(Ttn(X)) | ≤ c|Tt(X)− Ttn(X)|,

and the last term converges to zero as tn goes to t.(ii) The first part of conditions (V) is satisfied since, for each x ∈ RN and t, s

in [0, τ ],

|V (t, x)− V (s, x)|

≤∣∣∣∣∂T

∂t(t, T−1

t (x))− ∂T

∂t(t, T−1

s (x))∣∣∣∣+∣∣∣∣∂T

∂t(t, T−1

s (x))− ∂T

∂t(s, T−1

s (x))∣∣∣∣

≤ c|T−1t (x)− T−1

s (x)| +∣∣∣∣∂T

∂t(t, T−1

s (x))− ∂T

∂t(s, T−1

s (x))∣∣∣∣ .


Thus from (T3) and (T1) t &→ V (t, x) is continuous at s = t, and hence for all xin RN, V (., x) ∈ C([0, τ ];RN). The Lipschitzian property follows directly from theLipschitzian properties (T1) and (T3): for all x and y in RN,

|V (t, y)− V (t, x)| =∣∣∣∣∂T

∂t(t, T−1

t (y))− ∂T

∂t(t, T−1

t (x))∣∣∣∣

≤ c∣∣T−1

t (y)− T−1t (x)

∣∣ ≤ cc′|y − x|.

This proves that V satisfies condition (V).(iii) From (T1) and (T2) for f(t) = Tt − I and t ≥ s,

f(t)(y)− f(t)(x) =∫ t

0

∂T

∂t(r, Tr(y))− ∂T

∂t(r, Tr(x)) dr,

|f(t)(y)− f(t)(x)| ≤∫ t

0c |Tr(y)− Tr(x)| dr ≤ c2 t |y − x|.

For τ ′ = minτ, 1/(2c2) and 0 ≤ t ≤ τ ′, c(f(t)) ≤ 1/2,

g(t) = T−1t − I = (I − Tt) T−1

t = −f(t) [I + g(t)],

(1− c(f(t))) c(g(t)) ≤ c(f(t)) ⇒ c(g(t)) ≤ 1 and c(T−1t ) ≤ 2,

and the second condition (T3) is satisfied on [0, τ ′]. The first one follows by thesame argument as in part (i). Therefore the conclusions of part (ii) are true on[0, τ ′].

This equivalence theorem says that we can start either from a family of velocityfields V (t) on RN or a family of transformations Tt of RN provided that themap V , V (t, x) = V (t)(x), satisfies (V) or the map T , T (t, X) = Tt(X), satisfies(T1) to (T3).

Starting from V , the family of homeomorphisms Tt(V ) generates the family

Ωtdef= Tt(V )(Ω) = Tt(V )(X) : X ∈ Ω (4.13)

of perturbations of the initial domain Ω. Interior (resp., boundary) points of Ω aremapped onto interior (resp., boundary) points of Ωt. This is the basis of the velocitymethod which will be used to define shape derivatives.

4.2 Perturbations of the IdentityIn examples it is usually possible to show that the transformation T satisfies as-sumptions (T1) to (T3) and construct the corresponding velocity field V definedin (4.9). For instance, consider perturbations of the identity to the first (A = 0) orsecond order: for t ≥ 0 and X ∈ RN,

Tt(X) def= X + tU(X) +t2

2A(X), (4.14)

where U and A are transformations of RN. It turns out that for Lipschitziantransformations U and A, assumptions (T1) to (T3) are satisfied in some interval[0, τ ].


Theorem 4.2. Let U and A be two uniform Lipschitzian transformations of RN:there exists c > 0 such that for all X, Y ∈ RN,

|U(Y )− U(X)| ≤ c|Y −X| and |A(Y )−A(X)| ≤ c|Y −X|.

There exists τ > 0 such that the map T given by (4.14) satisfies conditions (T1) to(T3) on [0, τ ]. The associated velocity V given by

(t, x) &→ V (t, x) = U(T−1

t (x))

+ tA(T−1

t (x))

: [0, τ ]×RN → RN (4.15)

satisfies conditions (V) on [0, τ ].

Remark 4.1.Observe that from (4.14) and (4.15)

V (0) = U, V (0)(x) =∂V

∂t(t, x)

∣∣t=0 = A− [DU ]U, (4.16)

where DU is the Jacobian matrix of U . The term V (0) is an acceleration at t = 0which will always be present even when A = 0, but it can be eliminated by choosingA = [DU ]U .

Proof. (i) By the definition of T in (4.14), t &→ T (t, X) and t &→ ∂T∂t (t, X) = U(X)+

tA(X) are continuous on [0,∞[. Moreover, for all X and Y ,

|T (t, Y )− T (t, X)| ≤[1 + tc +

t2

2c

]|Y −X|

and∣∣∣∣∂T

∂t(t, Y )− ∂T

∂t(t, X)

∣∣∣∣ ≤ [c + tc] |Y −X|.

Thus conditions (T1) are satisfied for any finite τ > 0. To check condition (T2),consider for any Y ∈ RN the mapping h(X) def= Y −[Tt(X)−X]. For any X1 and X2

|h(X2)− h(X1)| ≤ t|U(X2)− U(X1)| +t2

2|A(X2)−A(X1)|

≤ t c |X2 −X1| +t2

2c |X2 −X1| = t c [1 + t/2] |X2 −X1|.

(4.17)

For τ = min1, 1/(4c), and any t, 0 ≤ t ≤ τ , t c [1 + t/2] < 1/2 and h is acontraction. So for all 0 ≤ t ≤ τ and Y ∈ RN, there exists a unique X ∈ RN

such that

Y − [Tt(X)−X] = h(X) = X ⇐⇒ Tt(X) = Y

and Tt is bijective. Therefore (T2) is satisfied in [0, τ ′]. The last part of the proofis the uniform Lipschitzian property of T−1

t . In view of (4.17), for all t, 0 ≤ t ≤ τ ,


t c [1 + t/2] < 1/2 and

|Tt(X2)−X2 − (Tt(X1)−X1)| = |h(X2)− h(X1)| ≤12

|X2 −X1|

⇒ |X2 −X1|− |Tt(X2)− Tt(X1)| ≤12

|X2 −X1|

⇒ |X2 −X1| ≤ 2 |Tt(X2)− Tt(X1)|.

In view of condition (T2) for all x and y,

|T−1t (y)− T−1

t (x)| ≤ 2 |Tt(T−1t (y))− Tt(T−1

t (x))| = 2 |y − x|. (4.18)

To complete our argument we prove the continuity with respect to t for each x. LetX = T−1

t (x). For any s in [0, τ ]

T−1s (x)− T−1

t (x) = T−1s (Tt(X))− T−1

t (Tt(X)) = T−1s (Tt(X))− T−1

s (Ts(X)),

and in view of (4.18)

|T−1s (x)− T−1

t (x)| ≤ 2|Tt(X)− Ts(X)|.

The continuity of T−1s (x) at s = t now follows from the continuity of Ts(X) at s = t.

Thus conditions (T3) are satisfied.

4.3 Equivalence for Special Families of Velocities

In this section we specialize Theorem 4.1 to velocities in Ck,1(RN,RN), Ck+10 (RN,

RN), and Ck+1(RN,RN), k ≥ 0. The following notation will be convenient:

f(t) def= Tt − I, f ′(t) =dTt

dt, g(t) def= T−1

t − I,

whenever T−1t exists and the identities

g(t) = −f(t) T−1t = −f(t) [I + g(t)],

V (t) =dTt

dt T−1

t = f ′(t) T−1t = f ′(t) [I + g(t)].

Recall also for a function F : RN → RN the notation

c(F ) def= supy =x

|F (y)− F (x)||y − x| and ∀k ≥ 1, ck(F ) def=

∑

|α|=k

c(∂αF ).


(i) Given τ > 0 and a velocity field V such that

V ∈ C([0, τ ];Ck(RN,RN)) and ck(V (t)) ≤ c (4.19)


for some constant c > 0 independent of t, the map T given by (4.4)–(4.6)satisfies conditions (T1), (T2), and

f ∈ C1([0, τ ];Ck(RN,RN)) ∩ C([0, τ ];Ck,1(RN,RN)), ck(f ′(t)) ≤ c, (4.20)

for some constant c > 0 independent of t. Moreover, conditions (T3) aresatisfied and there exists τ ′ > 0 such that

g ∈ C([0, τ ′];Ck(RN,RN)), ck(g(t)) ≤ ct, (4.21)

for some constant c independent of t.

(ii) Given τ > 0 and T : [0, τ ] × RN → RN satisfying conditions (4.20) andT (0, ·) = I, there exists τ ′ > 0 such that the velocity field V (t) = f ′(t) T−1

t

satisfies conditions (V) and (4.19) in [0, τ ′].

Proof. We prove the theorem for k = 0. The general case is obtained by inductionover k.

(i) By assumption on V , the conditions (V) given by (4.2) are satisfied andby Theorem 4.1 the corresponding family T satisfies conditions (T1) to (T3).

Conditions (4.20) on f . For any x and s ≤ t

Tt(x)− Ts(x) =∫ t

sV (r) Tr(x) dr,

|Tt(x)− Ts(x)| ≤∫ t

sc|Tr(x)− Ts(x)| + |V (r) Ts(x)| dr,

|f(t)(x)− f(s)(x)| ≤∫ t

sc|f(r)(x)− f(s)(x)| + ∥V (r)∥C dr.

By assumption on V and Gronwall’s inequality,

∀t, s ∈ [0, τ ], ∥f(t)− f(s)∥C ≤ c |t− s|

for another constant c independent of t. Moreover,

|(f(t)− f(s))(y)− (f(t)− f(s))(x)| = |(Tt − Ts)(y)− (Tt − Ts)(x)|

≤∫ t

s|V (r) Tr(y)− V (r) Tr(x)| dr

≤∫ t

sc|(Tr − Ts)(y)− (Tr − Ts)(x)| + c|Ts(y)− Ts(x)| dr

≤∫ t

sc|(f(r)− f(s))(y)− (f(r)− f(s))(x)| + cc′|y − x| dr

for some other constant c′ by the second condition (T1). Again by Gronwall’sinequality there exists another constant c such that

|(f(t)− f(s))(y)− (f(t)− f(s))(x)| ≤ c|t− s| |y − x|⇒ c(f(t)− f(s)) ≤ c |t− s|

⇒ f ∈ C([0, τ ];C0,1(RN,RN)) and ∥f(t)− f(s)∥C0,1 ≤ c |t− s|. (4.22)


Moreover, f ′(t) = V (t) Tt and

|f ′(t)(x)− f ′(s)(x)| ≤ |V (t)(Tt(x))− V (s)(Tt(x))|+ |V (s)(Tt(x))− V (s)(Ts(x))|

≤ ∥V (t)− V (s)∥C + c(V (s)) ∥Tt − Ts∥C≤ ∥V (t)− V (s)∥C + c ∥f(t)− f(s)∥C .

Finally,

|f ′(t)(y)− f ′(t)(x)| ≤ |V (t)(Tt(y))− V (t)(Tt(x))|≤ c(V (t)) |Tt(y)− Ts(x)| ≤ c c(Tt) |y − x|

and c(f ′(t)) ≤ c for some new constant c independent of t. Therefore,

f ∈ C1([0, τ ];C(RN,RN)) and c(f ′(t)) ≤ c.

Conditions (4.21) on g. Since conditions (T1) and (T2) are satisfied thereexists τ ′ > 0 such that conditions (T3) are satisfied by Theorem 4.1 (iii). Moreover,from conditions (4.20)

|g(t)(y)− g(t)(x)| ≤ |f(t)(T−1t (y))− f(t)(T−1

t (x))|≤ c(f(t))|T−1

t (y)− T−1t (x)|

≤ c(f(t)) (|g(t)(y)− g(t)(x)| + |y − x|)⇒ (1− c(f(t))) |g(t)(y)− g(t)(x)| ≤ c(f(t)) |y − x| ≤ ct |y − x|.

Choose a new τ ′′ = minτ ′, 1/(2c). Then for 0 ≤ t ≤ τ ′′, c(g(t)) ≤ 2ct. Now

g(t)− g(s) = −f(t) [I + g(t)] + f(s) [I + g(s)],∥g(t)− g(s)∥C ≤∥f(t) [I + g(t)]− f(t) [I + g(s)]∥C

+ ∥f(t) [I + g(s)]− f(s) [I + g(s)]∥C≤c(f(t))∥g(t)− g(s)∥C + ∥f(t)− f(s)∥C≤ct ∥g(t)− g(s)∥C + ∥f(t)− f(s)∥C .

For t in [0, τ ′′], ct ≤ 1/2, and

∥g(t)− g(s)∥C ≤ 2∥f(t)− f(s)∥C

⇒ g ∈ C([0, τ ′′];C(RN,RN)) and c(g(t)) ≤ 2ct.

The conditions (4.20) on f are satisfied for k = 0. For k = 1 we start from theequation

DTt −DTs =∫ t

sDV (r) Tr DTr dr


and use the fact that DT−1t = [DTt]−1 T−1

t in connection with the identity

Dg(t) = −Df(t) T−1t DT−1

t = −(Df(t)[DTt]−1) T−1t .

(ii) From conditions (4.20) on f , the transformation T satisfies conditions (T1).To check condition (T2) we consider two cases: k ≥ 1 and k = 0. For k ≥ 1 thefunction t &→ Df(t) = DTt − I : [0, τ ] → Ck−1(RN,RN)N is continuous. Hencet &→ det DTt : [0, τ ] → R is continuous and detDT0 = 1. So there exists τ ′ > 0such that Tt is invertible for all t in [0, τ ′] and (T2) is satisfied in [0, τ ′]. In thecase k = 0 consider for any Y the map h(X) = Y − f(t)(X). For any X1 and X2,|h(X2) − h(X1)| ≤ c(f(t)) |X2 − X1|. But by assumption f ∈ C([0, τ ]; C0,1(RN))and c(f(0)) = 0 since f(0) = 0. Hence there exists τ ′ > 0 such that c(f(t)) ≤ 1/2for all t in [0, τ ′] and h is a contraction. So for all Y in RN there exists a unique Xsuch that

Y − [Tt(X)−X] = h(X) = X ⇐⇒ Tt(X) = Y,

Tt is bijective, and condition (T2) is satisfied in [0, τ ′]. By Theorem 4.1 (iii) from(T1) and (T2), there exists another τ ′ > 0 for which conditions (T3) on g and (V)on V (t) = f ′(t) T−1

t are also satisfied. Moreover, we have seen in the proof ofpart (i) that conditions (4.21) on g follow from (T2) and (4.20). Using conditions(4.20) and (4.21),

|V (t)(y)− V (t)(x)| ≤ |f ′(t)(T−1t (y))− f ′(t)(T−1

t (x))|≤ c(f ′(t)) |T−1

t (y)− T−1t (x)|

≤ c(f ′(t)) [1 + c(g(t))] |y − x| ≤ c′|y − x|

and c(V (t)) ≤ c′. Also

|V (t)(x)− V (s)(x)| = |f ′(t)(T−1t (x))− f ′(s)(T−1

s (x))|≤ |f ′(t)(T−1

t (x))− f ′(t)(T−1s (x))|

+ |f ′(t)(T−1s (x))− f ′(s)(T−1

s (x))|≤ c(f ′(t)) |T−1

t (x)− T−1s (x)| + ∥f ′(t)− f ′(s)∥C

≤ c ∥g(t)− g(s)∥C + ∥f ′(t)− f ′(s)∥C .

Therefore, since both g and f ′ are continuous,

V ∈ C([0, τ ′];C(RN,RN)) and c(V (t)) ≤ c

for some constant c independent of t. This proves the result for k = 0. As in part(i), for k = 1 we use the identity

DV (t) = Df ′(t) T−1t DT−1

t = (Df ′(t)[DTt]−1) T−1t

and proceed in the same way. The general case is obtained by induction over k.


We now turn to the case of velocities in Ck0 (RN,RN). As in Chapter 2, it will

be convenient to use the notation Ck0 for the space Ck

0 (RN,RN), Ck(RN) for thespace Ck(RN,RN), and Ck,1(RN) for the space Ck,1(RN,RN).



V ∈ C([0, τ ];Ck0 (RN,RN)), (4.23)

the map T given by (4.4)–(4.6) satisfies conditions (T1), (T2), and

f ∈ C1([0, τ ];Ck0 (RN,RN)). (4.24)

Moreover, conditions (T3) are satisfied and there exists τ ′ > 0 such that

g ∈ C([0, τ ′];Ck0 (RN,RN)). (4.25)


t

satisfies conditions (V) and (4.23) on [0, τ ′].

Proof. As in the proof of Theorem 4.3, we prove only the theorem for k = 1. Thegeneral case is obtained by induction on k, the various identities on f , g, f ′, and V ,and the techniques of Lemma 2.4, Theorems 2.11 and 2.12, and Lemmas 2.5 and2.6 of section 2.5 in Chapter 3.

(i) By the embedding C10 (RN,RN) ⊂ C1(RN,RN) ⊂ C0,1(RN,RN), it follows

from (4.23) that V ∈ C([0, τ ];C0,1(RN,RN)) and condition (4.19) of Theorem 4.3are satisfied. Therefore, conditions (4.20) and (4.21) of Theorem 4.3 are also satis-fied in some interval [0, τ ′], τ ′ > 0.

Conditions (4.24) on f . It remains to show that f(t) and f ′(t) belong to thesubspace C0(RN,RN) of C(RN,RN) and to prove the appropriate properties forDf(t) and Df ′(t). Recall from the proof of the previous theorems that there existsc > 0 such that

|f(t)(x)| ≤ c

∫ t

0|V (r)(x)| dr ≤ c

∫ t

0|(V (r)− V (0))(x)| dr + c t |V (0)(x)|.

By assumption on V (0), for ε > 0 there exists a compact set K such that

∀x ∈ !K, |V (0)(x)| ≤ ε/(2c)

and there exists δ, 0 < δ < 1, such that

∀t, 0 ≤ t ≤ δ, ∥V (r)− V (0)∥C ≤ ε/(2c)⇒ ∀t, 0 ≤ t ≤ δ, ∀x ∈ !K, |f(t)(x)| ≤ ε ⇒ f(t) ∈ C0.

Proceeding in this fashion from the interval [0, δ] to the next interval [δ, 2δ] usingthe inequality

|(f(t)− f(s))(x)| ≤ c

∫ t

s|(V (r)− V (δ))(x)| dr + c |t− δ| |V (δ)(x)|,


the uniform continuity of V

∀t, s, |t− s| < δ, ∥V (t)− V (s)∥C ≤ ε/(2c),

and the fact that V (δ) ∈ C0, that is, that there exists a compact set K(δ) such that

∀x ∈ !K(δ), |V (δ)(x)| ≤ ε/(2c),

we get f(t) ∈ C0, δ ≤ t ≤ 2δ, and hence f ∈ C([0, τ ]; C0). For f ′(t) we makeuse of the identity f ′(t) = V (t) Tt. Again by assumption for any ε > 0 thereexists a compact set K(t) such that |V (t)(x)| ≤ ε on !K(t). Thus by choosing thecompact set K ′

t = T−1t (K(t)), |f ′(t)(x)| ≤ ε on !K ′

t, and f ′ ∈ C([0, τ ]; C0). In orderto complete the proof, it remains to establish the same properties for Df(t) andDf ′(t). The matrix Df(t) is solution of the equations

d

dtDf(t) = DV (t) Tt DTt, Df(0) = 0

⇒ Df ′(t) = DV (t) Tt Df(t) + DV (t) Tt. (4.26)

From the proofs of Theorems 2.11 and 2.12 in section 2.5 of Chapter 3, for each tthe elements of the matrix

A(t) def= DV (t) Tt = DV (t) [I + f(t)]

belong to C0 since DV (t) and f(t) do. By assumption, V ∈ C([0, τ ]; Ck0 ) and V and

all its derivatives ∂αV are uniformly continuous in [0, τ ]×RN. Therefore, for eachε > 0 there exists δ > 0 such that

∀ |t− s| < δ, ∀ |y′ − x′| < δ, |DV (t)(y′)−DV (s)(x′)| < ε.

Pick 0 < δ′ < δ such that

∀ |t− s| < δ′, ∥Tt − Ts∥C = ∥f(t)− f(s)∥C < δ

⇒ ∀x, ∀ |t− s| < δ′, |Tt(x)− Ts(x)| < δ

⇒ |DV (t)(Tt(x))−DV (s)(Ts(x))| < ε

⇒ ∥A(t)−A(s)∥C < ε ⇒ A ∈ C([0, τ ]; (C0)N ).

For each x, Df(t)(x) is the unique solution of the linear matrix equation (4.26). Toshow that Df(t) ∈ (C0)N we first show that Df(t)(x) is uniformly continuous for xin RN. For any x and y

|Df(t)(y)−Df(t)(x)| ≤∫ t

0|V (r, Tr(y))− V (r, Tr(x))| dr

≤∫ t

0c|Tr(y)− Tr(x)| dr

≤ c

∫ t

0|f(r)(y)− f(r)(x)| + |y − x| dr.


But f ∈ C([0, τ ]; C0) is uniformly continuous in (t, x): for each ε > 0 there exists δ,0 < δ < ε/(2cτ), such that

∀ |t− s| < δ, ∀ |y − x| < δ, |f(t)(y)− f(s)(x)| < ε/(2cτ).

Substituting in the previous inequality for each ε > 0, there exists δ > 0 such that

∀ t, ∀ |y − x| < δ, |Df(t)(y)−Df(t)(x)| < ε.

Hence Df(t) is uniformly continuous in RN. Furthermore, from (4.26) we have thefollowing inequality:

|Df(t)(x)| ≤∫ t

0|DV (r)(Tr(x))| |Df(r)(x)| + |DV (r)(Tr(x))| dr

≤ c

∫ t

0(|Df(r)(x)| + 1) dr

(4.27)

since V ∈ C([0, τ ]; C10). By Gronwall’s inequality |Df(t)(x)| ≤ c t for some other

constant c independent of t. Hence Df(t) ∈ C(RN,RN)N . Finally to show thatDf(t) vanishes at infinity we start from the integral form of (4.26):

Df(t)(x) =∫ t

0DV (r)(Tr(x))DTr(x) dr,

|Df(t)(x)| ≤ c

∫ t

0|DV (r)(Tr(x))−DV (r)(x)| + |DV (r)(x)| dr

≤ c′∫ t

0|f(r)(x)| + |DV (r)(x)| dr.

By the same technique as for f(t), it follows that the elements of Df(t) belong toC0 since both f(s) and DV (r) do. Finally for the continuity with respect to t

Df(t)−Df(s) =∫ t

sA(r)Df(r) + A(r) dr,

∥Df(t)−Df(s)∥C ≤∫ t

s∥A(r)∥C ∥Df(r)−Df(s)∥C+ ∥A(r)∥C (1 + ∥Df(r)∥C) dr.

Again, by Gronwall’s inequality, there exists another constant c such that

∥Df(t)−Df(s)∥C ≤ c |t− s|.

Therefore, Df ∈ C([0, τ ]; (C0)N ) and f ∈ C([0, τ ]; C10). For Df ′ we repeat the proof

for f ′ using the identity

Df ′(t) = DV (t) Tt Df(t) + DV (t) Tt

to get

Df ′ ∈ C([0, τ ]; (C0)N ) ⇒ f ′ ∈ C([0, τ ]; C10).


Conditions (4.25) on g. From the remark at the beginning of part (i) of theproof, the conclusions of Theorem 4.3 are true for g, and it remains to check theremaining properties for g and Dg using the identities

g(t) = −f(t) [I + g(t)], Dg(t) = −Df(t) [I + g(t)] (I + Dg(t)).

From the proof of Theorems 2.11 and 2.12 in section 2.5 of Chapter 3, g(t) ∈ C0since Df(t) and g(t) belong to C0. Therefore, g(t) ∈ C1

0 . The continuity follows bythe same argument as for f ′ and g ∈ C([0, τ ]; C1

0).(ii) By assumption from conditions (4.24), conditions (T1) are satisfied. For

(T2) observe that for k ≥ 1 the function t &→ Df(t) = DTt − I : [0, τ ] →Ck−1(RN,RN)N is continuous. Hence t &→ det DTt : [0, τ ] → R is continuousand detDT0 = 1. So there exists τ ′ > 0 such that Tt is invertible for all t in [0, τ ′]and (T2) is satisfied in [0, τ ′]. Furthermore, from the proof of part (i) conditions(T3) and (4.25) on g are also satisfied in some interval [0, τ ′], τ ′ > 0. Therefore,the velocity field

V (t) = f ′(t) T−1t = f ′(t) [I + g(t)]

satisfies the conditions (V) specified by (4.2) in [0, τ ′]. From the proof of The-orems 2.11 and 2.12 in section 2.5 of Chapter 3, V (t) ∈ Ck

0 since f ′(t) and g(t)belong to Ck

0 . By assumption, f ∈ C1([0, τ ]; Ck0 ). Hence f ′ and all its derivatives

∂αf ′, |α| ≤ k, are uniformly continuous on [0, τ ] ×RN; that is, given ε > 0, thereexists δ > 0 such that

∀t, s, |t− s| < δ, ∀y′, x′, |y′ − x′| < δ, |∂αf ′(t)(y′)− ∂αf ′(s)(x′)| < ε.

Similarly g ∈ C([0, τ ′]; Ck0 ) and there exists 0 < δ′ ≤ δ such that

∀t, s, |t− s| < δ′, ∀y, x, |y − x| < δ′, |∂αg(t)(y)− ∂αg(s)(x)| < δ.

Therefore, for |t− s| < δ′

∥T−1t − T−1

s ∥C = ∥g(t)− g(s)∥C < δ,

and since δ′ < δ

∀x, |f ′(t)(T−1t (x))− f ′(t)(T−1

s (x))| < ε

⇒ ∥V (t)− V (s)∥C < ε ⇒ V ∈ C([0, τ ′]; C0).

We then proceed to the first derivative of V ,

DV (t) = Df ′(t) T−1t DT−1

t = Df ′(t) [I + g(t)] [I + Dg(t)],

and by uniform continuity of the right-hand side V ∈ C([0, τ ′]; C10). By induction

on k, we finally get V ∈ C([0, τ ′]; Ck0 ).

The proof of the last theorem is based on the fact that the vector functionsinvolved are uniformly continuous. The fact that they vanish at infinity is not anessential element of the proof. Therefore, the theorem is valid with Ck(RN,RN) inplace of Ck

0 (RN,RN).

5. Constrained Families of Domains 193



V ∈ C([0, τ ];Ck(RN,RN)), (4.28)

the map T given by (4.4)–(4.6) satisfies conditions (T1), (T2), and

f ∈ C1([0, τ ];Ck(RN,RN)). (4.29)

Moreover, conditions (T3) are satisfied and there exists τ ′ > 0 such that

g ∈ C([0, τ ′];Ck(RN,RN)). (4.30)


t

satisfies conditions (V) and (4.28) on [0, τ ′].

5 Constrained Families of DomainsWe now turn to the case where the family of admissible domains Ω is constrainedto lie in a fixed larger subset D of RN or its closure. For instance, D can be anopen set or a closed submanifold of RN.

5.1 Equivalence between Velocities and TransformationsGiven a nonempty subset D of RN, consider a family of transformations

T : [0, τ ]×D → RN (5.1)

for some τ = τ(T ) > 0 with the following properties:

(T1D)∀X ∈ D, T (·, X) ∈ C1([0, τ ];RN) and ∃c > 0,

∀X, Y ∈ D, ∥T (·, Y )− T (·, X)∥C1([0,τ ];RN) ≤ c|Y −X|,(T2D) ∀t ∈ [0, τ ], X &→ Tt(X) = T (t, X) : D → D is bijective,

(T3D)∀x ∈ D, T−1(·, x) ∈ C

([0, τ ];RN) and ∃c > 0,

∀x, y ∈ D, ∥T−1(·, y)− T−1(·, x)∥C([0,τ ];RN) ≤ c|y − x|,

(5.2)

where under assumption (T2D) T−1 is defined from the inverse of Tt as

(t, x) &→ T−1(t, x) def= T−1t (x) : [0, τ ]×D → RN . (5.3)

These three properties are the analogue for D of the same three propertiesobtained for RN. In fact, Theorem 4.1 extends from RN to D by adding oneassumption to (V). Specifically, consider a velocity field

V : [0, τ ]×D → RN (5.4)


for which there exists τ = τ(V ) > 0 such that

(V1D)∀x ∈ D, V (·, x) ∈ C

([0, τ ];RN), ∃c > 0,

∀x, y ∈ D, ∥V (·, y)− V (·, x)∥C([0,τ ];RN) ≤ c|y − x|,(V2D) ∀x ∈ D, ∀t ∈ [0, τ ], ±V (t, x) ∈ TD(x),

(5.5)

where TD(x) is Bouligand’s contingent cone6 to D at the point x in D

TD(X) def=⋂

ε>0

⋂

α>0

⋃

0<h<α

[1h

(D −X) + εB

](5.6)

and B is the unit disk in RN (cf. J.-P. Aubin and A. Cellina [1, p. 176]). Thisdefinition is equivalent to

TD(X) = lim supt0

D −X

t

=

v

∣∣∣∣lim inft0

1tdD(X + tv) = 0

(5.7)

(cf. J.-P. Aubin and H. Frankowska [1, pp. 121–122, 17, 21]). Note that whenD is bounded in RN,

V (·) ∈ C([0, τ ];C(D;RN) ∩ Lip (D;RN)) = C([0, τ ];C0,1(D;RN)).

When D is equal to RN, TD(x) = RN for all x and condition (V2D) can be dropped.When D is equal to the boundary ∂A of a set A of class C1,1 in RN, ∂A is a C1,1-submanifold of RN and

∀x ∈ ∂A, ±V (t, x) ∈ T∂A(x) ⇐⇒ ∀x ∈ ∂A, V (t, x) ·∇bA(x) = 0;

that is, at each point of ∂A, the velocity field is tangent to ∂A: it belongs to thetangent linear space of ∂A.

The next theorem is a generalization of Theorem 4.1 from RN to an arbitraryset D which shows the equivalence between velocity and transformation viewpoints.

Theorem 5.1.

(i) Let τ > 0 and let V be a family of velocity fields satisfying conditions (V1D)and (V2D) and consider the family of transformations

(t, X) &→ T (t, X) = x(t;X) : [0, τ ]×D → RN, (5.8)

where x(·, X) is the solution of

dx

dt(t) = V

(t, x(t)

), 0 ≤ t ≤ τ, x(0) = X. (5.9)

Then the family of transformations T satisfies conditions (T1D) to (T3D).6This is an equivalent characterization of Bouligand’s contingent cone of Definition 2.4 in

section 2.4 of Chapter 2.


(ii) Conversely, given a family of transformations T satisfying conditions (T1D)to (T3D), the family of velocity fields

(t, x) &→ V (t, x) =∂T

∂t

(t, T−1

t (x))

: [0, τ ]×D → RN (5.10)

satisfies conditions (V1D) and (V2D). If, in addition, T (0, ·) = I, then T (·, X)is the solution of (5.9) for that V .

(iii) Given a real number τ > 0 and a map T : [0, τ ]×D → D satisfying assump-tions (T1D) and (T2D) and T (0, ·) = I, then there exists τ ′ > 0 such that theconclusions of part (ii) hold on [0, τ ′].

Remark 5.1.Assumption (V 2D) is a double viability condition. M. Nagumo [1]’s usual viabilitycondition

V (t, x) ∈ TD(x), ∀t ∈ [0, τ ],∀x ∈ D (5.11)

is a necessary and sufficient condition for a viable solution to (5.9):

∀t ∈ [0, τ ],∀X ∈ D, x(t;X) ∈ D(or Tt(D) ⊂ D

)(5.12)

(cf. J.-P. Aubin and A. Cellina [1, p. 174 and p. 180]). Condition (V 2D),

∀t ∈ [0, τ ],∀x ∈ D, ±V (t, x) ∈ TD(x), (5.13)

is a strict viability condition which says that Tt maps D into D and

∀t ∈ [0, τ ], Tt : D → D is a homeomorphism. (5.14)

In particular it maps interior points onto interior points and boundary points ontoboundary points (cf. J. Dugundji [1, pp. 87–88]).

Remark 5.2.Condition (V 2D) is a generalization to an arbitrary set D of the following conditionused by J.-P. Zolesio [12] in 1979: for all x in ∂D,

V (t, x) · n(x) = 0, if the outward normal n(x) exists,0, otherwise.

Proof of Theorem 5.1. (i) Existence and uniqueness of viable solutions to (5.9). Ap-ply M. Nagumo [1]’s theorem to the augmented system on [0, τ ]

⎧⎪⎨

⎪⎩

dx

dt(t) = V

(t, x(t)

), x(0) = X ∈ D,

dx0

dt(t) = 1, x0(0) = 0,

(5.15)

that is, ⎧⎨

⎩

dx

dt(t) = V

(x(t)

),

x(0) = (0, X) ∈ Ddef= R+×D,

(5.16)


where x(t) =(x0(t), x(t)

)∈ RN+1, V (x) =

(1, V (x)

), and

V (x0, x) =

V (x0, x), 0 ≤ x0 ≤ τV (τ, x), τ < x0

, x ∈ D. (5.17)

It is easy to check that systems (5.15) and (5.16) are equivalent on [0, τ ] and thatx(t) =

(t, x(t)

). The new velocity field on V ⊂ RN+1 is continuous at each point

x ∈ D by the first assumption (V 1D) since

V (y)− V (x) =(0, V (y0, y)− V (x0, x)

),

and for 0 ≤ x0, y0 ≤ τ

|V (y0, y)− V (x0, x)| ≤ |V (y0, y)− V (y0, x)| + |V (y0, x)− V (x0, x)|≤ n|y − x| + |V (y0, x)− V (x0, x)|.

In addition,

TD(x) = TR+(x0)× TD(x)

⇒ V (x) =(1, V (x)

)∈ TR+(x0)× TD(x).

Moreover V (D) is bounded and RN+1 is finite-dimensional. By using the versionof M. Nagumo[1]’s theorem given in J.-P. Aubin and A. Cellina [1, Thm. 3,part b), pp. 182–183], there exists a viable solution x to (5.16) for all t ≥ 0. Inparticular,

∀t ∈ [0, τ ], x(t) ∈ D = R+×D,

which is necessarily of the form x(t) = (t, x(t)). Hence there exists a viable solutionx, x(t) ∈ D on [0, τ ], to (5.9). The uniqueness now follows from the Lipschitz con-dition (V 1D). The Lipschitzian continuity (T1D) can be established by a standardargument.

Condition (T2D). Associate with X in D the function

y(s) = Tt−s(X), 0 ≤ s ≤ t.

Thendy

ds(s) = −V

(t− s, y(s)

), 0 ≤ s ≤ t, y(0) = Tt(X). (5.18)

For each x ∈ D, the differential equation

dy

ds(s) = −V

(t− s, y(s)

), 0 ≤ s ≤ t, y(0) = x ∈ D (5.19)

has a unique viable solution in C1([0, t];RN):

∀s ∈ [0, t], y(s) ∈ D, (5.20)

since by assumption (V 2D)

∀t ∈ [0, τ ],∀x ∈ D, −V (t, x) ∈ TD(x).


The proof is the same as above. The solutions of (5.19) define a Lipschitzianmapping

x &→ St(x) = y(t) : D → D

such that

∃c > 0,∀t ∈ [0, τ ],∀x, y ∈ D, |St(y)− St(x)| ≤ c|y − x|. (5.21)

Now in view of (5.18) and (5.19)

St

(Tt(X)

)= y(t) = Tt−t(X) = X ⇒ St Tt = I on D.

To obtain the other identity, consider the function

z(r) = y(t− r;x),

where y(·, x) is the solution of (5.19). By definition,

dz

dr(r) = V

(r, z(r)

), z(0) = y(t, x)

and necessarily

x = y(0;x) = z(t) = Tt

(y(t;x)

)= Tt

(St(x)

)

⇒ Tt St = I on D ⇒ St = T−1t : D → D.

Condition (T3D). The uniform Lipschitz continuity in (T3D) follows from(5.21) and (T2D), and we need only show that

∀x ∈ D, T−1(·, x) ∈ C([0, τ ];RN).

Given t in [0, τ ] choose an arbitrary sequence tn, tn → t. Then for each x ∈ Dthere exists X ∈ D such that

Tt(X) = x and Ttn(X)→ Tt(X) = x

from (T1D). But

T−1tn

(x)− T−1t (x) = T−1

tn

(Tt(X)

)− T−1

t

(Tt(X)

)

= T−1tn

(Tt(X)

)− T−1

tn

(Ttn(X)

).

By the uniform Lipschitz continuity of T−1t

|T−1tn

(x)− T−1t (x)| = |T−1

tn

(Tt(X)

)− T−1

tn

(Ttn(X)

)|

≤ c|Tt(X)− Ttn(X)|,

and the last term converges to zero as tn goes to t.


(ii) The first condition (V1D) is satisfied since for each x ∈ D and t, s in [0, τ ]

|V (t, x)− V (s, x)| ≤∣∣∣∣∂T

∂t

(t, T−1

t (x))− ∂T

∂t

(t, T−1

s (x))∣∣∣∣

+∣∣∣∣∂T

∂t

(t, T−1

s (x))− ∂T

∂t

(s, T−1

s (x))∣∣∣∣

≤ c|T−1t (x)− T−1

s (x)|

+∣∣∣∣∂T

∂t

(t, T−1

s (x))− ∂T

∂t

(s, T−1

s (x))∣∣∣∣ .

The second condition (V1D) follows from (T1D) and (T3D) and the following in-equality: for all x and y in D

|V (t, y)− V (t, x)| =∣∣∣∣∂T

∂t

(t, T−1

t (y))− ∂T

∂t

(t, T−1

t (x))∣∣∣∣

≤ c∣∣T−1

t (y)− T−1t (x)

∣∣ ≤ cc′|y − x|.

To check condition (V2D), recall definition (5.6) of the Bouligand contingent cone:

TD(X) =⋂

ε>0

⋂

α>0

⋃

0<h<α

[1h

(D −X) + εB

],

where B is the unit disk in RN. We first show that

∀x ∈ D, V (t, x) =∂T

∂t

(t, T−1

t (x))∈ TD(x).

By (T2D) Tt is bijective. So it is equivalent to show that

∀X ∈ D,∂T

∂t(t, X) ∈ TD

(Tt(X)

).

For simplicity we use the notation

x(t) = Tt(X) = T (t, X) and x′(t) =∂T

∂t(t, X). (5.22)

By the definition of TD(x(t)), we must prove that

∀ε > 0,∀α > 0,∃h ∈ ]0,α[ ,∃usuch that x′(t) ∈ u + εB and x(t) + hu ∈ D.

Choose δ, 0 < δ < α, such that

∀s, |s− t| < δ ⇒ |x′(s)− s′(t)| < ε.

Then fix t′. For all t such that 0 < t′ − t < δ,

x(t′)− x(t) =∫ t′

t

[x′(s)− x′(t)

]ds + (t′ − t)x′(t)

⇒∣∣∣∣x(t′)− x(t)

t′ − t− x′(t)

∣∣∣∣ < ε.


Therefore choose u = [x(t′)− x(t)]/(t′ − t). Now

x(t) + hu = x(t) +h

t′ − t

[x(t′)− x(t)

]

and choose h = t′ − t since 0 < t′ − t < δ < α:

T (t, X) + (t′ − t)u = T (t, X) +[T (t′, X)− T (t, X)

]= T (t′, X) ∈ D

by assumption on T . This proves (5.22). The second part of (V2D) is

∀x ∈ D, −V (t, x) = −∂T

∂t

(t, T−1

t (x))∈ TD(x),

which is equivalent to proving that

∀X ∈ D, −∂T

∂t(t, X) ∈ TD

(Tt(X)

),

or with the simplified notation

−x′(t) ∈ TD

(x(t)

). (5.23)

We proceed exactly as in the proof of (5.22) except that we choose t′ such that0 < t− t′ < δ, h = t− t′, and u = −[x(t)− x(t′)]/(t− t′). Then

|u + x′(t)| < ε

and x(t) + hu = x(t) + (t− t′)[−(

x(t)− x(t′)t− t′

)]= x(t′) ∈ D

and we get (5.23).(iii) From (T1D) and (T2D)

f(t)(y)− f(t)(x) =∫ t

0

∂T

∂t(r, Tr(y))− ∂T

∂t(r, Tr(x)) dr,

|f(t)(y)− f(t)(x)| ≤∫ t

0c |Tr(y)− Tr(x)| dr ≤ c2 t |y − x|.

For τ ′ = minτ, 1/(2c2) and 0 ≤ t ≤ τ ′, c(f(t)) ≤ 1/2,

g(t) = T−1t − I = (I − Tt) T−1

t = −f(t) [I + g(t)],

(1− c(f(t))) c(g(t)) ≤ c(f(t)) ⇒ c(g(t)) ≤ 1 and c(T−1t ) ≤ 2,

and the second condition (T3) is satisfied on [0, τ ′]. The first one follows by thesame argument as in part (i). Therefore, the conclusions of part (ii) are true on[0, τ ′]. This completes the proof of the theorem.


5.2 Transformation of Condition (V2D) into a Linear ConstraintCondition (V2D) is equivalent to

∀t ∈ [0, τ ],∀x ∈ D, V (t, x) ∈ −TD(x) ∩ TD(x) (5.24)

since TD(x) = TD(x). If TD(x) were convex, then the above intersection wouldbe a closed linear subspace of RN. This is true when D is convex. In that caseTD(x) = CD(x), where CD(x) is the Clarke tangent cone and

LD(x) = −CD(x) ∩ CD(x) (5.25)

is a closed linear subspace of RN. This means that (V2D) reduces to

∀t ∈ [0, τ ],∀x ∈ D, V (t, x) ∈ LD(x). (5.26)

It turns out that for continuous vector fields V (t, ·), the equivalence of (V2D) and(5.26) extends to arbitrary domains D. This equivalence generally fails for discon-tinuous vector fields. Other equivalences might be possible between TD and someintermediary convex cone between CD and TD, but there is no evidence so far ofthat fact. For smooth bounded open domains Ω, the two cones coincide and thecondition reduces to V · n = 0, n the normal to ∂Ω, and V (t, x) belongs to thetangent space to ∂Ω in each point of ∂Ω.

Theorem 5.2.

(i) Given a velocity field V satisfying (V1D), condition (V2D) is equivalent to

(V2C) ∀t ∈ [0, τ ], ∀x ∈ D, V (t, x) ∈ LD(x) = −CD(x) ∩ CD(x),

where CD(x) is the (closed convex ) Clarke tangent cone to D at x,

CD(x) =

⎧⎨

⎩v ∈ RN : limt0

y−→D

x

dD(y + tv)t

= 0⎫⎬

⎭ , (5.27)

and −→D

denotes the convergence in D.

(ii) LD(x) is a closed linear subspace of RN.

Proof. (i) The equivalence of (V2D) and (V2C) is a direct consequence of the fol-lowing lemma.

Lemma 5.1. Given a vector field W ∈ C(D;RN), the following two conditions areequivalent:

∀x ∈ D, W (x) ∈ TD(x), (5.28)

∀x ∈ D, W (x) ∈ CD(x). (5.29)


(ii) The set LD(x) is closed as the intersection of two closed sets. To showthat it is linear, we show that for all α ∈ R and V ∈ LD(x), αV ∈ LD(x), and forall V and W in LD(x), V + W ∈ LD(x). Since ±CD(x) are cones,

∀α ∈ R, ∀V ∈ LD(x), ±|α|V ∈ CD(x)⇒ ±αV ∈ CD(x)⇒ αV ∈ LD(x).

By convexity of ±CD(x)

∀V, W ∈ LD(x), ±(V + W ) ∈ CD(x)⇒ V + W ∈ LD(x).

This completes the proof of the theorem.

Proof of Lemma 5.1. Assume that (5.29) is verified. By definition CD(x) ⊂ TD(x)and (5.29) ⇒ (5.28). Conversely, either x is an isolated point and TD(x) = 0 =CD(x) or there are points x = y ∈ D such that y −→

Dx. In the latter case we

know that

lim infy−→

D

xTD(y) = CD(x)

(cf., for instance, J.-P. Aubin and H. Frankowska [1, Thm. 4.1.10, sect. 4.1.5,p. 130]). Since W is continuous in D and (5.28) is satisfied, then for each x ∈ D

W (x) = limy−→

D

xW (y) ∈ lim inf

y−→D

xTD(y) = CD(x)

and (5.28) implies (5.29).

Remark 5.3.Lemma 5.1 essentially says that for continuous vector fields we can relax the condi-tion of M. Nagumo [1]’s theorem from (V2D) involving the Bouligand contingentcone to (V2C) involving the smaller Clarke convex tangent cone. In dimensionN = 3, LD(x) is 0, a line, a plane, or the whole space.

Notation 5.1.In what follows, it will be convenient to introduce the spaces and subspaces

L =V : [0, τ ]×RN → RN : V satisfies (V) on RN , (5.30)

and for an arbitrary domain D in RN

LD = V : [0, τ ]×D → RN : V satisfies (V1D) and (V2C) on D

. (5.31)

For any integers k ≥ 0 and m ≥ 0 and any compact subset K of RN define thefollowing subspaces of L:

Vm,kK = Cm

([0, τ ],Dk

(K,RN)

)∩ L, (5.32)

where Dk(K,RN) is the space of k-times continuously differentiable transformationsof RN with compact support in K. In all cases Vm,k

K ⊂ LK . As usual D∞(K,RN)will be written D(K,RN).


6 Continuity of Shape Functions along Velocity FlowsThroughout this section, assume that Ω is a closed subset or an open crack-free7

subset of RN and that B is a Banach space.Since the construction of the Courant metric is relatively abstract, can the

continuity of a shape functional be characterized in a more direct way? The answeris yes thanks to the sharp theorems giving the equivalence between transformationsand velocities in the previous section without restricting the analysis to the subgroupGΘ defined by (2.7) in section 2. This is due to the fact that continuity andsemiderivatives are local properties and that the velocity character naturally popsout as a sequence of transformations goes to identity.

In this section, we give a characterization via velocities of the continuity of ashape functional of the form

Ω′ &→ J(Ω′) : X (Ω)→ B, X (Ω) = F (Ω) : ∀F ∈ F(Θ) (6.1)

with respect to the Courant metric topology of the quotient F(Θ)/G(Ω) using theequivalence Theorems 4.3, 4.4, and 4.5. Checking the continuity along flows of avelocity is easier and more natural. As in Definition 3.1, assume that J verifies thecompatibility condition of Definition 3.1:

∀H, F ∈ F(Θ) such that H(F (Ω)) = F (Ω), J(H(F (Ω))) = J(F (Ω)).

We specifically consider the continuity of shape functionals with respect to theCourant metric associated with Θ equal to Ck+1

0 (RN,RN), Ck+1(RN,RN), andCk,1(RN,RN), k ≥ 0, but similar equivalences are true for the other spaces ofChapter 3.

Remark 6.1.An obvious consequence of Theorems 6.1, 6.2, and 6.3 is that when a shape func-tional is semidifferentiable in the sense of Definition 3.2 (Hadamard), it is continuousat that point for the Courant metric topology in complete analogy with the classicalEuclidean calculus.

We begin with the space Ck0 (RN) = Ck

0 (RN,RN) of A. M. Micheletti [1].

Theorem 6.1. Let k ≥ 1 be an integer, B be a Banach space, and Ω be a nonemptyopen subset of RN. Consider a shape functional J : NΩ([I]) → B defined in aneighborhood NΩ([I]) of [I] in F(Ck

0 (RN,RN))/G(Ω). Then J is continuous at Ωfor the Courant metric if and only if

limt0

J(Tt(Ω)) = J(Ω) (6.2)

for all families of velocity fields V (t) : 0 ≤ t ≤ τ satisfying the condition

V ∈ C([0, τ ];Ck0 (RN,RN)). (6.3)

7Cf. Definition 7.1 (ii) of Chapter 8.

6. Continuity of Shape Functions along Velocity Flows 203

Proof. It is sufficient to prove the theorem for a real-valued function J . The Banachspace case is readily obtained by considering the new real-valued function j(T ) =|J(T (Ω))− J(Ω)|.

(i) If J is dG-continuous at Ω, then for all ε > 0 there exists δ > 0 such that

∀T, [T ] ∈ NΩ([I]), dG([T ], [I]) < δ, |J(T (Ω))− J(Ω)| < ε.

Condition (6.3) on V coincides with condition (4.23) of Theorem 4.4, which impliesconditions (4.24) and (4.25):

f ∈ C1([0, τ ];Ck0 (RN,RN)) and g ∈ C([0, τ ];Ck

0 (RN,RN))

⇒ ∥Tt − I∥Ck(RN) → 0 and ∥T−1t − I∥Ck(RN) → 0 as t→ 0.

But by definition of the metric dG

d([Tt], [I]) ≤ ∥T−1t − I∥Ck + ∥Tt − I∥Ck → 0 as t→ 0,

and we get the convergence (6.2) of the function J(Tt(Ω)) to J(Ω) as t goes to zerofor all V satisfying (6.3).

(ii) Conversely, it is sufficient to prove that for any sequence [Tn] such thatdG([Tn], [I]) goes to zero, there exists a subsequence such that

J(Tnk(Ω))→ J(I(Ω)) = J(Ω) as k →∞.

Indeed let

ℓ = lim infn→∞

J(Tn(Ω)) and L = lim supn→∞

J(Tn(Ω)).

By definition of the liminf, there is a subsequence, still indexed by n, such thatℓ = lim infn→∞ J(Tn(Ω)). But since there exists a subsequence Tnk of Tn suchthat J(Tnk(Ω))→ J(Ω), then necessarily ℓ = J(Ω). The same reasoning applies tothe limsup, and hence the whole sequence J(Tn(Ω)) converges to J(Ω) and we havethe continuity of J at Ω.

We prove that we can construct a velocity V associated with a subsequence ofTn verifying conditions (4.23) of Theorem 4.4 and hence conditions (6.3). By thesame techniques as in Theorems 2.8 and 2.6 of Chapter 3, associate with a sequenceTn such that dG([Tn], [I])→ 0 a subsequence, still denoted by Tn, such that

∥fn∥Ck + ∥gn∥Ck = ∥T−1n − I∥Ck + ∥Tn − I∥Ck ≤ 2−2(n+2).

For n ≥ 1 set tn = 2−n and observe that tn−tn+1 = −2−(n+1). Define the followingC1-interpolation in (0, 1/2]: for t in [tn+1, tn],

Tt(X) def= Tn(X) + p

(tn+1 − t

tn+1 − tn

)(Tn+1(X)− Tn(X)), T0(X) def= X,

where p ∈ P 3[0, 1] is the polynomial of order 3 on [0, 1] such that p(0) = 1 andp(1) = 0 and p(1)(0) = 0 = p(1)(1).


Conditions on f . By definition, for all t, 0 ≤ t ≤ 1/2, f(t) = Tt−I ∈ Ck0 (RN).

Moreover, for 0 < t ≤ 1/2

Ttn(X) = Tn(X), Ttn+1(X) = Tn+1(X),∂T

∂t(tn, X) = 0 =

∂T

∂t(tn+1, X),

∂T

∂t(t, X) =

Tn+1(X)− Tn(X)|tn − tn+1|

p(1)(

tn+1 − t

tn+1 − tn

),

f ′(t) = ∂T/∂t(t, ·) ∈ Ck0 (RN), and f(·)(X) = T (·, X) − I ∈ C1((0, 1/2];RN). By

definition, f(0) = 0. For each 0 < t ≤ 1/2 there exists n ≥ N such that tn+1 ≤ t ≤tn and

∥f(t)− f(0)∥Ck = ∥f(t)∥Ck = ∥fn + p

(tn+1 − t

tn+1 − tn

)(fn+1 − fn)∥Ck

≤ 2∥fn∥Ck + ∥fn+1∥Ck ≤ 2 2−2(n+2) + 2−2(n+3) ≤ 2−(n+1) ≤ t.

Define at t = 0, f ′(t) = 0. By the same technique, there exists a constant c > 0,and for each 0 < t ≤ 1/2 there exists n ≥ N such that tn+1 ≤ t ≤ tn and

∥f ′(t)− f ′(0)∥Ck = ∥f ′(t)∥Ck

=∥∥∥∥∂T

∂t(t, ·)

∥∥∥∥Ck

≤ c∥Tn+1 − Tn∥Ck

|tn+1 − tn| = c∥fn+1 − fn∥Ck

2−(n+1)

≤ c 2 2−2(n+2)/2−(n+1) ≤ c 2−12−(n+1) ≤ c 2−(n+1) ≤ c t

⇒ ∥f ′(t)∥Ck ≤ ct.

So for each X the functions t &→ f(t)(X) and t &→ Tt(X) belong to C1([0, 1/2];RN).By uniform Ck-continuity of the Tn’s and the continuity with respect to t for eachX, it follows that f ∈ C1([0, 1/2]; Ck

0 (RN)) and the condition (4.24) of Theorem 4.4is satisfied. Hence the corresponding velocity V satisfies conditions (4.23). FinallyV satisfies conditions (6.3) and by (6.2) for all ε > 0 there exists δ > 0 such that

∀t, 0 ≤ t ≤ δ, |J(Tt(Ω))− J(Ω)| < ε.

In particular there exists N > 0 such that for all n ≥ N , tn ≤ δ, and

∀n ≥ N, |J(Tn(Ω))− J(Ω)| = |J(Ttn(Ω))− J(Ω)| < ε,

and this proves the d-continuity for the subsequence Tn.

The case of the Courant metric associated with the space Ck(RN) = Ck(RN,RN)is a corollary to Theorem 6.1.

Theorem 6.2. Let k ≥ 1 be an integer, B be a Banach space, and Ω be a nonemptyopen subset of RN. Consider a shape functional J : NΩ([I]) → B defined in aneighborhood NΩ([I]) of [I] in F(Ck(RN,RN))/G(Ω).


Then J is continuous at Ω for the Courant metric if and only if

limt0

J(Tt(Ω)) = J(Ω) (6.4)

for all families of velocity fields V (t) : 0 ≤ t ≤ τ satisfying the condition

V ∈ C([0, τ ];Ck(RN,RN)). (6.5)

The proof of the theorem for the Courant metric topology associated with thespace Ck,1(RN) = Ck,1(RN,RN) is similar to the proof of Theorem 6.1 with obviouschanges.

Theorem 6.3. Let k ≥ 0 be an integer, Ω be a nonempty open subset of RN, andB be a Banach space. Consider a shape functional J : NΩ([I]) → B defined in aneighborhood NΩ([I]) of [I] in F(Ck,1(RN,RN))/G(Ω). Then J is continuous at Ωfor the Courant metric if and only if

limt0

J(Tt(Ω)) = J(Ω) (6.6)

for all families V (t) : 0 ≤ t ≤ τ of velocity fields in Ck,1(RN,RN) satisfying theconditions

V ∈ C([0, τ ];Ck(RN,RN)) and ck(V (t)) ≤ c (6.7)

for some constant c independent of t.

Proof. As in the proof of Theorem 6.1, it is sufficient to prove the theorem for areal-valued function J .

(i) If J is dG-continuous at Ω, then for all ε > 0 there exists δ > 0 such that

∀T, [T ] ∈ NΩ([I]), dG([T ], [I]) < δ, |J(T (Ω))− J(Ω)| < ε.

Under condition (6.7) from Theorem 4.3

f = T − I ∈ C([0, τ ]; Ck,1(RN)) and ∥Tt − I∥Ck,1 → 0 as t→ 0,

g(t) = T−1t − I ∈ Ck,1(RN) and ∥T−1

t − I∥Ck,1 ≤ ct→ 0 as t→ 0.

But by definition of the metric dG

dG([Tt], [I]) ≤ ∥T−1t − I∥Ck,1 + ∥Tt − I∥Ck,1 → 0 as t→ 0,

and we get the convergence (6.6) of the function J(Tt(Ω)) to J(Ω) as t goes to zerofor all V satisfying (6.7).

(ii) Conversely, as in the proof of Theorem 6.1, it is sufficient to prove thatgiven any sequence [Tn] such that δ([Tn], [I])→ 0 there exists a subsequence suchthat

J(Tnk(Ω))→ J(I(Ω)) = J(Ω) as k →∞.


By the same techniques as in Theorems 2.8 and 2.6 of Chapter 3, associate witha sequence Tn such that dG([Tn], [I]) → 0 a subsequence, still denoted by Tn,such that

∥fn∥Ck,1 + ∥gn∥Ck,1 = ∥T−1n − I∥Ck,1 + ∥Tn − I∥Ck,1 ≤ 2−2(n+2).

For n ≥ 1 set tn = 2−n and observe that tn−tn+1 = −2−(n+1). Define the followingC1-interpolation in (0, 1/2]: for t in [tn+1, tn],

Tt(X) def= Tn(X) + p

(tn+1 − t

tn+1 − tn

)(Tn+1(X)− Tn(X)), T0(X) def= X,

where p ∈ P 3[0, 1] is the polynomial of order 3 on [0, 1] such that p(0) = 1 andp(1) = 0 and p(1)(0) = 0 = p(1)(1).

Conditions on f . By definition for all t, 0 ≤ t ≤ 1/2, f(t) = Tt−I ∈ Ck,1(RN).Moreover, for 0 < t ≤ 1/2

Ttn(X) = Tn(X), Ttn+1(X) = Tn+1(X),∂T

∂t(tn, X) = 0 =

∂T

∂t(tn+1, X),

∂T

∂t(t, X) =

Tn+1(X)− Tn(X)|tn − tn+1|

p(1)(

tn+1 − t

tn+1 − tn

),

f ′(t) = ∂T/∂t ∈ Ck,1(RN), and f(·)(X) = T (·, X) − I ∈ C1((0, 1/2];RN). Bydefinition, f(0) = 0. For each 0 < t ≤ 1/2 there exists n ≥ N such that tn+1 ≤t ≤ tn and

∥f(t)− f(0)∥Ck,1 = ∥f(t)∥Ck,1 =∥∥∥∥fn + p

(tn+1 − t

tn+1 − tn

)(fn+1 − fn)

∥∥∥∥Ck,1

≤ 2∥fn∥Ck,1 + ∥fn+1∥Ck,1 ≤ 2 2−2(n+2) + 2−2(n+3) ≤ 2−(n+1) ≤ t.

Define at t = 0, f ′(t) = 0. By the same technique there exists a constant c > 0,and for each 0 < t ≤ 1/2 there exists n ≥ N such that tn+1 ≤ t ≤ tn and

∥f ′(t)− f ′(0)∥Ck,1 = ∥f ′(t)∥Ck,1

=∥∥∥∥∂T

∂t(t, ·)

∥∥∥∥Ck,1

≤ c∥Tn+1 − Tn∥Ck,1

|tn+1 − tn| = c∥fn+1 − fn∥Ck,1

2−(n+1)

≤ c 2 2−2(n+2)/2−(n+1) ≤ c 2−12−(n+1) ≤ c 2−(n+1) ≤ c t

⇒ ∥f ′(t)∥Ck,1 ≤ ct and ck(f ′(t)) ≤ ct

and for each X the functions t &→ f(t)(X) and t &→ Tt(X) belong to C1([0, 1/2];RN).By uniform Ck-continuity of the Tn’s and the continuity with respect to t for eachX, it follows that f ∈ C1([0, 1/2]; Ck(RN)). Moreover, it can be shown that

ck(f ′(t)) ≤ ct ⇒ ∀t, s ∈ [0, τ ], ck(f(t)− f(s)) ≤ c′|t− s|

for some c′ > 0. The result is straightforward for k = 0 and then the general casefollows by induction on k. As a result f ∈ C([0, 1/2]; Ck,1(RN)) and the condition


(4.20) of Theorem 4.3 is satisfied. Hence the corresponding velocity V satisfiesconditions (4.19). Finally the velocity field V satisfies conditions (6.7), and by (6.6)for all ε > 0 there exists δ > 0 such that

∀t ≤ δ, |J(Tt(Ω))− J(Ω)| < ε.

In particular there exists N > 0 such that for all n ≥ N , tn ≤ δ and

∀n ≥ N, |J(Tn(Ω))− J(Ω)| = |J(Ttn(Ω))− J(Ω)| < ε,

and we have the dG-continuity for the subsequence Tn.

Remark 6.2.The conclusions of Theorems 6.1, 6.2, and 6.3 are generic. They also have theircounterpart in the constrained case. The difficulty lies in the second part of thetheorem, which requires a special construction to make sure that the family oftransformations Tt : 0 ≤ t ≤ τ constructed from the sequence Tn are homeo-morphisms of D.

Chapter 5

Metrics via CharacteristicFunctions

1 IntroductionThe constructions of the metric topologies of Chapter 3 are limited to families ofsets which are the image of a fixed set by a family of homeomorphisms or diffeo-morphisms. If it is connected, bounded, or Ck, k ≥ 1, then the images will havethe same properties under Ck-transformations of RN. In this chapter we consider-ably enlarge the family of available sets by relaxing the smoothness assumption tothe mere Lebesgue measurability and even just measurability to include Hausdorffmeasures. This is done by identifying Ω ⊂ RN with its characteristic function

χΩ(x) def=

1, if x ∈ Ω0, if x /∈ Ω

if Ω = ∅ and χ∅

def= 0. (1.1)

We first introduce an Abelian group structure1 on characteristic functions of measur-able2 subsets of a fixed holdall D in section 2 and then construct complete met-ric spaces of equivalence classes of measurable characteristic functions via the Lp-norms, 1 ≤ p <∞, that turn out to be algebraically and topologically equivalent.

Starting with section 3, we specialize to the Lebesgue measure in RN. Thecomplete topology generated by the metric defined as the Lp-norm of the differenceof two characteristic functions is called the strong topology in section 3.1. In sec-tion 3.2 we consider the weak Lp-topology on the family of characteristic functions.Weak limits of sequences of characteristic functions are functions with values in [0, 1]which belong to the closed convex hull of the equivalence classes of measurable char-acteristic functions. This occurs in optimization problems where the function to beoptimized depends on the solution of a partial differential equation on the variabledomain, as was shown, for instance, by F. Murat [1] in 1971. It usually corre-sponds to the appearance of a microstructure or a composite material in mechanics.

1Cf. the group structure on subsets of a holdall D in section 2.2 of Chapter 2.2Measurable in the sense of L. C. Evans and R. F. Gariepy [1]. It simultaneously covers

Hausdorff and Lebesgue measures.

209

210 Chapter 5. Metrics via Characteristic Functions

One important example in that category was the analysis of the optimal thicknessof a circular plate by K.-T. Cheng and N. Olhoff [2, 1] in 1981. The readeris referred to the work of F. Murat and L. Tartar [3, 1], initially publishedin 1985, for a comprehensive treatment of the calculus of variations and homog-enization. Section 3.3 deals with the question of finding a nice representative inthe equivalence class of sets. Can it be chosen open? We introduce the measuretheoretic representative and characterize its interior, exterior, and boundary. It willbe used later in section 6.4 to prove the compactness of the family of Lipschitziandomains verifying a uniform cone property. Section 3.4 shows that the family ofconvex subsets of a fixed bounded holdall is closed in the strong topology.

The use of the Lp-topologies is illustrated in section 4 by revisiting the optimaldesign problem studied by J. Cea and K. Malanowski [1] in 1970. By relaxingthe family of characteristic functions to functions with values in the interval [0, 1]a saddle point formulation is obtained. Functions with values in the interval [0, 1]can also be found in the modeling of a dam by H. W. Alt [1] and H. W. Altand G. Gilardi [1], where the “liquid saturation” in the soil is a function withvalues in [0, 1]. Another problem amenable to that formulation is the buckling ofcolumns as will be illustrated in section 5 using the work of S. J. Cox and M. L.Overton [1]. It is one of the very early optimal design problems, formulated byLagrange in 1770.

The Caccioppoli or finite perimeter sets of the celebrated Plateau problem arerevisited in section 6. Their characteristic function is a function of bounded varia-tion. They provide the first example of compact families of characteristic functionsin Lp(D)-strong, 1 ≤ p < ∞. This was developed mainly by R. Caccioppoli [1]and E. De Giorgi [1] in the context of J. A. F. Plateau [1]’s problem of minimalsurfaces. Section 6.3 exploits the embedding3 of BV(D) ∩ L∞(D) into the Sobolevspaces W ε,p(D), 0 < ε < 1/p, p ≥ 1, to introduce a cascade of complete metricsbetween Lp(D) and BV(D) on characteristic functions. In section 6.4 we show thatthe family of Lipschitzian domains in a fixed bounded holdall verifying the uniformcone property of section 6.4.1 in Chapter 2 is also compact. This condition natu-rally yields a uniform bound on the perimeter of the sets in the family and hencecan be viewed as a special case of the first compactness theorem. Section 7 gives anexample of the use of the perimeter in the Bernoulli free boundary problem and inparticular for the water wave. There the energy associated with the surface tensionof the water is proportional to the perimeter, that is, the surface area of the freeboundary.

2 Abelian Group Structure on MeasurableCharacteristic Functions

2.1 Group Structure on Xµ(RN)From section 2.2 in Chapter 2, the Abelian group structure on the subsets of RN

extends to the corresponding family of their characteristic functions.

3BV(D) is the space of functions of bounded variations in D.

2. Abelian Group Structure on Measurable Characteristic Functions 211

Theorem 2.1. Let ∅ = D ⊂ RN. The space of characteristic functions X(D)endowed with the multiplication and the neutral multiplicative element χ∅ = 0

χA χBdef= |χA − χB | = χ(A∩!B)∪(B∩!A) = χAB (2.1)

is an Abelian group, that is,

χA χB = χB χA, χA χ∅ = χA, χA χA = χ∅, χ−1A = χA. (2.2)

Proof. Note that the underlying group structure on subsets of D is the one ofsection 2.2 of Chapter 2. The product is well-defined in X(D) since it is the charac-teristic function of a subset of A∪B in D. The first two identities (2.2) are obvious.As for the inverse

χA χB = χ∅ ⇔ |χA − χB | = 0 ⇔ B = A,

and there is a unique multiplicative inverse χ−1A = χA or χA χA = χ∅.

Remark 2.1.We chose the product terminology , but we could have chosen to call that algebraicoperation an addition ⊕.

2.2 Measure SpacesDefinition 2.1.Let X be a set and P(X) be the collection of subsets of X.

(i) A mapping µ : P(X)→ [0,∞] is called a measure4 on X if

(a) µ(∅) = 0, and(b) µ(A) ≤

∑∞k=1 µ(Ak), whenever A ⊂ ∪∞

k=1Ak.

(ii) A subset A ⊂ X is σ-finite with respect to µ if it can be written A = ∪∞k=1Bk,

where Bk is µ-measurable and µ(Bk) <∞ for k = 1, 2, . . . .

(iii) A measure µ on RN is a Radon measure if µ is Borel regular and µ(K) <∞for each compact set K ⊂ RN.

The Lebesgue measure mN on RN is a Radon measure since RN is σ-finitewith respect to mN . The s-dimensional Hausdorff measure Hs, 0 ≤ s < N , on RN

(cf. section 3.2.3 in Chapter 2) is Borel regular but not necessarily a Radon measuresince bounded Hs-measurable subsets of RN are not necessarily σ-finite with respectto Hs. Yet, a Borel regular measure µ can be made σ-finite by restricting it to aµ-measurable set A ⊂ RN such that µ(A) < ∞ or, more generally, to a σ-finiteµ-measurable set A ⊂ RN. For smooth submanifolds of dimension s, s an integer,Hs gives the same area as the integral of the canonical density in section 3.2.2 ofChapter 2.

4A measure in the sense of L. C. Evans and R. F. Gariepy [1, Chap. 1]. It is called an outermeasure in most texts.


Given a µ-measurable subset D ⊂ RN, let Lp(D, µ) denote the Banach space ofequivalence classes of µ-measurable functions such that

∫|f |p dµ <∞, 1 ≤ p <∞,

and let L∞(D, µ) be the space of equivalence classes of µ-measurable functions suchthat ess supD|f | <∞. Denote by [A]µ the equivalence class of µ-measurable subsetsof D that are equal almost everywhere and

Xµ(D) def= χA : A ⊂ D is µ-measurable .

Since A B is µ-measurable for A and B µ-measurable, [A]µ : A ⊂ D is µ-measur-able is a subgroup of P(D) and

Xµ(D) ∩ L1(D, µ) = χA : A ⊂ D is µ-measurable and µ(A) <∞ (2.3)

is a group in Lp(D, µ) with respect to . When µ = mN , we drop the subscript µ.

2.3 Complete Metric for Characteristic Functions inLp-Topologies

Now that we have revealed the underlying group structure of Xµ(D) ∩ L1(D, µ),we construct a metric on that group. We have a structure similar to the Courantmetric of A. M. Micheletti [1] in Chapter 3 on groups of transformations of RN.We identify equivalence classes of µ-measurable subsets of D and characteristicfunctions via the bijection [A]µ ↔ χA.

Theorem 2.2. Let 1 ≤ p < ∞, let µ be a measure on RN, and let ∅ = D ⊂ RN

be µ-measurable.

(i) Xµ(D) ∩ L1(D, µ) is closed in Lp(D, µ). The function

ρD([A2]µ, [A1]µ) def= ∥χA2 − χA1∥Lp(D,µ)

defines a complete metric structure on the Abelian group Xµ(D)∩L1(D, µ) thatmakes it a topological group. If µ(D) <∞, then Xµ(D)∩L1(D, µ) = Xµ(D).

(ii) If, in addition, D is σ-finite with respect to µ for a family Dk of µ-measurablesubsets of D such that µ(Dk) < ∞, for all k ≥ 1, then Xµ(D) is closed inLp

loc(D, µ) and

ρ([A2]µ, [A1]µ) def=∞∑

k=1

12n

∥χA2 − χA1∥Lp(Dk,µ)

1 + ∥χA2 − χA1∥Lp(Dk,µ)

defines a complete metric structure on the Abelian group Xµ(D) that makes ita topological group. When µ is a Radon measure on RN, the assumption thatD is σ-finite with respect to µ can be dropped.

Remark 2.2.For the Lebesgue measure mN and D measurable (resp., measurable and bounded),X(D) is a topological Abelian group in Lp

loc(D,mN ) (resp., Lp(D,mN )). For Haus-dorff measures Hs, 0 ≤ s < N , the theorem does not say more.

2. Abelian Group Structure on Measurable Characteristic Functions 213

Proof. (i) Let χAn be a Cauchy sequence of µ-measurable characteristic functionsin Xµ(D) ∩ L1(D, µ) ⊂ Lp(D, µ) converging to some f ∈ Lp(D, µ). There existsa subsequence, still denoted by χAn, such that χAn(x) → f(x) in D except fora subset Z of zero µ-measure. Hence 0 = χAn(x) (1 − χAn(x)) → f(x) (1 − f(x)).Define the set

Adef= x ∈ D\Z : f(x) = 1.

Clearly A is µ-measurable and χA = f on D\Z since f(x) (1 − f(x)) = 0 onD\Z. Hence f = χA almost everywhere on D, χAn → χA in Lp(D, µ), and χA ∈Xµ(D) ∩ L1(D, µ). Finally, the metric is compatible with the group structure

ρD([Ω2]µ, [Ω1]µ) def= ∥χΩ2 − χΩ1∥Lp = ∥χΩ2 χ−1Ω1∥Lp = ∥χΩ1 χ−1

Ω2∥Lp

as in Chapter 3 and Xµ(D) ∩ L1(D, µ) is a topological group.(ii) From part (i) using the Frechet topology associated with the family of

seminorms qk(f) = ∥f∥Lp(Dk), k ≥ 1.

For p = 1 and µ-measurable subsets A1 and A2 in D, the metric ρD([A2]µ, [A1]µ)is the µ-measure of the symmetric difference

A1∆A2 = ("A1 ∩A2) ∪ ("A2 ∩A1)

and for 1 ≤ p <∞ all the topologies are equivalent on Xµ(D) ∩ L1(D, µ).

Theorem 2.3. Let 1 ≤ p < ∞, let µ be a measure on RN, and let ∅ = D ⊂ RN

be µ-measurable.

(i) The topologies induced by Lp(D, µ) on Xµ(D)∩L1(D, µ) are all equivalent for1 ≤ p <∞.

(ii) If, in addition, D is σ-finite with respect to µ for a family Dk of µ-measurablesubsets of D such that µ(Dk) <∞, for all k ≥ 1, then the topologies inducedby Lp

loc(D, µ) on Xµ(D) are all equivalent for 1 ≤ p <∞.

Proof. For D µ-measurable and 1 ≤ p <∞, Xµ(D)∩Lp(D, µ) = Xµ(D)∩L1(D, µ),and for any χ and χ in Xµ(D) ∩ L1(D, µ)

∥χ− χ∥pLp(D,µ) =∫

D|χ− χ(x)|p dµ =

∫

D|χ− χ(x)| dµ = ∥χ− χ∥L1(D,µ),

since χ(x) and χ(x) are either 0 or 1 almost everywhere in D. Therefore the linear(identity) mapping between Xµ(D)∩Lp(D, µ) and Xµ(D)∩L1(D, µ) is bicontinuousand the topologies are equivalent.

Remark 2.3.At this stage, it is not clear what the tangent space to this topological Abeliangroup is. If µ is a Radon measure, then for all ϕ ∈ D(RN)

1µ(Bε(x))

∫

RN

[χA χBε(x)

] χ−1

A ϕ dµ =1

µ(Bε(x))

∫

RNχBε(x) ϕ dµ


and by the Lebesgue–Besicovitch differentiation theorem

1µ(Bε(x))

∫

RN

[χA χBε(x)

] χ−1

A ϕ dµ→ ϕ(x), µ a.e. in RN .

3 Lebesgue Measurable Characteristic FunctionsIn this section we specialize to the Lebesgue measure and the corresponding familyof characteristic functions

X(RN) def=χΩ : ∀Ω Lebesgue measurable in RN . (3.1)

Clearly X(RN) ⊂ L∞(RN), and for all p ≥ 1, X(RN) ⊂ Lploc(R

N). Also associatewith ∅ = D ⊂ RN the set

X(D) def= χΩ : ∀Ω Lebesgue measurable in D . (3.2)

The above definitions are special cases of the definitions of section 2.3, and Theo-rems 2.2 and 2.3 apply to mN as a Radon measure.

3.1 Strong Topologies and C∞-ApproximationsIn view of the equivalence Theorem 2.3, we introduce the notion of strong conver-gence.

Definition 3.1. (i) If D is a bounded measurable subset of RN, a sequence χnin X(D) is said to be strongly convergent in D if it converges in Lp(D)-strongfor some p, 1 ≤ p <∞.

(ii) A sequence χn in X(RN) is said to be locally strongly convergent if it con-verges in Lp

loc(RN)-strong for some p, 1 ≤ p <∞.

The following approximation theorem will also be useful

Theorem 3.1. Let Ω be an arbitrary Lebesgue measurable subset of RN. Thereexists a sequence Ωn of open C∞-domains in RN such that

χΩn → χΩ in L1loc(R

N).

Proof. The construction of the family of C∞-domains Ωn can be found in manyplaces (cf., for instance, E. Giusti [1, sect. 1.14, p. 10 and Lem. 1.25, p. 23]).Associate with χ = χΩ the sequence of convolutions fn = χ∗ρn for a sequence ρnof symmetric mollifiers. By construction, 0 ≤ fn ≤ 1. For t, 0 < t < 1, define

Fntdef=x ∈ RN : fn(x) > t

.

By definition, fn − χ > t in Fnt\Ω, χ− fn > 1− t in Ω\Fnt, and

|χ− χFnt | ≤1

mint, 1− t |χ− fn| a.e. in RN .

3. Lebesgue Measurable Characteristic Functions 215

By Sard’s Theorem 4.3 of Chapter 2, the set Fnt is a C∞-domain for almost allt in (0, 1). Fix α, 0 < α < 1/2, and choose a sequence tn such that, for all n,α ≤ tn ≤ 1− α, and define Ωn

def= Fntn . Therefore

|χΩ − χΩn | ≤ 1mintn, 1− tn |χΩ − fn| ≤ 1

α|χΩ − fn| a.e. in RN

and for any bounded measurable subset D of RN

∥χΩ − χΩn∥L1(D) ≤ α−1 ∥χΩ − fn∥L1(D),

which goes to zero as n goes to infinity.

3.2 Weak Topologies and MicrostructuresSome shape optimization problems lead to apparent paradoxes. Their solution isno longer a geometric domain associated with a characteristic function, but a fuzzydomain associated with the relaxation of a characteristic function to a function withvalues ranging in [0, 1]. The intuitive notion of a geometric domain is relaxed to thenotion of a probability distribution of the presence of points of the set. When theunderlying problem involves two different materials characterized by two constantsk1 = k2, the occurrence of such a solution can be interpreted as the mixing orhomogenization of the two materials at the microscale. This is also referred to asa composite material or a microstructure. Somehow this is related to the fact thatthe strong convergence needs to be relaxed to the weak Lp-convergence and thespace X(D) needs to be suitably enlarged.

Even if X(D) is strongly closed and bounded in Lp(D), it is not stronglycompact. However, for 1 < p <∞ its closed convex hull co X(D) is weakly compactin the reflexive Banach space Lp(D). In fact,

co X(D) = χ ∈ Lp(D) : χ(x) ∈ [0, 1] a.e. in D . (3.3)

Indeed, by definition co X(D) ⊂ χ ∈ Lp(D) : χ(x) ∈ [0, 1] a.e. in D. Converselyany χ that belongs to the right-hand side of (3.3) can be approximated by a sequenceof convex combinations of elements of X(D). Choose

χn =n∑

m=1

1nχBnm , Bnm =

x : χ(x) ≥ m

n

,

for which |χn(x) − χ(x)| < 1/n. The elements of co X(D) are not necessarilycharacteristic functions of a domain; that is, the identity

χ(x) (1− χ(x)) = 0 a.e. in D

is not necessarily satisfied.We first give a few basic results and then consider a classical example from

the theory of homogenization of differential equations.


Lemma 3.1. Let D be a bounded open subset of RN, let K be a bounded subsetof R, and let

K def= k : D → R : k is measurable and k(x) ∈ K a.e. in D .

(i) For any p, 1 ≤ p < ∞, and any sequence kn ⊂ K the following statementsare equivalent:

(a) kn converges in L∞(D)-weak⋆;

(b) kn converges in Lp(D)-weak;

(c) kn converges in D(D)′,

where D(D)′ is the space of scalar distributions on D.

(ii) If K is bounded, closed, and convex, then K is convex and compact in L∞(D)-weak⋆, Lp(D)-weak, and D(D)′.

The above results remain true in the vectorial case when K is the set of mappingsk : D → K for some bounded subset K ⊂ Rp and a finite integer p ≥ 1.

Proof. (i) It is clear that (a) ⇒ (b) ⇒ (c). To prove that (c) ⇒ (a) recall thatsince K is bounded, there exists a constant c > 0 such that K ⊂ cB, where Bdenotes the unit ball in RN. By density of D(D) in L1(D), any ϕ in L1(D) can beapproximated by a sequence ϕm ⊂ D(D) such that ϕm → ϕ in L1(D). So foreach ε > 0 there exists M > 0 such that

∀m ≥M, ∥ϕm − ϕ∥L1(D) ≤ε

4c.

Moreover, there exists N > 0 such that

∀n ≥ N,∀ℓ ≥ N,

∣∣∣∣∫

DϕM (kn − kℓ) dx

∣∣∣∣ ≤ε

2.

Hence for each ε > 0, there exists N such that for all n ≥ N and ℓ ≥ N∣∣∣∣∫

Dϕ(kn − kℓ) dx

∣∣∣∣ ≤∣∣∣∣∫

DϕM (kn − kℓ) dx

∣∣∣∣+∣∣∣∣∫

D(ϕ− ϕM )(kn − kℓ) dx

∣∣∣∣

≤ ε

2+ 2c∥ϕ− ϕM∥L1(D) ≤ ε.

(ii) When K is bounded, closed, and convex, K is also bounded, closed,and convex. Since L2(D) is a Hilbert space, K is weakly compact. In view ofthe equivalences of part (i), K is also sequentially compact in all the other weaktopologies.5

5In a metric space the compactness is equivalent to the sequential compactness. For the weaktopology we use the fact that if E is a separable normed space, then, in its topological dual E′, anyclosed ball is a compact metrizable space for the weak topology. Since K is a bounded subset of thenormed reflexive separable Banach space Lp(D), 1 ≤ p < ∞, the weak compactness of K coincideswith its weak sequential compactness (cf. J. Dieudonne [1, Vol. 2, Chap. XII, sect. 12.15.9, p. 75]).


In view of this equivalence we adopt the following terminology.

Definition 3.2.Let ∅ = D ⊂ RN be bounded open. A sequence χn in X(D) is said to be weaklyconvergent if it converges for some topology between L∞(D)-weak⋆ and D(D)′.

It is interesting to observe that working with the weak convergence makessense only when the limit element is not a characteristic function.

Theorem 3.2. Let the assumptions of the theorem be satisfied. Let χn and χbe elements of X(RN) (resp., X(D)) such that χn χ weakly in L2(D) (resp.,L2

loc(RN)). Then for all p, 1 ≤ p <∞,

χn → χ strongly in Lp(D)(resp., Lp

loc(RN)).

Proof. It is sufficient to prove the result for D. The strong L2(D)-convergencefollows from the property

∫

D|χn|2 dx =

∫

Dχn1 dx→

∫

Dχ1 dx =

∫

D|χ|2 dx

⇒∫

D|χn − χ|2dx =

∫

D|χn|2 − 2χχn + |χ|2dx→

∫

D|χ|2 − 2χχ+|χ|2dx = 0.

The convergence for any p ≥ 1 now follows from Theorem 2.2.

Working with the weak convergence creates new phenomena and difficulties.For instance when a characteristic function is present in the coefficient of the higher-order term of a differential equation the weak convergence of a sequence of charac-teristic functions χn to some element χ of co X(D),

χn χ in L2(D)-weak ⇒ χn χ in L∞(D)-weak⋆,

does not imply the weak convergence in H1(D) of the sequence y(χn) of solutionsto the solution of the differential equation corresponding to y(χ),

y(χn) y(χ) in H1(D)-weak.

By compactness6 of the injection of H1(D) into L2(D) this would have impliedconvergence in L2(D)-strong:

y(χn)→ y(χ) in L2(D)-strong.

This fact was pointed out in 1971 by F. Murat [1] in the following example,which will be rewritten to emphasize the role of the characteristic function.

6This is true since D is a bounded Lipschitzian domain. Examples of domains between twospirals can be constructed where the injection is not compact.


Example 3.1.Consider the following boundary value problem:

⎧⎨

⎩

− d

dx

(k

dy

dx

)+ ky = 0 in D = ]0, 1[ ,

y(0) = 1 and y(1) = 2,(3.4)

where

K = k = k1(x)χ+ k2(x)(1− χ) : χ ∈ X(D) (3.5)

with

k1(x) = 1−√

12− x2

6, k2(x) = 1 +

√12− x2

6. (3.6)

This is equivalent to

K = k ∈ L∞(0, 1) : k(x) ∈ k1(x), k2(x) a.e. in [0, 1] . (3.7)

Associate with the integer p ≥ 1 the function

kp(x) =

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

1−√

12− x2

6,

m

p< x ≤ 2m + 1

2p,

1 +

√12− x2

6,

2m + 12p

< x ≤ m + 1p

,

0 ≤ m ≤ p− 1, (3.8)

and the characteristic function

χp(x) =

⎧⎪⎪⎨

⎪⎪⎩

1,m

p< x ≤ 2m + 1

2p,

0,2m + 1

2p< x ≤ m + 1

p,

0 ≤ m ≤ p− 1. (3.9)

It is readily seen that kp = k1 χp + k2 (1 − χp). He shows that for each p, kp ∈ K(resp., χp ∈ X(D)) and that

kp k∞ = 1(

resp., χp 12

)in L∞(0, 1)-weak⋆, (3.10)

1kp

12

[1k1

+1k2

]=

11/2 + x2/6

in L∞(0, 1)-weak ⋆ . (3.11)

Moreover,

yp y in H1(0, 1)-weak, (3.12)

where yp denotes the solution of (3.4) corresponding to k = kp and y the solutionof the boundary value problem

⎧⎪⎨

⎪⎩

− d

dx

[(12

+x2

6

)dy

dx

]+ y = 0 in ]0, 1[ ,

y(0) = 1, y(1) = 2.

(3.13)


Define the function

kH(x) =12

+x2

6, (3.14)

which corresponds to

χH(x) =12

[1 +

√12− x2

6

]∈ co X(D). (3.15)

Notice that kH appears in the second-order term and k∞ in the zeroth-order termin (3.13):

− d

dx

[kH

dy

dx

]+ k∞y = 0 in ]0, 1[ . (3.16)

It is easy to check that

y(x) = 1 + x2 in [0, 1], (3.17)

which is not equal to the solution y∞ of (3.4) for the weak limit k∞ = 1:

y∞(x) =2(ex − e−x) + e1−x − e−(1−x)

e− e−1 . (3.18)

To our knowledge this was the beginning of the theory of homogenization inFrance. This is only one part of F. Murat [1]’s example. He also constructs anobjective function for which the lower bound is not achieved by an element of K.This is a nonexistence result.

The above example uses space varying coefficients k1(x) and k2(x). However,it is still valid for two positive constants k1 > 0 and k2 > 0. It is easy to show that

k∞ =k1 + k2

2, χ∞ =

12, and

1kH

=12

[1k1

+1k2

], χH =

k2

k1 + k2,

and the solution y of the boundary value problem (3.16) is given by

y(x) =2 sinh cx + sinh c(1− x)

sinh c, where c =

k1 + k2

2√

k1k2≥ 1,

and the solution y∞ by

y∞(x) =2 sinhx + sinh(1− x)

sinh 1.

Thus for k1 = k2, or equivalently c > 1, y = y∞.In fact, using the same sequence of χp’s, the sequence yp = y(χp) of solutions

of (4.5) weakly converges to yH = y(χH), which is different from the solutiony∞ = y(χ∞) for k1 = k2. This is readily seen by noticing that since χ∞ and χH

are constant

−∆(kH yH) = χ∞ f = −∆(k∞ y∞)

⇒ kH yH = k∞ y∞ ⇒ yH =(k1 + k2)2

4k1 k2y∞ = y∞, for k1 = k2.


Despite this example, we shall see in the next section that in some cases we obtainthe existence (and uniqueness) of a maximizer in co X(D) which belongs to X(D).

3.3 Nice or Measure Theoretic RepresentativeSince the strong and weak Lp-topologies are defined on equivalence classes [Ω] of(Lebesgue) measurable subsets Ω of RN, it is natural to ask if there is a nicerepresentative that is generic of the class [Ω]. For instance we have seen in Chapter 2that within the equivalence class of a set of class Ck there is a unique open and aunique closed representative and that all elements of the class have the same interior,boundary, and exterior. The same question will again arise for finite perimeter sets.As an illustration of what is meant by a nice representative, consider the smilingand the expressionless suns in Figure 5.1. The expressionless sun is obtained byadding missing points and lines “inside” Ω and removing the rays “outside” Ω.This “restoring/cleaning” operation can be formalized as follows.

Ω Ω

Figure 5.1. Smiling sun Ω and expressionless sun Ω.

Definition 3.3.Associate with a Lebesgue measurable set Ω in RN the sets

Ω0def=x ∈ RN : ∃ρ > 0 such that m(Ω ∩B(x, ρ)

)= 0

,

Ω1def=x ∈ RN : ∃ρ > 0 such that m

(Ω ∩B(x, ρ)

)= m

(B(x, ρ)

),

Ω•def=x ∈ RN : ∀ρ > 0 such that 0 < m

(Ω ∩B(x, ρ)

)< m

(B(x, ρ)

)

and the measure theoretic exterior O, interior I, and boundary ∂∗Ω

O def=

x ∈ RN : limr0

m(B(x, r) ∩ Ω)m(B(x, r))

= 0

,

I def=

x ∈ RN : limr0

m(B(x, r) ∩ Ω)m(B(x, r))

= 1

,

∂∗Ω def=

x ∈ RN : 0 ≤ lim inf

r0

m(B(x, r) ∩ Ω)m(B(x, r))

< lim supr0

m(B(x, r) ∩ Ω)m(B(x, r))

≤ 1

.

We shall say that I is the nice or measure theoretic representative of Ω.


The six sets Ω0, Ω1, Ω•, O, I, and ∂∗Ω are invariant for all sets in the equiv-alence class [Ω] of Ω. They are two different partitions of RN

Ω0 ∪ Ω1 ∪ Ω• = RN and O ∪ I ∪ ∂∗Ω = RN (3.19)

like int "Ω ∪ int Ω ∪ ∂Ω = RN. The next theorem links the six invariant sets anddescribes some of the interesting properties of the measure theoretic representative.

Theorem 3.3. Let Ω be a Lebesgue measurable set in RN and let Ω0, Ω1, Ω•, O,I, and ∂∗Ω be the sets constructed from Ω in Definition 3.3.

(i) The sets O, I, and ∂∗Ω are invariants of the equivalence class [Ω]. Moreover,they are Borel measurable and

χI = χΩ, χO = χ!Ω, and χ∂∗Ω = 0 a.e. in RN .

(ii) The sets Ω0, Ω1, and Ω• are invariants of the equivalence class [Ω]. The setsΩ0 and Ω1 are open, Ω• is closed, and

int Ω ⊂ Ω1 = int I, int "Ω ⊂ Ω0 = int O, ∂Ω ⊃ "I ∩ "O = Ω• ⊃ ∂∗Ω, (3.20)∂∗Ω ∪ ∂O ∪ ∂I ⊂ Ω• ⊂ ∂Ω. (3.21)

(iii) The condition m(∂Ω) = 0 implies m(∂I) = 0. When m(∂I) = 0, then int Iand I can be chosen as the respective open and closed representatives of Ω. Inparticular, this is true for Lipschitzian sets and hence for convex sets.

Proof. (i) All the sets involved depend only on χΩ almost everywhere. They areinvariants of the equivalence class [Ω]. From L. C. Evans and R. F. Gariepy [1,Lem. 2, sect. 5.11, p. 222, Cor. 3, sect. 1.7.1, p. 45], I, O, and ∂∗Ω are Borelmeasurable and m(I\Ω ∪ Ω\I) = 0, and

limr0

m(B(x, r) ∩ Ω)m(B(x, r))

= 1 a.e. in Ω, limr0

m(B(x, r) ∩ Ω)m(B(x, r))

= 0 a.e. in "Ω.

Therefore, m(∂∗Ω) = 0, χI = χΩ, and χO = χ!Ω almost everywhere in RN.(ii) We use the proof given by E. Giusti [1, Prop. 4.1, pp. 42–43] for Ω0

and Ω1. To show that Ω0 is open, pick any point x in Ω0. By construction,there exists ρ > 0 such that m

(Ω ∩ B(x, ρ)

)= 0. For any y ∈ B(x, ρ) (that

is, |y − x| < ρ), choose ρ0 = ρ − |x − y| > 0. Then B(y, ρ0) ⊂ B(x, ρ) andm(B(y, ρ0)∩Ω

)≤ m

(B(x, ρ)∩Ω

)= 0. So B(x, ρ) ⊂ Ω0 and Ω0 is open. Similarly,

by repeating the argument for "Ω, we obtain that Ω1 is open. By complementarity,Ω• is closed.

By definition, int Ω ⊂ Ω1 ⊂ I and, since Ω1 is open, Ω1 ⊂ int I. For allx ∈ int I, there exists r > 0 such that B(x, r) ⊂ I and

m(B(x, r)

)= m

(B(x, r) ∩ I

)= m

(B(x, r) ∩ Ω

).

Hence int I ⊂ Ω1 and necessarily int I = Ω1. Similarly, int "Ω ⊂ Ω0 = int O. Bytaking the union term by term of the two chains int Ω ⊂ int I ⊂ Ω1 = int I ⊂ I


and int "Ω ⊂ Ω0 = int O ⊂ O and then the complement, we get ∂Ω ⊃ "I ∩ "O =Ω• ⊃ ∂∗Ω. From the first two relations (3.20), Ω1 = int I implies "Ω1 = "I ⊂ O andΩ0 = int O implies "Ω0 = "O ⊂ I. Therefore, Ω• = "Ω1 ∩ "Ω0 ⊃ "I ∩ I = ∂I andsimilarly Ω• = "Ω1 ∩ "Ω0 ⊃ "O ∩O = ∂O.

(iii) From part (ii).

The inclusions (3.20) indicate that the measure theoretic interior and exteriorare enlarged and that the measure theoretic boundary is reduced. In general thoseoperations do not commute with the set theoretic operations. For instance, they donot commute with the closure, as can be seen from the following simple example.

Example 3.2.Consider the set Ω of all rational numbers in [0, 1]. Then

O = Ω0 = RI = Ω1 = ∅

∂∗Ω = Ω• = ∅

∣∣∣∣∣∣∣⇒ I(Ω) = ∅,

O(Ω) = (Ω)0 = R \[0, 1]

I(Ω) = (Ω)1 = ]0, 1[

∂∗(Ω) = (Ω)• = 0, 1

∣∣∣∣∣∣∣⇒ I(Ω) = ]0, 1[ .

However, the complement operation commutes with the other set of operationsand determines the same sets

("Ω)0 = Ω1, ("Ω)1 = Ω0, ("Ω)• = Ω•, (3.22)I("Ω) = O(Ω), I(Ω) = O("Ω), ∂∗(Ω) = ∂∗("Ω). (3.23)

The next theorem is a companion to Lemma 5.1 in Chapter 2.

Theorem 3.4. The following conditions are equivalent:

(i) int Ω = Ω and int "Ω = "Ω.

(ii) int Ω = Ω1, int "Ω = Ω0, ∂Ω = ∂Ω0 = ∂Ω1 (= Ω•).

If either of the above two conditions is satisfied, then

int I = int Ω, int O = int "Ω, "I ∩ "O = ∂Ω = ∂Ω = ∂"Ω.

Proof. (i) ⇒ (ii). First note that Ω1 = "(Ω0 ∪ Ω•) = "Ω0 ∩ "Ω• ⊂ "Ω0. Similarly,Ω0 ⊂ "Ω1. From the first two relations (3.20), int Ω ⊂ Ω1 and int "Ω ⊂ Ω0.Combining them with the two identities in (i), Ω = int Ω ⊂ Ω1 ⊂ "Ω0 and "Ω =int "Ω ⊂ Ω0 ⊂ "Ω1. By taking the intersection of the two chains, ∂Ω ⊂ "Ω0∩"Ω1 ="(Ω0 ∪ Ω1) = Ω•. But int Ω ∪ int "Ω ∪ ∂Ω and Ω1 ∪ Ω0 ∪ Ω• are two disjointpartitions of RN such that int Ω ⊂ Ω1, int "Ω ⊂ Ω0, and ∂Ω ⊂ Ω•. We concludethat int Ω = Ω1, int "Ω = Ω0, ∂Ω = Ω•, and ∂Ω = Ω• = ∂Ω0 = ∂Ω1 since,


by Lemma 5.1 in Chapter 2, int Ω = Ω and int "Ω = "Ω imply ∂ int Ω = ∂Ω =∂ int "Ω.

(ii) ⇒ (i). By assumption,

Ω = int Ω ∪ ∂Ω = Ω1 ∪ ∂Ω1 = Ω1 = int Ω,

"Ω = int "Ω ∪ ∂Ω = Ω0 ∪ ∂Ω0 = Ω0 = int "Ω

⇒ ∂Ω = Ω ∩ "Ω = int Ω ∩ "Ω = ""Ω ∩ "Ω = ∂ "Ω

⇒ ∂Ω = Ω ∩ "Ω = Ω ∩ int "Ω = Ω ∩ "Ω = ∂ Ω.

3.4 The Family of Convex SetsConvex sets will play a special role here and in the subsequent chapters. We shallsay that an equivalence class [Ω] of Lebesgue measurable subsets of D is convex ifthere exists a convex Lebesgue measurable subset Ω∗ of D such that [Ω] = [Ω∗]. Wealso introduce the notation

C(D) def= χΩ : Ω convex subset of D . (3.24)

Theorem 3.5. Let Ω = ∅ be a convex subset of RN.

(i) Ω and int Ω are convex.7

(ii) "(int Ω) = "Ω = "Ω = int "Ω.If int Ω = ∅, then int Ω = Ω.If int Ω = ∅ and Ω = RN, then ∂Ω = ∅ and int "Ω = ∅.

(iii) Given a measurable (resp., bounded measurable) subset D in RN

∀1 ≤ p <∞, C(D) is closed in Lploc(D) (resp., Lp(D)).

Proof. (i) For all x and y in Ω, there exist xn and yn in Ω such that xn → xand yn → y. Then for all λ ∈ [0, 1]

Ω ∋ λxn + (1− λ) yn → λx + (1− λ) y ∈ Ω.

If int Ω = ∅, then for all x and y in int Ω, there exist rx > 0 and ry > 0 such thatB(x, rx) ⊂ Ω and B(y, ry) ⊂ Ω. So for all λ ∈ [0, 1]

xλdef= λx + (1− λ)y ∈B(xλ,λrx + (1− λ)ry)

⊂ λB(x, rx) + (1− λ)B(y, ry) ⊂ Ω

and xλ ∈ int Ω.(ii) By definition, int Ω = ""Ω and "int Ω = "Ω. If Ω = RN, "Ω ⊂ "Ω = ∅.

This implies int "Ω = "Ω = "Ω = ∅ and int Ω = Ω. If Ω = RN and int Ω = ∅,7By convention ∅ is convex.


from Theorem 5.6 in Chapter 2, Ω is locally Lipschitzian, and from Theorem 5.4 inChapter 2, ∂Ω = ∅, int "Ω = ∅, int Ω = Ω, and "Ω = "Ω. Finally, if int Ω = ∅,then "Ω = RN and Ω belongs to a linear subspace L of RN of dimension strictlyless than N . Therefore "Ω ⊃ "Ω ⊃ "L, RN = "Ω ⊃ "Ω ⊃ "L = RN, and "Ω = "Ω.

(iii) It is sufficient to prove the result for D bounded. For any Cauchy sequenceχn in C(D), there exist a sequence of convex sets Ωn and χΩ in X(D) such thatχΩn → χΩ in Lp(D). In particular, there exists a subsequence χk = χΩnk

such thatχk(x)→ χΩ(x) almost everywhere in D. Define

Ω∗ def= x ∈ D : χk(x)→ 1 as k →∞.

To show that Ω∗ is convex consider x and y in Ω∗. There exists K ≥ 1 such that

∀k ≥ K, |χk(x)− 1| < 1/2, |χk(y)− 1| < 1/2,

and, since χk is either 0 or 1, for all k ≥ K, χk(x) = 1 = χk(y). By convexity forall λ ∈ [0, 1], xλ = λx + (1− λ)y ∈ Ωnk and

∀k ≥ K, χk(λx + (1− λ)y) = 1→ 1 ⇒ xλ ∈ Ω∗.

But χk(x)→ χΩ(x) almost everywhere and

limk→∞

χk(x) ∈ 0, 1 a.e.

So for x ∈ D\Ω∗, χk(x) → 0 = χΩ∗(x) almost everywhere in D. By the Lebesgue-dominated convergence theorem, χk → χΩ∗ in Lp(D) and necessarily χΩ = χΩ∗ .This means that [Ω] is a convex equivalence class.

We shall show in section 6.1 (Corollary 1) that for bounded domains D, C(D)is also compact for the Lp(D) topology, 1 ≤ p <∞.

3.5 Sobolev Spaces for Measurable DomainsThe lack of a priori smoothness on Ω may introduce technical difficulties in theformulation of some boundary value problems. However, it is possible to relaxsuch boundary value problems from smooth bounded open connected domains Ω tomeasurable domains (cf. J.-P. Zolesio [7]). For instance consider the homogeneousDirichlet boundary value problem

−y = f in Ω, y = 0 on Γ (3.25)

over a bounded open connected domain Ω with a boundary Γ of class C1 andassociate with its solution y = y(Ω) the objective function and volume constraint

J(Ω) =12

∫

Ω|y − g|2 dx,

∫

Ωdx = π. (3.26)

There is a priori no reason to assume that an optimal (minimizing) domain Ω∗ isof class C1 or is connected. So the problem must be suitably relaxed to a large


enough class of domains, which preserves the meaning of the underlying functionspaces, the well-posedness of the original problem, and the volume.

To extend problem (3.25)–(3.26) to Lebesgue measurable sets, we first haveto make sense of the Sobolev space for measurable subsets Ω of D.

Theorem 3.6. Let D be an open domain in RN. For any Lebesgue measurablesubset Ω of RN, the spaces

H1• (Ω; D) def=

ϕ ∈ H1

0 (D) : (1− χΩ)∇ϕ = 0 a.e. in D

, (3.27)

H1⋄ (Ω; D) def=

ϕ ∈ H1

0 (D) : (1− χΩ)ϕ = 0 a.e. in D

(3.28)

are closed subspaces of H10 (D) and hence Hilbert spaces.8 Similarly, for any χ ∈

co X(D),

H1• (χ;D) def=

ϕ ∈ H1

0 (D) : (1− χ)∇ϕ = 0 a.e. in D

, (3.29)

H1⋄ (χ;D) def=

ϕ ∈ H1

0 (D) : (1− χ)ϕ = 0 a.e. in D

(3.30)

are also closed subspaces of H10 (D) and hence Hilbert spaces. Furthermore9

H1⋄ (Ω; D) ⊂ H1

• (Ω; D), H1⋄ (χ;D) ⊂ H1

• (χ;D).

Proof. We give only the proof for H1• (Ω; D). Let ϕn in H1

• (Ω; D) be a Cauchysequence. It converges to an element ϕ in the H1

0 (D)-topology. Hence ∇ϕnconverges to ∇ϕ in L2(D). But for all n

(1− χΩ)∇ϕn = 0 in L2(D).

By Schwarz’s inequality the map ϕ 8→ (1− χΩ)∇ϕ is continuous and

(1− χΩ)∇ϕ = 0 in L2(D).

So finally ϕ ∈ H1• (Ω; D) and H1

• (Ω; D) is a closed subspace of H10 (D).

Assuming that D is bounded, the variational problems⎧⎨

⎩

find y = y(Ω) ∈ H1• (Ω; D) such that

∀ϕ ∈ H1• (Ω; D),

∫

D∇y ·∇ϕ dx =

∫

DχΩfϕ dx,

(3.31)

⎧⎨

⎩

find y = y(Ω) ∈ H1⋄ (Ω; D) such that

∀ϕ ∈ H1⋄ (Ω; D),

∫

D∇y ·∇ϕ dx =

∫

DχΩfϕ dx

(3.32)

8Observe that for two open domains in the same equivalence class, the spaces defined by (3.27)–(3.28) coincide. Therefore their functions do not see cracks in the underlying domain. The caseof cracks will be handled in Chapter 8 by capacity methods.

9From L. C. Evans and R. F. Gariepy [1, Thm. 4, p. 130], ∇ϕ = 0 a.e. on ϕ = 0.


now make sense and have unique solutions for measurable subsets Ω of D (or evenχ ∈ co X(D)), and the associated objective function

J(Ω) = h(χΩ, y(Ω)

), h(χ,ϕ) def=

12

∫

Dχ|ϕ− g|2 dx (3.33)

can be minimized over all measurable subsets Ω of D (or all χ ∈ co X(D)) withfixed measure m(Ω) = π.

The above problems are now well-posed, and their restriction to smoothbounded open connected domains coincides with the initial problem (3.25)–(3.26).Indeed, if Ω is a connected open domain with a boundary Γ of class C1, then Γhas a zero Lebesgue measure and the definition (3.28) of H1

• (Ω; D) coincides withH1

⋄ (Ω; D). Therefore, the problem specified by (3.31)–(3.33) is a well-defined ex-tension of problem (3.25)–(3.26). In general, when Ω contains holes, the elementsof H1

• (Ω; D) are not necessarily equal to zero on the boundary of each hole and canbe equal to different constants from hole to hole, as in physical problems involvinga potential.

Example 3.3.Let D = ]−2, 2[ , Ω = ]−2,−1[ ∪ ]1, 2[ , and f = 1 on D. The solution of (3.31) isgiven by

⎧⎪⎨

⎪⎩

− d2y

dx2 = 1 on ]−2,−1[ ,

y(−2) = 0,dy

dx(−1) = 0,

⎧⎪⎨

⎪⎩

− d2y

dx2 = 1 on ]1, 2[ ,

y(2) = 0,dy

dx(1) = 0,

y =12

on [−1, 1],

⇒ y(x) =

⎧⎪⎨

⎪⎩

−(x + 2)x2 in ]−2,−1[ ,

12 in [−1, 1],−(x− 2)x

2 in ]1, 2[ .

Example 3.4.Let D = B(0, 2), the open ball of radius 2 centered in 0 in R2, H = B(0, 1),Ω = "DH, and f = 1. In polar coordinates the solution of (3.31) is given by

⎧⎪⎪⎨

⎪⎪⎩

− 1r

d

dr

(rdy

dr

)= 1 in ]1, 2[ ,

y(2) = 0,dy

dr(1) = 0,

andy(r) =12

ln12

+34

in [0, 1],

or explicitly

y(r) =

12 ln r

2 + 1− r2

4 in ]1, 2[ ,12 ln 1

2 + 34 in [0, 1].


The last example is a special case that retains the characteristics of the one-dimensional example. In higher dimensions the normal derivative is not necessarilyzero on the “internal boundary” of Ω.

Example 3.5.Let D = B(0, 1) in R2, let H be a bounded open connected hole in D such thatH ⊂ D, and let Ω = "DH. Then it can be checked that the solution of (3.31) is ofthe form

−∆y = f in Ω, y = 0 on ∂D,

where the constant c on H is determined by the condition

∀ϕ ∈ H1• (Ω; D),

∫

∂Ω

∂y

∂nϕ dγ = 0,

or equivalently ∫

∂H

∂y

∂ndγ = 0.

Example 3.6.Let D = ]−2, 2[ × ]−2, 2[ and let Ω1 and Ω2 be two open squares in G = ]−1, 1[ ×]−1, 1[ such that Ω1 ⊂ G, Ω2 ⊂ G, and Ω1 ∩ Ω2 = ∅. Define the domain as

Ω = Ω0 ∪ Ω1 ∪ Ω2, Ω0 = D\G (cf. Figure 5.2).

Then the solution of (3.31) is characterized by

−∆y = f in Ω, y = 0 on ∂D, y = c on G\[Ω1 ∪ Ω2

],

where the constant c is determined by the condition∫

∂Ωint0

∂y

∂ndγ +

∫

∂Ω1

∂y

∂ndγ +

∫

∂Ω2

∂y

∂ndγ = 0

and ∂Ωint0 = ∂G, the interior boundary of Ω0.

Ω2

Ω1

Ω0 = D\G

Figure 5.2. Disconnected domain Ω = Ω0 ∪ Ω1 ∪ Ω2.


4 Some Compliance Problems with Two MaterialsAs a first example consider the optimal compliance problem where the optimizationvariable is the distribution of two materials with different physical characteristicswithin a fixed domain D. It cannot a priori be assumed that the two regions areseparated by a smooth boundary and that each region is connected. The optimalsolution may lead to a nonsmooth interface and even to the mixing of the twomaterials. This type of solution occurs in control or optimization problems over abounded nonconvex subset of a function space. In general, their relaxed solution liesin the closed convex hull of that subset. In control theory, the phenomenon is knownas chattering control. To illustrate this approach it is best to consider a genericexample. In section 4.1, we consider a variation of the optimal design problemstudied by J. Cea and K. Malanowski [1] in 1970 and then discuss its originalversion in section 4.2. The variation of the problem is constructed in such a way thatthe form of the associated variational equation is similar to the one of Example 3.1,which provided a counterexample to the weak continuity of the solution. Yet themaximization of the minimum energy yields as solution a characteristic functionin both cases. However, this is no longer true for the minimization of the sameminimum energy as was shown by F. Murat and L. Tartar [1, 3].

4.1 Transmission Problem and ComplianceLet D ⊂ RN be a bounded open domain with Lipschitzian boundary ∂D. Assumethat the domain D is partitioned into two subdomains Ω1 and Ω2 separated by asmooth boundary ∂Ω1 ∩ ∂Ω2 as illustrated in Figure 5.3. Domain Ω1 (resp., Ω2) ismade up of a material characterized by a constant k1 > 0 (resp., k2 > 0). Let y bethe solution of the transmission problem

⎧⎨

⎩

− k1y = f in Ω1, −k2y = 0 in Ω2,

y = 0 on ∂D, k1∂y

∂n1+ k2

∂y

∂n2= 0 on ∂Ω1 ∩ ∂Ω2,

(4.1)

where n1 (resp., n2) is the unit outward normal to Ω1 (resp., Ω2) and f is a givenfunction in L2(D). Our objective is to maximize the equivalent of the compliance

J(Ω1) = −∫

Ω1

fy dx (4.2)

Ω2Ω1

Figure 5.3. Fixed domain D and its partition into Ω1 and Ω2.

4. Some Compliance Problems with Two Materials 229

over all domains Ω1 in D. In mechanics the compliance is associated with the totalwork of the body forces f .

Denote by Ω the domain Ω1. By complementarity Ω2 = "DΩ and let χ = χΩ.Problem (4.1) can be rewritten in the following variational form:

find y = y(χ) ∈ H10 (D) such that ∀ϕ ∈ H1

0 (D),∫

D[k1χ+ k2(1− χ)]∇y ·∇ϕ dx =

∫

Dχfϕ dx.

(4.3)

For k1 > 0 and k2 > 0 and all χ in X(D)

k(x) def= k1 χ(x) + k2 (1− χ(x)),0 < mink1, k2 ≤ k(x) ≤ maxk1, k2 a.e. in D.

(4.4)

By the Lax–Milgram theorem, the variational equation (4.3) still makes sense andhas a unique solution y = y(χ) in H1

0 (D) that coincides with the solution of theboundary value problem

−div (k∇y) = χf in D, y = 0 on ∂D. (4.5)

Notice that the high-order term has the same form as the term in Example 3.1. Asfor the objective function, it can be rewritten as

J(χ) = −∫

Dχfy(χ) dx. (4.6)

Thus the initial boundary value problem (4.1) has been transformed into the varia-tional problem (4.3), and the initial objective function (4.2) into (4.6). Both makesense for χ in X(D) and even in co X(D). The family of characteristic functionsX(D) and its closed convex hull co X(D) have been defined and characterized in(3.2) and (3.3).

The objective function (4.6) can be further rewritten as a minimum

J(χ) = minϕ∈H1

0 (D)E(χ,ϕ) (4.7)

for the energy function

E(χ,ϕ) def=∫

D(k1 χ+ k2 (1− χ)) |∇ϕ|2 − 2χfϕ dx (4.8)

associated with the variational problem (4.3). The initial optimal design problembecomes a max-min problem

maxχ∈X(D)

J(χ) = maxχ∈X(D)

minϕ∈H1

0 (D)E(χ,ϕ), (4.9)

where X(D) can be considered as a subset of Lp(D) for some p, 1 ≤ p <∞.


This problem can also be relaxed to functions χ with value in [0, 1]

maxχ∈co X(D)

J(χ) = maxχ∈co X(D)

minϕ∈H1

0 (D)E(χ,ϕ). (4.10)

We shall refer to this problem as the relaxed problem.These formulations have been introduced by J. Cea and K. Malanowski

[1], who used the variable k(x) and the following equality constraint on its integral:∫

Dk(x) dx = γ or

∫

Dχ(x) dx =

γ − k2 m(D)k1 − k2

for some appropriate γ > 0. Moreover the force f in (4.1)–(4.3) was exerted every-where in D and not only in Ω. So the objective function (4.2)–(4.6) was the integralof fy over all of D. We shall see in section 4.2 that the fact that the support of fis Ω or all of D does not affect the nature of the results.

In both maximization problems (4.1)–(4.2) and (4.3)–(4.6) it is necessary tointroduce a volume constraint in order to avoid a trivial solution. Notice that byusing (4.3) with ϕ = y(χ) the objective function (4.6) becomes

J(χ) = −∫

D(k1χ+ k2(1− χ)) |∇y(χ)|2 dx ≤ 0.

So maximizing J(χ) is equivalent to minimizing the integral of k(x) |∇y|2. There-fore, χ = 0 (only material k2) is a maximizer since the corresponding solutionof (4.3) is y = 0. In order to eliminate this situation we introduce the followingconstraint on the volume of material k1:

∫

Dχ dx ≥ α > 0 (4.11)

for some α, 0 < α ≤ m(D). The case α = m(D) yields the unique solution χ = 1(only material k1). So we can further assume that α < m(D).

For 0 < α < m(D), the optimal design problem becomes

maxχ∈X(D)∫Dχ dx≥α

J(χ) = maxχ∈X(D)∫Dχ dx≥α

minϕ∈H1

0 (D)E(χ,ϕ) (4.12)

and its relaxed version

maxχ∈co X(D)∫

Dχ dx≥α

J(χ) = maxχ∈co X(D)∫

Dχ dx≥α

minϕ∈H1

0 (D)E(χ,ϕ). (4.13)

We shall now show that problem (4.13) has a unique solution χ∗ in co X(D) andthat χ∗ is a characteristic function, χ∗ ∈ X(D), for which the inequality constraintis saturated:

χ∗ ∈ X(D) and∫

Dχ∗ dx = α. (4.14)


At this juncture it is advantageous to incorporate the volume inequality con-straint into the problem formulation by introducing a Lagrange multiplier λ ≥ 0.The formulation of the relaxed problem becomes

maxχ∈co X(D)

minϕ∈H1

0 (D)λ≥0

G(χ,ϕ,λ), G(χ,ϕ,λ) def= E(χ,ϕ) + λ

(∫

Dχ dx− α

)(4.15)

and E(χ,ϕ) is the energy function given in (4.7).We first establish the existence of saddle point solutions to problem (4.15).

We use a general result in I. Ekeland and R. Temam [1, Prop. 2.4, p. 164].The set co X(D) is a nonempty bounded closed convex subset of L2(D) and the setH1

0 (D)× [R+∪0] is trivially closed and convex. The function G is concave-convexwith the following properties:

⎧⎨

⎩

∀χ ∈ co X(D), (ϕ,λ) 8→ G(χ,ϕ,λ) is convex, continuous, and∃χ0 ∈ co X(D) such that lim

∥ϕ∥H10+|λ|→∞

G(χ0 ,ϕ,λ) = +∞; (4.16)

∀ϕ ∈ H1

0 (D), ∀λ ≥ 0, the map χ 8→ G(χ,ϕ,λ) is affine and

continuous for L2(D)-strong.(4.17)

For the first condition recall that 0 < α < m(D) and pick χ0 = 1 on D. To check thesecond condition pick any sequence χn in co X(D) which converges to some χ inL2(D)-strong. Then the sequence also converges in L2(D)-weak and by Lemma 3.1in L∞(D)-weak∗. Hence G(χn,ϕ,λ) converges to G(χ,ϕ,λ).

The set of saddle points (χ, y, λ) is of the form X×Y ⊂ co X(D)× H10 (D)×

[R+ ∪ 0] and is completely characterized by the following variational equationand inequalities (cf. I. Ekeland and R. Temam [1, Prop. 1.6, p. 157]):

∀ϕ ∈ H01(D),∫

D[k2 + (k1 − k2)χ]∇y ·∇ϕ− χfϕ dx = 0, (4.18)

∀χ ∈ co X(D),∫

D

[(k1 − k2)|∇y|2 − 2fy + λ

](χ− χ) dx ≤ 0, (4.19)

(∫

Dχ dx− α

)λ = 0,

∫

Dχ dx− α ≥ 0, λ ≥ 0. (4.20)

But for each χ ∈ co X(D) there exists a unique y(χ) solution of (4.18) and then

∀χ ∈ X, χ× Y ⊂ χ×y(χ)× [R+ ∪ 0]

.

Therefore, y(χ) is independent of χ ∈ X, that is, Y = y× Λ and

∀χ ∈ X, λ ∈ Λ, y(χ) = y.

For each λ, inequality (4.19) is completely equivalent to the following charac-terization of the maximizer χ ∈ X:

χ(x) =

⎧⎪⎨

⎪⎩

1, if (k1 − k2)|∇y|2 − 2fy + λ > 0,

∈ [0, 1], if (k1 − k2)|∇y|2 − 2fy + λ = 0,

0, if (k1 − k2)|∇y|2 − 2fy + λ < 0.

(4.21)


Associate with an arbitrary λ ≥ 0 the sets

D+(λ) = x ∈ D : (k1 − k2)|∇y|2 − 2fy + λ > 0,

D0(λ) = x ∈ D : (k1 − k2)|∇y|2 − 2fy + λ = 0.(4.22)

In order to complete the characterization of the optimal triplets, we need the fol-lowing general result.

Lemma 4.1. Consider a function

G : A×B → R (4.23)

for some sets A and B. Define

gdef= inf

x∈Asupy∈B

G(x, y), A0def=

x ∈ A : supy∈B

G(x, y) = g

, (4.24)

hdef= sup

y∈Binfx∈A

G(x, y), B0def=

y ∈ B : infx∈A

G(x, y) = h

. (4.25)

When g = h the set of saddle points (possibly empty) will be denoted by

Sdef= (x, y) ∈ A×B : g = G(x, y) = h . (4.26)

Then the following hold.

(i) In general h ≤ g and

∀(x0, y0) ∈ A0 ×B0, h ≤ G(x0, y0) ≤ g. (4.27)

(ii) If h = g, then S = A0 ×B0.

Proof. (i) If A0 × B0 = ∅, there is nothing to prove. If there exist x0 ∈ A0 andy0 ∈ B0, then by definition

h = infx∈A

G(x, y0) ≤ G(x0, y0) ≤ supy∈B

G(x0, y) = g. (4.28)

(ii) If h = g, then in view of (4.26)–(4.27), A0 × B0 ⊂ S. Conversely if thereexists (x0, y0) ∈ S, then h = G(x0, y0) = g and by the definitions of A0 and B0,(x0, y0) ∈ A0 ×B0.

Associate with an arbitrary solution (χ, y, λ), the characteristic function

χλ =

χD+(λ), if D+(λ) = ∅,

0, if D+(λ) = ∅.(4.29)

Then from (4.21)

χ

(k1 − k2)|∇y|2 − 2fy + λ

= χλ

(k1 − k2)|∇y|2 − 2fy + λ

a.e. in D


and

G(χ, y, λ) =∫

D[χk1 + (1− χ)k2] |∇y|2 − 2χfy dx + λ

(∫

Dχ dx− α

)

=∫

Dk2|∇y|2 + χ

[(k1 − k2)|∇y|2 − 2fy + λ

]dx− λα

=∫

Dk2|∇y|2 + χλ

[(k1 − k2)|∇y|2 − 2fy + λ

]dx− λα

= G(χλ, y, λ).

From Lemma 4.1, (χλ, y, λ) is also a saddle point. So there exists a maximizerχλ ∈ X(D) that is a characteristic function and necessarily

maxχ∈X(D)

minϕ∈H1

0 (D)λ≥0

G(χ,ϕ,λ) = maxχ∈co X(D)

minϕ∈H1

0 (D)λ≥0

G(χ,ϕ,λ).

But we can show more than that. If there exists λ ∈ Λ such that λ > 0, then byconstruction χ ≥ χλ and by (4.20)

α =∫

Dχ dx ≥

∫

Dχλ dx = α.

Since χ = χλ = 1 almost everywhere in D+(λ), then χ = χλ. But, always byconstruction, χλ is independent of χ. Thus the maximizer is unique, it is a charac-teristic function, and its integral is equal to α.

The case Λ = 0 is a degenerate one. Set ϕ = y in (4.18) and regroup theterms as follows:

0 =∫

D(k2 + (k1 − k2)χ) |∇y|2 − fyχ dx

=∫

D

(k2 +

k1 − k2

2χ

)|∇y|2 dx +

∫

D

k1 − k2

2χ |∇y|2 − fy

χ dx.

The integrand of the first integral is positive. From the characterization of χ, theintegrand of the second one is also positive. Hence they are both zero almosteverywhere in D. As a result

m(D+(0)) = 0 and ∇y = 0.

Therefore y = 0 in D. Hence the saddle points are of the general form (χ, y, λ) =(χ, 0, 0), χ ∈ X. In particular, D+(0) = ∅ and from our previous considerationsχ∅; that is, χ = 0 is a solution. But this is impossible since

∫D χ dx ≥ α > 0.

Therefore λ = 0 cannot occur.In conclusion, there exists a (unique for α > 0) maximizer χ∗ in co X(D) that

is in fact a characteristic function, and necessarily


minϕ∈H1

0 (D)E(χ,ϕ) = max

χ∈co X(D)∫Dχ dx≥α

minϕ∈H1

0 (D)E(χ,ϕ). (4.30)


Moreover, for 0 < α ≤ m(D),∫

Dχ∗ dx = α (4.31)

and χ∗ is the unique solution of the problem with an equality constraint:

maxχ∈X(D)∫Dχ dx=α

minϕ∈H1

0 (D)E(χ,ϕ) = max


minϕ∈H1

0 (D)E(χ,ϕ). (4.32)

As a numerical illustration of the theory consider (4.1) over the diamond-shapeddomain

D = (x, y) : |x| + |y| < 1 (4.33)

with the function f of Figure 5.4. This function has a sharp peak in (0, 0) whichhas been scaled down in the picture. The variational form (4.3) of the boundaryvalue problem was approximated by continuous piecewise linear finite elements oneach triangle, and the function χ by a piecewise constant function on each triangle.The constant on each triangle was constrained to lie between 0 and 1 together withthe global constraint on its integral over the whole domain D. Figure 5.5 shows theoptimal partition (the domain has been rotated by 45 degrees to save space). Thegrey triangles correspond to the region D0(λm), where χ ∈ [0, 1]. The presence ofthis grey zone in the approximated problem is due to the fact that equality for thetotal area where χ = 1 could not be exactly achieved with the chosen triangulationof the domain. Thus the problem had to adjust the value of χ between 0 and 1 in

−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

−1−0.8

−0.6−0.4

−0.20

0.20.4

0.60.8

1

0

10

20

30

40

50

60

Figure 5.4. The function f(x, y) = 56 (1− |x|− |y|)6.


Figure 5.5. Optimal distribution and isotherms with k1 = 2 (black) andk2 = 1 (white) for the problem of section 4.1.

a few triangles in order to achieve equality for the integral of χ. For this examplethe Lagrange multiplier associated with the problem is strictly positive.

4.2 The Original Problem of Cea and MalanowskiWe now put the force f everywhere in the fixed domain D. The same techniqueand the same conclusion can be drawn: there exists at least one maximizer of thecompliance, which is a characteristic function. For completeness we give the mainelements below.

Fix the bounded open Lipschitzian domain D in RN and let Ω be a smoothsubset of D. Let y be the solution of the transmission problem

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

− k1y = f in Ω, −k2y = f in D\Ω,

y = 0 on ∂D,

k1∂y

∂n1+ k2

∂y

∂n2= 0 on ∂Ω ∩D,

(4.34)

where n1 (resp., n2) is the unit outward normal to Ω (resp., "DΩ) and f is agiven function in L2(D). Again problem (4.34) can be reformulated in terms of thecharacteristic function χ = χΩ:

find y = y(χ) ∈ H10 (D) such that ∀ϕ ∈ H1

0 (D),∫

D[k1χ+ k2(1− χ)] ∇y ·∇ϕ dx =

∫

Dfϕ dx,

(4.35)

with the objective function

J(χ) = −∫

Dfy(χ) dx (4.36)


to be maximized over all χ ∈ X(D). As in the previous case, the function J(χ) canbe rewritten as the minimum of the energy function

E(χ,ϕ) def=∫

D[k1χ+ k2(1− χ)] |∇ϕ|2 − 2fϕ dx (4.37)

over H10 (D),

J(χ) = minϕ∈H1

0 (D)E(χ,ϕ), (4.38)

and we have the relaxed max-min problem

maxχ∈co X(D)

minϕ∈H1

0 (D)E(χ,ϕ). (4.39)

Without constraint on the integral of χ, the problem is trivial and χ = 1(resp., χ = 0) if k1 > k2 (resp., k2 > k1). In other words it is optimal to use onlythe strong material. In order to make the problem nontrivial assume that the strongmaterial is k1, that is, k1 > k2, and put an upper bound on the volume of materialk1 which occupies the part Ω of D:

∫

Dχ dx ≤ α, 0 < α < m(D). (4.40)

The case α = 0 trivially yields χ = 0. Under assumption (4.40) the case χ = 1with only the strong material k1 is no longer admissible. Thus we consider for0 < α < m(D) the problem

maxχ∈co X(D)∫

Dχ dx≤α

minϕ∈H1

0 (D)E(χ,ϕ). (4.41)

We shall now show that problem (4.41) has a unique solution χ∗ in co X(D) andthat in fact χ∗ is a characteristic function for which the inequality constraint issaturated:

∫

Dχ∗ dx = α. (4.42)

This solves the original problem of Cea and Malanowski with the equality constraint:


minϕ∈H1

0 (D)E(χ,ϕ). (4.43)

As in section 4.1 it is convenient to reformulate the problem with a Lagrangemultiplier λ ≥ 0 for the constraint inequality (4.40):

maxχ∈X(D)

minϕ∈H1

0 (D)λ≥0

G(χ,ϕ,λ), G(χ,ϕ,λ) = E(χ,ϕ)− λ[ ∫

Dχ dx− α

]. (4.44)


We then relax the problem to co X(D):

maxχ∈co X(D)

minϕ∈H1

0 (D)λ≥0

G(χ,ϕ,λ). (4.45)

By the same arguments as the ones used in section 4.1 we have the existence ofsaddle points (χ, y, λ) which are completely characterized by

∀ϕ ∈ H10 (D),

∫

D[k2 + (k1 − k2)χ]∇y ·∇ϕ− fϕ dx = 0, (4.46)

∀χ ∈ co X(D),∫

D

[(k1 − k2)|∇y|2 − λ

](χ− χ) dx ≤ 0, (4.47)

[∫

Dχ dx− α

]λ = 0,

∫

Dχ dx− α ≤ 0, λ ≥ 0. (4.48)

As before y is unique and the set of saddle points of G is the closed convex set

X ×y× Λ

⊂ X(D)×

y× λ : λ ≥ 0

.

The closed convex set Λ has a minimal element λm ≥ 0.For each λ, each maximizer χ ∈ X is necessarily of the form

χ(x) =

⎧⎪⎨

⎪⎩

1, if (k1 − k2)|∇y|2 − λ > 0,

∈ [0, 1], if (k1 − k2)|∇y|2 − λ = 0,

0, if (k1 − k2)|∇y|2 − λ < 0.

(4.49)

Associate with each λ > 0 the sets

D+(λ) =x ∈ D : (k1 − k2)|∇y|2 − λ > 0

, (4.50)

D0(λ) =x ∈ D : (k1 − k2)|∇y|2 − λ = 0

, (4.51)

D−(λ) =x ∈ D : (k1 − k2)|∇y|2 − λ < 0

. (4.52)

Define the characteristic function χm = χD+(λm). By construction all χ ∈ X are ofthe form

χm ≤ χ,

∫

Dχ dx ≤ α. (4.53)

Again it is easy to show that (χm, y, λm) is also a saddle point of G. Therefore, wehave a maximizer χm ∈ X(D) over co X(D) which is a characteristic function and

∫

Dχm dx ≤

∫

Dχ dx ≤ α.

If there exists λ > 0 in Λ, then for all χ ∈ X∫

Dχm dx = α =

∫

Dχ dx.


As a result the maximizer χm is unique, it is a characteristic function, and itsintegral is equal to α.

The case Λ = 0 cannot occur since the triplet (1, y1, 0) would be a saddlepoint, where y1 is the solution of the variational equation (4.34) for χ = 1. To seethis, first observe that for k1 > k2

∀χ,ϕ, G(χ,ϕ, 0) = E((χ,ϕ) ≤ E((1,ϕ) = G(1,ϕ, 0).

As a result

infϕ,λ

G(χ,ϕ,λ) ≤ infϕ

G(χ,ϕ, 0) ≤ infϕ

G(1,ϕ, 0),

infϕ

G(1,ϕ, 0) = G(1, y1, 0) ≤ supχ

infϕ

G(χ,ϕ, 0)

⇒ supχ

infϕ,λ

G(χ,ϕ,λ) ≤ supχ

infϕ

G(χ,ϕ, 0) = G(1, y1, 0).

But we know that there exists a unique y ∈ H10 (D) such that

supχ

infϕ

G(χ,ϕ, 0) ≤ supχ

G(χ, y, 0) = infϕ,λ

supχ

G(χ,ϕ,λ).

So from the above two inequalities,

supχ

infϕ,λ

G(χ,ϕ,λ) = G(1, y1, 0) = infϕ,λ

supχ

G(χ,ϕ,λ),

(1, y1, 0) is a saddle point of G, and

α ≥∫

Dχ dx = m(D).

This contradicts the fact that α < m(D).In conclusion in all cases there exists a (unique when 0 ≤ α < m(D)) maxi-

mizer χ∗ in co X(D), which is in fact a characteristic function, and


minϕ∈H1

0 (D)E(χ,ϕ) = max


minϕ∈H1

0 (D)E(χ,ϕ).

Moreover, for 0 ≤ α < m(D)∫

Dχ∗ dx = α.

This is precisely the solution of the original problem of Cea and Malanowski and


minϕ∈H1

0 (D)E(χ,ϕ) = max


minϕ∈H1

0 (D)E(χ,ϕ).

Again, as an illustration of the theoretical results, consider (4.34) over the diamond-shaped domain D defined in (4.33) with the function f of Figure 5.4 in section 4.1.


The variational form (4.35) of the boundary value problem was approximated inthe same way as the variational form (4.3) of (4.1) in section 4.1. Figure 5.6shows the optimal partition of the domain (rotated 45 degrees). The grey trianglescorrespond to the region D0(λm), where χ ∈ [0, 1]. The black region correspondsto the points where |∇y|2 > λm/(k1 − k2) and the white region to the ones where|∇y|2 < λm/(k1 − k2), as can be readily seen in Figure 5.6. For this example theLagrange multiplier associated with the problem is strictly positive. It is interestingto compare this computation with the one of Figure 5.5 in section 4.1, where thesupport of the force was restricted to Ω.

Figure 5.6. Optimal distribution and isotherms with k1 = 2 (black) andk2 = 1 (white) for the problem of Cea and Malanowski.

Remark 4.1.The formulation and the results remain true if the generic variational equation isreplaced by a variational inequality (unilateral problem)


minϕ∈ψ∈H1

0 (D):ψ≥0 in DE(χ,ϕ).

4.3 Relaxation and HomogenizationIn sections 4.1 and 4.2 the possible homogenization phenomenon predicted in Ex-ample 3.1 of section 3.2 did not take place, and in both examples the solution wasa characteristic function. Yet, it occurs when the maximization is changed to aminimization in the problem of J. Cea and K. Malanowski [1] of section 4.2:

infχ∈X(D)∫Dχ dx=α

minϕ∈H1

0 (D)E(χ,ϕ).


In 1985 F. Murat and L. Tartar [1] gave a general framework to study this classof problems by relaxation. It is based on the use of L. C. Young [2]’s generalizedfunctions (measures). They present a fairly complete analysis of the homogenizationtheory of second-order elliptic problems of the form

−∑

ij

∂

∂xi

(aij(x)

∂u

∂xj

)= f.

They give as examples the maximization and minimization versions of the problemof section 4.2. This material, which would have deserved a whole chapter in thisbook, is fortunately available in English (cf. F. Murat and L. Tartar [3]). Moreresults on composite materials can be found in the book edited by A. Cherkaevand R. Kohn [1], which gathers a selection of translations of key papers originallywritten in French and Russian.

5 Buckling of ColumnsOne of the very early optimal design problem was formulated by Lagrange in 1770(cf. I. Todhunter and K. Pearson [1]) and later studied by T. Clausen in 1849.It consists in finding the best profile of a vertical column to prevent buckling. Thisproblem and other problems related to columns have been revisited in a series ofpapers by S. J. Cox [1], S. J. Cox and M. L. Overton [1], S. J. Cox [2], and S.J. Cox and C. M. McCarthy [1]. Since Lagrange many authors have proposedsolutions, but a complete theoretical and numerical solution for the buckling of acolumn was given only in 1992 by S. J. Cox and M. L. Overton [1].

Consider a normalized column of unit height and unit volume. Denote by ℓ themagnitude of the normalized axial load and by u the resulting transverse displace-ment. Assume that the potential energy is the sum of the bending and elongationenergies

∫ 1

0EI |u′′|2 dx− ℓ

∫ 1

0|u′|2 dx,

where I is the second moment of area of the column’s cross section and E is itsYoung’s modulus. For sufficiently small load ℓ the minimum of this potential energywith respect to all admissible u is zero. Euler’s buckling load λ of the column is thelargest ℓ for which this minimum is zero. This is equivalent to finding the followingminimum:

λdef= inf

0 =u∈V

∫ 10 EI |u′′|2 dx∫ 10 |u′|2 dx

, (5.1)

where V = H20 (0, 1) corresponds to the clamped case, but other types of bound-

ary conditions can be contemplated. This is an eigenvalue problem with a specialRayleigh quotient.

Assume that E is constant and that the second moment of area I(x) of thecolumn’s cross section at the height x, 0 ≤ x ≤ 1, is equal to a constant c times its


cross-sectional area A(x),

I(x) = c A(x) ⇒∫ 1

0A(x) dx = 1.

Normalizing λ by cE and taking into account the engineering constraints

∃0 < A0 < A1, ∀x ∈ [0, 1], 0 < A0 ≤ A(x) ≤ A1,

we finally get

supA∈A

λ(A), λ(A) def= inf0 =u∈V

∫ 10 A |u′′|2 dx∫ 10 |u′|2 dx

, (5.2)

A def=

A ∈ L2(0, 1) : A0 ≤ A ≤ A1 and∫ 1

0A(x) dx = 1

. (5.3)

This problem can also be reformulated by rewriting

A(x) = A0 + χ(x) (A1 −A0),∫ 1

0χ(x) dx = α

def=1−A0

A1 −A0

for some χ ∈ co X([0, 1]). Clearly the problem makes sense only for 0 < A0 ≤ 1.Then

supχ∈co X([0,1])∫ 1

0χ(x) dx=α

λ(χ), λ(χ) def= inf0 =u∈V

∫ 10 [A0 + (A1 −A0)χ] |u′′|2 dx

∫ 10 |u′|2 dx

. (5.4)

Rayleigh’s quotient is not a nice convex concave function with respect to (v, A),and its analysis necessitates tools different from the ones of section 4. One ofthe original elements of the paper of S. J. Cox and M. L. Overton [1] wasto replace Rayleigh’s quotient by G. Auchmuty [1]’s dual variational principlefor the eigenvalue problem (5.1). We first recall the existence of solution to theminimization of Rayleigh’s quotient. In what follows we shall use the norm ∥u′∥L2

for the space H10 (0, 1) and ∥u′′∥L2 for the space H2

0 (0, 1).

Theorem 5.1. There exists at least one nonzero solution u ∈ V to the minimizationproblem

λ(A) def= inf0 =u∈V

∫ 10 A |u′′|2 dx∫ 10 |u′|2 dx

. (5.5)

Then λ(A0) > 0 and for all A ∈ A, λ(A) ≥ λ(A0), and

∀v ∈ V,

∫ 1

0|v′|2 dx ≤ λ(A0)−1

∫ 1

0A |v′′|2 dx. (5.6)

The solutions are completely characterized by the variational equation:

∃u ∈ V, ∀v ∈ V,

∫ 1

0A u′′ v′′ dx = λ(A)

∫ 1

0u′ v′ dx. (5.7)


Proof. The infimum is bounded below by 0 and is necessarily finite. Let unbe a minimizing sequence such that ∥u′

n∥L2 = 1. Then the sequence ∥u′′n∥L2(D) is

bounded. Hence un is a bounded sequence in H2(0, 1) and there exist u ∈ H2(0, 1)and a subsequence, still indexed by n, such that un u in H2

0 (0, 1)-weak. Thereforethe subsequence strongly converges in H1

0 (0, 1) and

1 =∫ 1

0|u′

n|2 dx→∫ 1

0|u′|2 dx and

∫ 1

0A |u′′|2 dx ≤ lim inf

n→∞

∫ 1

0A |u′′

n|2 dx

⇒∫ 10 A |u′′|2 dx∫ 10 |u′|2 dx

≤ lim infn→∞

∫ 10 A |u′′

n|2 dx∫ 10 |u′

n|2 dx= λ(A),

and, by the definition of λ(A), u ∈ H20 (0, 1) is a minimizing element. Notice that

∀A ∈ A, λ(A0) = inf

∫ 10 A0 |u′′|2 dx∫ 10 |u′|2 dx

: ∀u ∈ H20 (0, 1), u = 0

≤ inf

∫ 10 A |u′′|2 dx∫ 10 |u′|2 dx

: ∀u ∈ H20 (0, 1), u = 0

= λ(A)

and similarly λ(A) ≤ λ(A1). If λ(A0) = 0, we repeat the above construction andend up with an element u ∈ H2

0 (0, 1) such that∫ 1

0|u′|2 dx = 1 and A0

∫ 1

0|u′′|2 dx =

∫ 1

0A0|u′′|2 dx = 0,

which is impossible in H20 (0, 1) since A0 = 0 and u′′ = 0 implies u = 0.

If u = 0, then Rayleigh’s quotient is differentiable and its directional semide-rivative in the direction v is given by

2∫ 10 A u′′ v′′ dx

∥u′∥2L2

− 2∫ 1

0A |u′′|2 dx

∫ 10 u′ v′ dx

∥u′∥4L2

.

A nonzero solution u of the minimization problem is necessarily a stationary point,and for all v ∈ H2

0 (0, 1)∫ 1

0A u′′ v′′ dx =

∫ 10 A0 |u′′|2 dx∫ 10 |u′|2 dx

∫ 1

0u′ v′ dx = λ(A)

∫

Du′ v′ dx.

Conversely any nonzero solution of (5.7) is necessarily a minimizer of Rayleigh’squotient.

The dual variational principle of G. Auchmuty [1] for the eigenvalue problem(5.5) can be chosen as

µ(A) def= infL(A, v) : v ∈ H2

0 (0, 1)

, (5.8)

L(A, v) def=12

∫ 1

0A |v′′|2 dx−

[∫ 1

0|v′|2 dx

]1/2

. (5.9)


Theorem 5.2. For each A ∈ A, there exists at least one minimizer of L(A, v),

µ(A) = − 12λ(A)

, (5.10)

and the set of minimizers of (5.8) is given by

E(A) def=

⎧⎪⎨

⎪⎩u ∈ H2

0 (0, 1) :u is solution of (5.7) and∫ 1

0|u′|2 dx

1/2

= 1/λ(A)

⎫⎪⎬

⎪⎭. (5.11)

Proof. The existence of the solution follows from the fact that v 8→ L(A, v) is weaklylower semicontinuous and coercive. From Theorem 5.1 the set E(A) is not empty,and for any u ∈ E(A)

µ(A) ≤ L(A, u) = − 12λ(A)

< 0. (5.12)

Therefore, the minimizers of (5.7) are different from the zero functions. For u = 0,the function L(A, u) is differentiable and its directional semiderivative is given by

dL(A, u; v) =∫ 1

0A u′′ v′′ dx− 1

∥u′∥L2

∫ 1

0u′ v′ dx, (5.13)

and any minimizer u of L(A, v) is a stationary point of dL(A, u; v), that is,

∀v ∈ H20 (0, 1),

∫ 1

0A u′′ v′′ dx− 1

∥u′∥L2

∫ 1

0u′ v′ dx = 0. (5.14)

Therefore, u is a solution of the eigenvalue problem with

λ′ =1

∥u′∥L2⇒ − 1

2λ′ = µ(A) ≤ − 12λ(A)

from inequality (5.12). By the minimality of λ(A), we necessarily have λ′ = λ(A),and this concludes the proof of the theorem.

Theorem 5.3.

(i) The set A is compact in the L2(0, 1)-weak topology.

(ii) The function A 8→ µ(A) is concave and upper semicontinuous with respect tothe L2(0, 1)-weak topology.

(iii) There exists A in A which maximizes µ(A) over A and, a fortiori, whichmaximizes λ(A) over A.

Proof. (i) A is convex, bounded, and closed in L2(0, 1). Hence it is compact inL2(0, 1) weak.


(ii) Concavity. For all λ in [0, 1] and A and A′ in A,

L(λA + (1− λ)A′, v) = λL(A, v) + (1− λ)L(A′, v)≥ λ inf

v∈VL(A, v) + (1− λ) inf

v∈VL(A′, v)

⇒ µ(λA + (1− λ)A′) = infv∈V

L(λA + (1− λ)A′, v) ≥ λµ(A) + (1−λ)µ(A′).

Upper semicontinuity. Let uA ∈ A be a minimizer of L(A, v) and An bea sequence converging to A in the L2(0, 1)-weak topology. By Lemma 3.1, Anconverges to A in the L∞(0, 1)-weak⋆ topology. Then

µ(An)− µ(A) ≤ L(An, uA)− L(A, uA) =12

∫ 1

0(An −A) |u′′

A|2 dx

⇒ lim supn→∞

µ(An)− µ(A) ≤ limn→∞

12

∫ 1

0(An −A) |u′′

A|2 dx = 0.

(iii) The existence of solution follows from the concavity and upper semicon-tinuity of µ(A). Hence it is weakly upper semicontinuous over the weakly compactsubset A of L2(0, 1). The existence of a minimizer of λ(A) follows from iden-tity 5.10.

6 Caccioppoli or Finite Perimeter SetsThe notion of a finite perimeter set has been introduced and developed mainly byR. Caccioppoli [1] and E. De Giorgi [1] in the context of J. A. F. Plateau [1]’sproblem, named after the Belgian physicist and professor (1801–1883), who didexperimental observations on the geometry of soap films. A modern treatment ofthis subject can be found in the book of E. Giusti [1]. One of the difficulties instudying the minimal surface problem is the description of such surfaces in the usuallanguage of differential geometry. For instance, the set of possible singularities isnot known. Finite perimeter sets provide a geometrically significant solution toPlateau’s problem without having to know ahead of time what all the possiblesingularities of the solution can be. The characterization of all the singularities ofthe solution is a difficult problem which can be considered separately.

This was a very fundamental contribution to the theory of variational prob-lems, where the optimization variable is the geometry of a domain. This point ofview has been expanded, and a variational calculus was developed by F. J. Alm-gren, Jr. [1]. This is the theory of varifolds.

Perhaps the one most important virtue of varifolds is that it is possibleto obtain a geometrically significant solution to a number of variationalproblems, including Plateau’s problem, without having to know ahead oftime what all the possible singularities of the solution can be.(cf. F. J. Almgren, Jr. [1, p. viii]).

The treatment of this topic is unfortunately much beyond the scope of this book,and the interested reader is referred to the above reference for more details.

6. Caccioppoli or Finite Perimeter Sets 245

6.1 Finite Perimeter SetsGiven an open subset D of RN, consider L1-functions f on D with distributionalgradient ∇f in the space M1(D)N of (vectorial) bounded measures; that is,

ϕ 8→ ⟨∇f, ϕ⟩Ddef= −

∫

Dfdiv ϕ dx : D1(D;RN)→ R

is continuous with respect to the topology of uniform convergence in D:

∥∇f∥M1(D)Ndef= sup

ϕ∈D1(D;RN)∥ϕ∥C≤1

⟨∇f, ϕ⟩D <∞, ∥ϕ∥Cdef= sup

x∈D|ϕ(x)|RN , (6.1)

where M1(D) def= D0(D)′ is the topological dual of D0(D), and

∇f ∈ L(D0(D;RN),R

)≡ L

(D0(D),R

)N = M1(D)N .

Such functions are known as functions of bounded variation. The space

BV(D) def=f ∈ L1(D) : ∇f ∈M1(D)N

(6.2)

endowed with the norm

∥f∥BV(D) = ∥f∥L1(D) + ∥∇f∥M1(D)N (6.3)

is a Banach space.

Definition 6.1 (L. C. Evans and R. F. Gariepy [1]).A function f ∈ L1

loc(U) in an open subset U of RN has locally bounded variation iffor each bounded open subset V of U such that V ⊂ U , f ∈ BV(V ). The set of allsuch functions will be denoted by BVloc(U).

Theorem 6.1. Given an open subset U of RN, f belongs to BVloc(U) if and onlyif for each x ∈ U there exists ρ > 0 such that B(x, ρ) ⊂ U and f ∈ BV(B(x, ρ)),B(x, ρ) the open ball of radius ρ in x.

Proof. If f ∈ BVloc(U), then the result is true by specializing to balls in each pointof U . Conversely, by assumption, the open set U is covered by a family F(U) ofballs B(x, ρx) such that B(x, ρx) ⊂ U and f ∈ BV(B(x, ρx)). Given a boundedopen subset V of U such that V ⊂ U , the compact subset V can be covered by afinite number of balls B(xi, ρi), 1 ≤ i ≤ n, in F(U) for points xi ∈ V , 1 ≤ i ≤ n,

V ⊂n⋃

i=1

Bi, Bidef= B(xi, ρxi).

Denote by ψi ∈ D(Bi)ni=1 a partition of unity for the family Bi such that

∀i, 0 ≤ ψi ≤ 1 in Bi, andn∑

i=1

ψi = 1 on V .


Then for each ϕ ∈ V1(V )N

ϕ =n∑

i=1

ϕi on V, ϕidef= ψi ϕ,

and by construction ϕi ∈ V1(Bi ∩ V )N . Therefore, for all ϕ ∈ V1(V )N such that∥ϕ∥C(V ) ≤ 1

−∫

Vf div ϕ dx = −

n∑

i=1

∫

Vf div ϕi dx = −

n∑

i=1

∫

V ∩Bi

f div ϕi dx

⇒∣∣∣∣∫

Vf div ϕ dx

∣∣∣∣ ≤n∑

i=1

∣∣∣∣∫

Bi∩Vf div ϕi dx

∣∣∣∣

⇒∣∣∣∣∫

Vf div ϕ dx

∣∣∣∣ ≤n∑

i=1

∥∇f∥M1(Bi∩V ) ≤n∑

i=1

∥∇f∥M1(Bi) <∞,

since V1(Bi ∩ V )N ⊂ V1(Bi)N and ∥ϕi∥C(Bi) ≤ 1. Therefore, ∥∇f∥M1(V ) is finiteand f ∈ BVloc(RN).

For more details and properties see also F. Morgan [1, p. 117], H. Fed-erer [5, sect. 4.5.9], L. C. Evans and R. F. Gariepy [1], W. P. Ziemer [1],and R. Temam [1].

As in the previous section, consider measurable subsets Ω of a fixed boundedopen subset D of RN. Their characteristic functions χΩ ∈ X(D) are L1(D)-functions

∥χΩ∥L1(D) =∫

DχΩ dx = m(Ω) ≤ m(D) <∞ (6.4)

with distributional gradient

∀ϕ ∈(D(D)

)N, ⟨∇χΩ, ϕ⟩D

def= −∫

DχΩdiv ϕ dx. (6.5)

When Ω is an open domain with boundary Γ of class C1, then by the Stokes diver-gence theorem

−∫

Ω∩Ddiv ϕ dx = −

∫

∂(Ω∩D)ϕ · n dΓ = −

∫

Γ∩Dϕ · n dΓ,

where n is the outward normal field along ∂(Ω ∩ D). Since Γ is of class C1 thenormal field n along Γ belongs to C0(Γ), and from the last identity the maximum is

PD(Ω) def= ∥∇χΩ∥M1(D) =∫

Γ∩D|n|2 dΓ =

∫

Γ∩DdΓ = HN−1(Γ ∩D),

the (N − 1)-dimensional Hausdorff measure of Γ ∩D. As a result

HN−1(Γ) = PD(Ω) + HN−1(Γ ∩ ∂D). (6.6)

Thus the norm of the gradient provides a natural relaxation of the notion of perime-ter to the following larger class of domains.


Definition 6.2.Let Ω be a Lebesgue measurable subset of RN.

(i) The perimeter of Ω with respect to an open subset D of RN is defined as

PD(Ω) def= ∥∇χΩ∥M1(D)N . (6.7)

Ω is said to have finite perimeter with respect to D if PD(Ω) is finite. Thefamily of measurable characteristic functions with finite measure and finiterelative perimeter in D will be denoted as

BX(D) def= χΩ ∈ X(D) : χΩ ∈ BV(D) .

(ii) Ω is said to have locally finite perimeter if for all bounded open subsets D ofRN, χΩ ∈ BV(D), that is, χΩ ∈ BVloc(RN).

(iii) Ω is said to have finite perimeter if χΩ ∈ BV(RN).

Theorem 6.2. Let Ω be a Lebesgue measurable subset of RN. Ω has locally finiteperimeter, that is, χΩ ∈ BVloc(RN), if and only if for each x ∈ ∂Ω there existsρ > 0 such that χΩ ∈ BV(B(x, ρ)), B(x, ρ) the open ball of radius ρ in x.

Proof. We use Theorem 6.1. In one direction the result is obvious. Conversely,if x ∈ int Ω, then there exists ρ > 0 such that B(x, ρ) ⊂ int Ω, and for all ϕ ∈D1(B(x, ρ))

−∫

B(x,ρ)χΩ div ϕ dx = −

∫

B(x,ρ)div ϕ dx = −

∫

∂B(x,ρ)ϕ · n dHN−1 = 0

since ϕ = 0 on the boundary of B(x, ρ). Thus χΩ ∈ BV(B(x, ρ)). If x ∈ int "Ω,then there exists ρ > 0 such that B(x, ρ) ⊂ int "Ω, and for all ϕ ∈ D1(B(x, ρ))

−∫

B(x,ρ)χΩ div ϕ dx = 0.

So again χΩ ∈ BV(B(x, ρ)). Finally, if x ∈ ∂Ω, then by assumption there exists ρ >0 such that χΩ ∈ BV(B(x, ρ)). Therefore, by Theorem 6.1, χΩ ∈ BVloc(RN).

The interest behind this construction is twofold. First, the notion of perimeterof a set is extended to measurable sets; second, this framework provides a firstcompactness theorem which will be useful in obtaining existence of optimal domains.

Theorem 6.3. Assume that D is a bounded open domain in RN with a Lipschitzianboundary ∂D. Let Ωn be a sequence of measurable domains in D for which thereexists a constant c > 0 such that

∀n, PD(Ωn) ≤ c. (6.8)

Then there exist a measurable set Ω in D and a subsequence Ωnk such that

χΩnk→ χΩ in L1(D) as k →∞ and PD(Ω) ≤ lim inf

k→∞PD(Ωnk) ≤ c. (6.9)


Moreover ∇χΩnk“converges in measure” to ∇χΩ in M1(D)N ; that is, for all ϕ in

D0(D,RN),

limk→∞

⟨∇χΩnk, ϕ⟩M1(D)N → ⟨∇χΩ, ϕ⟩M1(D)N . (6.10)

Proof. This follows from the fact that the injection of the space BV(D) endowedwith the norm (6.3) into L1(D) is continuous and compact (cf. E. Giusti [1,Thm. 1.19, p. 17], V. G. Maz’ja [1, Thm. 6.1.4, p. 300, Lem. 1.4.6, p. 62], C. B.Morrey, Jr. [1, Thm. 4.4.4, p. 75], and L. C. Evans and R. F. Gariepy [1]).

Example 6.1 (The staircase).In (6.9) the inequality can be strict, as can be seen from the following example(cf. Figure 5.7). For each n ≥ 1, define the set

Ωn =n⋃

j=1

[j

n,j + 1

n

]×[0, 1− j

n

].

Its limit is the set

Ω = (x, y) : 0 ≤ x ≤ 1, 0 ≤ y ≤ x.

It is easy to check that the sets Ωn are contained in the holdall D = ]−1, 2[× ]−1, 2[and that

∀n ≥ 1, PD(Ωn) = 4, PD(Ω) = 2 +√

2,

χΩn → χΩ in Lp(D), 1 ≤ p <∞.

Each Ωn is uniformly Lipschitzian, but the Lipschitz constant and the two neigh-borhoods of Definition 3.2 in Chapter 2 cannot be chosen independently of n.

1/n

ΩΩn

Figure 5.7. The staircase.

Corollary 1. Let D be open bounded and Lipschitzian. There exists a constantc > 0 such that for all convex domains Ω in D

PD(Ω) ≤ c, m(Ω) ≤ c

and the set C(D) of convex subsets of D is compact in Lp(D) for all p, 1 ≤ p <∞.


Proof. If Ω is a convex set with zero volume, m(Ω) = 0, then its perimeterPD(Ω) = 0 (cf. E. Giusti [1, Rem. 1.7 (iii), p. 6]). If m(Ω) > 0, then int Ω = ∅.For convex sets with a nonempty interior the perimeter and the volume enjoy avery nice monotonicity property. If C• denotes the set of nonempty closed convexsubsets of RN, then the map Ω 8→ m(Ω) : C• → R is strictly increasing:

∀A, B ∈ C•, A " B =⇒ m(A) < m(B)

(cf. M. Berger [1, Vol. 3, Prop. 12.9.4.3, p. 141]). Similarly if P (Ω) denotesthe perimeter of Ω in RN, the map Ω 8→ P (Ω) : C• → R is strictly increasing(cf. M. Berger [1, Vol. 3, Prop. 12.10.2, p. 144]). As a result

∀n, m(Ωn) ≤ m(D) <∞ and PD(Ωn) ≤ P (Ωn) ≤ P (co D) <∞,

since the perimeter of a bounded convex set is finite. Then the conditions of thetheorem are satisfied and the conclusions of the theorem follow. But we have seenin Theorem 3.5 (iii) that the set C(D) is closed in L1(D). Therefore, Ω can bechosen convex in the equivalence class. This completes the proof.

This theorem and the lower (resp., upper) semicontinuity of the shape functionΩ 8→ J(Ω) will provide existence results for domains in the class of finite perimetersets in D. One example is the transmission problem (4.1) to (4.3) of section 4.1with the objective function

J(Ω) =12

∫

Ω|y(Ω)− g|2 dx + αPD(Ω), α > 0, (6.11)

for some g ∈ L2(D). We shall come back later to the homogeneous Dirichlet bound-ary value problem (3.25)–(3.26).

It is important to recall that even if a set Ω in D has a finite perimeterPD(Ω), its relative boundary Γ ∩ D can have a nonzero N -dimensional Lebesguemeasure. To illustrate this point consider the following adaptation of Example 1.10in E. Giusti [1, p. 7].

Example 6.2.Let D = B(0, 1) in R2 be the open ball in 0 of radius 1. For i ≥ 1, let xi be anordered sequence of all points in D with rational coordinates. Associate with eachi the open ball

Bi = x ∈ D : |x− xi| < ρi, 0 < ρi ≤ min2−i, 1− |xi|.

Define the new sequence of open subsets of D,

Ωn =n⋃

i=1

Bi,

and notice that for all n ≥ 1,

m(∂Ωn) = 0, PD(Ωn) ≤ 2π,


where m = m2 is the Lebesgue measure in R2 and ∂Ωn is the boundary of Ωn.Moreover, since the sequence of sets Ωn is increasing,

χΩn → χΩ in L1(D), Ω =∞⋃

i=1

Bi, PD(Ω) ≤ lim infn→∞

PD(Ωn) ≤ 2π.

Observe that Ω = D and that ∂Ω = Ω ∩ "Ω ⊃ D ∩ "Ω. Thus

m(∂Ω) = m(D ∩ "Ω) ≥ m(D)−m(Ω) ≥ 2π3

since

m(D) = π and m(Ω) ≤∞∑

1=1

π2−2i =π

3.

Recall that

m(Ωn) ≤n∑

i=1

m(Bi) ≤n∑

i=1

π2−2i ⇒ m(D) ≤∞∑

i=1

m(Bi) ≤∞∑

i=1

π2−2i.

For p, 1 ≤ p < ∞, the sequence of characteristic functions χΩn converges to χΩin Lp(D)-strong. However, for all n, m(∂Ωn) = 0, but m(∂Ω) > 0.

In fact we can associate with the perimeter PD(Ω) the reduced boundary,∂∗Ω, which is the set of all x ∈ ∂Ω for which the normal n(x) exists. We quotethe following interesting theorems from W. H. Fleming [1, p. 455] (cf. also E.De Giorgi [2], L. C. Evans and R. F. Gariepy [1, Thm. 1, sect. 5.1, p. 167,Notation, pp. 168–169, Lem. 1, p. 208], and H. Federer [2]).

Theorem 6.4. Let χΩ ∈ BVloc(RN). There exist a Radon measure ∥∂Ω∥ on RN

and a ∥∂Ω∥-measurable function νΩ : RN → RN such that

(i) |νΩ(x)| = 1, ∥∂Ω∥ almost everywhere and

(ii)∫RN χΩ divϕ dx = −

∫RN ϕ · νΩ d∥∂Ω∥, for all ϕ ∈ C1

c (RN;RN).

Definition 6.3.Let χΩ ∈ BVloc(RN). The point x ∈ RN belongs to the reduced boundary ∂∗Ω if

(i) ∥∂Ω∥(Br(x)) > 0, for all r > 0,

(ii) limr01

m(Br(x))

∫Br(x) νΩ d∥∂Ω∥ = νΩ(x), and

(iii) |νΩ(x)| = 1.

Theorem 6.5. Let Ω have finite perimeter P (Ω). Let ∂∗Ω denote the reducedboundary of Ω. Then

(i) ∂∗Ω ⊂ ∂∗Ω ⊂ Ω• ⊂ ∂Ω and ∂∗Ω = ∂Ω,

(ii) P (Ω) = HN−1(∂∗Ω) (cf. E. Giusti [1, Chap. 4]),


(iii) the Gauss–Green theorem holds with ∂∗Ω, and

(iv) m(∂∗Ω) = m(∂∗Ω) = 0, HN−1(∂∗Ω\∂∗Ω) = 0.

We quote the following density theorem (cf. E. Giusti [1, Thm. 1.24 andLem. 1.25, p. 23]), which complements Theorem 3.1.

Theorem 6.6. Let Ω be a bounded measurable domain in RN with finite perimeter.Then there exists a sequence Ωj of C∞-domains such that as j goes to ∞

∫

RN|χΩj − χΩ| dx→ 0 and P (Ωj)→ P (Ω).

6.2 Decomposition of the Integral along Level Sets

We complete this section by giving some useful theorems on the decompositionof the integral along the level sets of a function. The first one was used by J.-P. Zolesio [6] (Grad’s model in plasma physics) in 1979 and [9, sect. 4.4, p. 95] in1981, by R. Temam [1] (monotone rearrangements) in 1979, by J. M. Rakotosonand R. Temam [1] in 1987, and more recently in 1995 by M. C. Delfour andJ.-P. Zolesio [19, 20, 21, 25] (intrinsic formulation of models of shells). We quotethe version given in L. C. Evans and R. F. Gariepy [1, Prop. 3, p. 118]. Theoriginal theorem can be found in H. Federer [3] and L. C. Young [1].

Theorem 6.7. Let f : RN → R be Lipschitz continuous with

|∇f | > 0 a.e.

For any Lebesgue summable function g : RN → R, we have the following decompo-sition of the integral along the level sets of f :

∫

f>tg dx =

∫ ∞

t

(∫

f=s

g

|∇f | dHN−1

)ds.

The second theorem (W. H. Fleming and R. Rishel [1]) uses a BV functioninstead of a Lipschitz function.

Theorem 6.8. Let D be an open subset of RN. For any f ∈ BV(D) and real t, letEt

def= x : f(x) < t. Then

∥∇f∥M1(D)N =∫ ∞

−∞P (Et) dt

(cf., for instance, H. Whitney [1, Chap. 11]).

This is known as the co-area formula (see also E. Giusti [1, Thm. 1.23, p. 20]and E. De Giorgi [4]).


6.3 Domains of Class W ε,p(D), 0 ≤ ε < 1/p, p ≥ 1, anda Cascade of Complete Metric Spaces

There is a general property enjoyed by functions in BV(D).

Theorem 6.9. Let D be a bounded open Lipschitzian domain in RN.

(i) BV(D) ⊂W ε,1(D), 0 ≤ ε < 1.

(ii) BV(D) ∩ L∞(D) ⊂W ε,p(D), 0 ≤ ε < 1p , 1 ≤ p <∞.

This theorem says that for a Caccioppoli set Ω in D,

∀p ≥ 1, 0 ≤ ε <1p, χΩ ∈W ε,p(D).

The special case p = 2 was proved by C. Baiocchi, V. Comincioli, E. Ma-genes, and G. A. Pozzi [1] in the context of the celebrated problem of the dam.They showed that the domain Ω is the hypograph of a continuous monotonicallydecreasing function on a closed interval is Hε(D) = W ε,2(D), 0 ≤ ε < 1/2, in R2.

Proof of Theorem 6.9. (i) Given f in BV(D) we want to show that f ∈ W ε,1(D),0 < ε < 1. This is equivalent to showing that the double integral

I =∫

Ddx

∫

Ddy

|f(y)− f(x)||y − x|N+ε

is finite. Given α > 0 small we break the above integral into two parts:

I1 =∫

Ddx

∫

D∩B(x,α)dy

|f(y)− f(x)||y − x|N+ε , I2 =

∫

Ddx

∫

!DB(x,α)dy

|f(y)− f(x)||y − x|N+ε .

The second integral I2 is bounded since |y − x| ≥ α. For the first one we changethe variable y to t = y − x,

I1 =∫

Ddx

∫

B(0,α)dt

|(χDf)(x + t)− (χDf)(x)||t|N+ε ,

and after a change in the order of integration,

I1 =∫

B(0,α)dt

1|t|N+ε

∫

Ddx |(χDf)(x + t)− (χDf)(x)|.

But the integral over D is bounded by |t| ∥∇f∥M1(D). The result is true for smoothfunctions (cf. V. G. Maz’ja [1, Lem. 1.4.6, p. 62]). Pick a sequence of smoothfunctions such that

fn → f in L1(D), ∇fn → ∇f in M1(D)-weak;

then

∥∇f∥M1(D) ≤ lim supn→∞

∥∇fn∥M1(D) = lim supn→∞

∥∇fn∥L1(D)


is bounded, and by going to the limit on both sides of the inequality,∫

Ddx |(χDfn)(x + t)− (χDfn)(x)| ≤ |t|∥∇fn∥L1(D),

we obtain the desired result. Now coming back to the estimate of I1,

I1 ≤∫

B(x,α)|t|1−N−ε dt ∥∇f∥M1(D) ≤ c(α)

α1−ε

1− ε∥∇f∥M1(D), 0 < ε < 1.

(ii) The function f belongs to W ε,p(D) if the integral

I =∫

Ddx

∫

Ddy

|f(y)− f(x)|p

|y − x|N+εp < +∞.

For each f ∈ BV(D) ∩ L∞(D), there exists M > 0 such that

|f(x)| ≤M a.e. in D.

Then

I = (2 M)p

∫

Ddx

∫

Ddy

∣∣∣ f(y)−f(x)2 M

∣∣∣p

|y − x|N+εp ,

and since∣∣∣∣f(y)− f(x)

2 M

∣∣∣∣ ≤ 1,

then for each p, 1 ≤ p <∞,∣∣∣∣f(y)− f(x)

2 M

∣∣∣∣p

≤∣∣∣∣f(y)− f(x)

2 M

∣∣∣∣ .

As a result

I ≤ (2 M)p−1∫

Ddx

∫

Ddy

|f(y)− f(x)||y − x|N+εp .

But we have seen in part (i) that for f ∈ BV(D), the double integral is finite if

0 ≤ ε p < 1 ⇒ 0 ≤ ε <1p.

This complete the proof.

For a characteristic function χΩ of a measurable set Ω and ε > 0, we have0 ≤ ε < 1/p for all p ≥ 1 and

∥χΩ∥pW ε,p(D) =∫

D

∫

D

|χΩ(x)− χΩ(y)||x− y|N+pε

dx dy + ∥χΩ∥pLp(D). (6.12)


The W ε,p(D)-norm is equivalent to the W ε′,p′(D) for all pairs (p′, ε′), p′ ≥ 1,

0 ≥ ε′ < 1/p′, such that p′ ε′ = p ε. At this stage, it is not clear whether that normis related to some Hausdorff measure of the relative boundary Γ ∩D of the set.

For p = 2 and a characteristic function, χΩ, of a measurable set Ω, we get

∥χΩ∥2Hε(D) = 2∫

Ω

∫

!DΩ|x− y|−(N+2ε) dx dy + m(Ω ∩D). (6.13)

The space X(D)∩W ε,2(D) is a closed subspace of the Hilbert space W ε,2(D) and thesquare of its norm is differentiable. A direct consequence for optimization problemsis that a penalization term of the form ∥χΩ∥2W ε,2(D) is now differentiable and canbe used in various minimization problems or to regularize the objective function toobtain existence of approximate solutions. For instance, to obtain existence resultswhen minimizing a function J(Ω) (defined for all measurable sets Ω in D), we canconsider a regularized problem in the following form:

Jα(Ω) = J(Ω) + α∥χΩ∥2Hε(D), α > 0. (6.14)

Section 6.1 provided the important compactness Theorem 6.3 for Caccioppolior finite perimeter sets. For a bounded open holdall D, this also provides a newcomplete metric topology on X(D) ∩ BV(D) when endowed with the metric

ρBV(χ1,χ2)def= ∥χ2 − χ1∥BV(D) = ∥χ2 − χ1∥L1(D) + ∥∇χ2 −∇χ1∥M1(D).

Similarly, the intermediate spaces of Theorem 6.9 between BV(D) and Lp(D) pro-vide little rougher metrics for p ≥ 1 and ε < 1/p on X(D)∩W ε,p(D) when endowedwith the metric

ρW ε,p(χ1,χ2)def= ∥χ2 − χ1∥W ε,p(D).

Theorem 6.10. Given a bounded open holdall D in RN, X(D)∩BV(D) and X(D)∩W ε,p(D), p ≥ 1, ε < 1/p, are complete metric spaces and

X(D) ∩ BV(D) ⊂ X(D) ∩W ε,p(D) ⊂ X(D).

6.4 Compactness and Uniform Cone PropertyIn Theorem 6.3 of section 6.1 we have seen a first compactness theorem for a familyof sets with a uniformly bounded perimeter. In this section we present a secondcompactness theorem for measurable domains satisfying a uniform cone propertydue to D. Chenais [1, 4, 6]. We provide a proof of this result that emphasizes thefact that the perimeter of sets in that family is uniformly bounded. It then becomesa special case of Theorem 6.3. This result will also be obtained as Corollary 2 toTheorem 13.1 in section 13 of Chapter 7. It will use completely different argumentsand apply to families of sets that do not even have a finite boundary measure.

Theorem 6.11. Let D be a bounded open holdall in RN with uniformly Lipschitzianboundary ∂D. For r > 0, ω > 0, and λ > 0 consider the family

L(D, r,ω,λ) def=

Ω ⊂ D : Ω is Lebesgue measurable and satisfiesthe uniform cone property for (r,ω,λ)

. (6.15)


For p, 1 ≤ p <∞, the set

X(D, r,ω,λ) def= χΩ : ∀Ω ∈ L(D, r,ω,λ) (6.16)

is compact in Lp(D) and there exists a constant p(D, r,ω,λ) > 0 such that

∀Ω ∈ L(D, r,ω,λ), HN−1(∂Ω) ≤ p(D, r,ω,λ). (6.17)

We first show that the perimeter of the sets of the family L(D, r,ω,λ) isuniformly bounded and use Theorem 6.8 to show that any sequence of X(D, r,ω,λ)has a subsequence converging to the characteristic function of some finite perimeterset Ω. We use the full strength of the compactness of the injection of BV(D)into L1(D) rather than checking directly the conditions under which a subset ofLp(D) is relatively compact. The proof will be completed by showing that the nicerepresentative (Definition 3.3) I of Ω satisfies the same uniform cone property. Theproof uses some elements of D. Chenais [4, 6]’s original proof.

Proof of Theorem 6.11. Each Ω ∈ L(D, r,ω,λ) satisfies the same uniform coneproperty (cf. Definition 6.3 (ii)) of Chapter 2 with parameters r, λ, and rλ.By Theorem 6.8 of Chapter 2 it is uniformly Lipschitzian in the sense of Defini-tion 5.2 (iii) of Chapter 2. As a result ρ, the largest radius such that BH(0, ρ) ⊂PH(y − x) : ∀y ∈ B(x, rλ) ∩ ∂Ω, and the neighborhoods V = BH(0, ρ) andU = B(0, rλ) ∩ y : PHy ∈ BH(0, ρ) can be chosen as specified in Theorem 6.6 (i)of Chapter 2. They are the same for all Ω ∈ L(D, r,ω,λ): for all Ω ∈ L(D, r,ω,λ)and all x ∈ ∂Ω, Vx = V , and there exists Ax ∈ O(N) such that U(x) = x + AxU .By construction

B(x, ρ) ⊂ B(x, rλ) ∩ y : PHA−1x (y − x) ∈ BH(0, ρ) = x + AxU = U(x).

The familyB (z, ρ/2) : z ∈ D

is an open cover of D. Since D is compact, there ex-

ists a finite subcover Bimi=1, Bi = B (zi, ρ/2), of D. Now for any Ω ∈ L(D, r,ω,λ),

∂Ω ⊂ D and there is a subcover BikKk=1 of Bim

i=1 such that

∂Ω ⊂K⋃

k=1

Bik and ∀k, 1 ≤ k ≤ K, ∂Ω ∩Bik = ∅.

Pick a sequence xkKk=1 such that xk ∈ ∂Ω ∩Bik , 1 ≤ k ≤ K, and notice that

Bik = B(zik ,

ρ

2

)⊂ B(xk, ρ)

or

∃K ≤ m, ∃xkKk=1 ⊂ ∂Ω, ∂Ω ⊂

K⋃

k=1

B(xk, ρ).


Thus from the estimate (5.37) in Theorem 5.7 of Chapter 2 with cx = tanω,

HN−1(∂Ω) ≤K∑

k=1

HN−1(∂Ω ∩B(xk, ρ)

)≤

K∑

k=1

HN−1(∂Ω ∩ U(xk)

)

≤K∑

k=1

ρN−1αN−1√

1 + (tanω)2 ≤ m ρN−1αN−1√

1 + (tanω)2,

where αN−1 is the volume of the unit (N − 1)-dimensional ball.So the right-hand side of the above inequality is a constant that is equal to

p = p(D, r,ω,λ) > 0 and is independent of Ω in L(D, r,ω,λ):

∀Ω ∈ L(D, r,ω,λ), PD(Ω) = HN−1(∂Ω) ≤ p.

Now from Theorem 6.3 for any sequence Ωn ⊂ L(D, r,ω,λ), there exists Ω suchthat χΩ ∈ BV(D) and a subsequence, still denoted by Ωn, such that

χΩn → χΩ in L1(D) and PD(Ω) ≤ p.

(ii) To complete the proof we consider the representative I of Ω (cf. Defini-tion 3.3) and show that I ∈ L(D, r,ω,λ). We need the following lemma.

Lemma 6.1. Let χΩn → χΩ in Lp(D), 1 ≤ p <∞, for Ω ⊂ D, Ωn ⊂ D, and let Ibe the measure theoretic representative of Ω. Then

∀x ∈ I, ∀R > 0, ∃N(x, R) > 0, ∀n ≥ N(x, R), m(B(x, R) ∩ Ωn

)> 0.

Moreover,

∀x ∈ ∂I, ∀R > 0, ∃N(x, R) > 0, ∀n ≥ N(x, R),m(B(x, R) ∩ Ωn

)> 0 and m

(B(x, R) ∩ "Ωn

)> 0

and B(x, R) ∩ ∂Ωn = ∅.

Proof. We proceed by contradiction. Assume that

∃x ∈ I, ∃R > 0, ∀N > 0, ∃n ≥ N, m(B(x, R) ∩ Ωn

)= 0.

So there exists a subsequence Ωnk, nk →∞, such that

m(B(x, R) ∩ Ω

)= lim

k→∞m(B(x, R) ∩ Ωnk

)= 0

⇒ ∃R > 0, m(B(x, R) ∩ Ω

)= 0 =⇒ x ∈ Ω0 = int "I = "I.

But this is in contradiction with the fact that x ∈ I. For x ∈ ∂I = I∩ "I, simultane-ously apply the first statement to I and χΩn → χΩ and to "I and χ!Ωn

→ χ!Ω sinceχ!Ω = χ!I almost everywhere by Theorem 3.3 (i) and choose the larger of the twointegers. As for the last property it follows from the fact that the open ball B(x, R)cannot be partitioned into two nonempty disjoint open subsets.


We wish to prove that

∀x ∈ ∂I, ∃Ax ∈ O(N), ∀y ∈ I ∩B(x, r), y + AxC(λ,ω) ⊂ int I.

Since Ωn is Lipschitzian, "Ωn is also Lipschitzian and

χΩn= χΩn and χ!Ωn

= χ!Ωn

almost everywhere. So we apply the second part of the lemma to x ∈ ∂I,χΩn→ χΩ

and χ!Ωn→ χ!Ω. So for each x ∈ ∂I, for all k ≥ 1, there exists nk ≥ k such that

B(x,

r

2k

)∩ ∂Ωnk = ∅.

Denote by xk an element of that intersection:

∀k ≥ 1, xk ∈ B(x,

r

2k

)∩ ∂Ωnk .

By construction, xk → x. Next consider y ∈ B(x, r)∩I. By Lemma 6.1, there existsa subsequence of Ωnk, still denoted by Ωnk, such that

∀k ≥ 1, B(y,

r

2k

)∩ Ωnk = ∅.

For each k ≥ 1 denote by yk a point of that intersection. By construction,

yk ∈ Ωnk → y ∈ I ∩B(x, r).

There exists K > 0 large enough such that for all k ≥ K, yk ∈ B(xk, r). To see thisnote that y ∈ B(x, r) and that

∃ρ > 0, B(y, ρ) ⊂ B(x, r) and |y − x| +ρ

2< r.

Now

|yk − xk| ≤ |yk − y| + |y − x| + |x− xk|

≤ r

2k+ r − ρ

2+

r

2k≤ r +

[ r

2k−1 −ρ

2

]< r

forr

2k−1 −ρ

2< 0 =⇒ k > 2 +

log(r/ρ)log 2

.

So we have constructed a subsequence Ωnk such that for k ≥ K,

xk ∈ ∂Ωnk → x ∈ ∂I and yk ∈ Ωnk ∩B(xk, r)→ y ∈ I ∩B(x, r).

For each k, there exists Ak ∈ O(N) such that

yk + AkC(λ,ω) ⊂ int Ωnk .


Pick another subsequence of Ωnk, still denoted by Ωnk, such that

∃A ∈ O(N), Ak → A,

since O(N) is compact. Now consider z ∈ y + AC(λ,ω). Since z is an interior point

∃ρ > 0, B(z, ρ) ⊂ y + AC(λ,ω).

So there exists K ′ ≥ K such that

∀k ≥ K ′, B(z,ρ

2

)⊂ yk + AkC(λ,ω) ⊂ int Ωnk .

Therefore, for k ≥ K ′

m(

B

(z,ρ

2

))= m

(B

(z,ρ

2

)∩ Ωnk

),

and, as k →∞,

m(

B

(z,ρ

2

))= m

(B

(z,ρ

2

)∩ Ω

),

and by Definition 3.3, z ∈ Ω1 and y + AC(λ,ω) ⊂ Ω1 = int I. This proves thatI ⊂ D satisfies the uniform cone property and I ∈ L(D,λ,ω, r).

7 Existence for the Bernoulli Free Boundary Problem7.1 An Example: Elementary Modeling of the Water WaveConsider a fluid in a domain Ω in R3 and assume that the velocity of the flow usatisfies the Navier–Stokes equations

ut + Du u− ν∆u +∇p = −ρ g in Ω, div u = 0 in Ω, (7.1)

where ν > 0 and ρ > 0 are the respective viscosity and density of the fluid, p isthe pressure, and g is the gravity constant. The second equation characterizes theincompressibility of the fluid. A standard example considered by physicists is thewater wave in a channel. The boundary conditions on ∂Ω are the sliding conditionsat the bottom S and on the free boundary Γ, that is,

u · n = 0 on S ∪ Γ. (7.2)

Assume that a stationary regime has been reached so that the velocity of the fluidis no longer a function of the time. Furthermore assume that the motion of the fluidis irrotational. By the classical Hodge decomposition, the velocity can be writtenin the form

u = ∇ϕ+ curl ξ. (7.3)

7. Existence for the Bernoulli Free Boundary Problem 259

As curl u = curl curl ξ = 0 we conclude that ξ = 0, since curl curl is a good isomor-phism. Then u = ∇ϕ and the incompressibility condition becomes

∆ϕ = 0 in Ω. (7.4)

Therefore the Navier–Stokes equation reduces to

D(∇ϕ)∇ϕ+ ν∇(∆ϕ) +∇p = −∇(ρgz), (7.5)

but

D(∇ϕ)∇ϕ =12∇(|∇ϕ|2), (7.6)

so that

∇(

12|∇ϕ|2 + p + ρgz

)= 0 in Ω. (7.7)

Then if Ω is connected, there exists a constant c such that

12|∇ϕ|2 + p + ρgz = c in Ω. (7.8)

In the domain Ω, that Bernoulli condition yields explicitly the pressure p as afunction of the velocity |∇ϕ| and the height z in the fluid. The flow is now assumedto be independent of the transverse variable y so that the initial three-dimensionalproblem in a perfect channel reduces to a two-dimensional one. Since the problemhas been reduced to a two-dimensional one, introduce the harmonic conjugate ψ,the so-called stream function, so that the boundary condition ∂ϕ/∂n = 0 takes theform ψ = constant on each connected component of the boundary ∂Ω.

On the free boundary at the top of the wave, the pressure is related to theexisting atmospheric pressure pa through the surface tension σ > 0 and the meancurvature H of the free boundary. Actually

p− pa = −σH (7.9)

on the free boundary of the stationary wave, where H is the mean curvature associ-ated with the fluid domain. Also, from the Cauchy conditions, we have |∇ψ| = |∇ϕ|so that the boundary condition (7.8) on the free boundary of the wave becomes

12|∇ψ|2 + ρgz + σH = pa. (7.10)

In order to simplify the presentation we replace the equation ∆ψ = 0 by∆ψ = f in Ω to avoid a forcing term on a part of the boundary, and we show thatthe resulting free boundary problem has the following shape variational formulation.Let D be a fixed, sufficiently large, smooth, bounded, and open domain in R2. Leta be a real number such that 0 < a < m(D). To find Ω ⊂ D, m(Ω) = a andψ ∈ H1

0 (D) such that

−∆ψ = f in Ω (7.11)


and ψ = constant and satisfies the boundary condition (7.10) on the free part∂Ω ∩D of the boundary.

For a fixed Ω, the solution of this problem is a minimizing element of thefollowing variational problem:

J(Ω) def= infϕ∈H1

⋄(Ω;D)

∫

Ω

(12|∇ϕ|2 − fϕ+ ρgz

)dx + σ PD(Ω), (7.12)

where

H1⋄ (Ω; D) def=

u ∈ H1

0 (D) : u(x) = 0 a.e. x in D \ Ω

(7.13)

is the relaxation of the definition of the Sobolev space H10 (Ω)10 for any measurable

subset Ω of D. Its properties were studied in Theorem 3.6.With that formulation there exists a measurable Ω∗ ⊂ D, |Ω∗| = a, such that

∀Ω ⊂ D, |Ω| = a, J(Ω∗) ≤ J(Ω). (7.14)

By using the methods of Chapter 10 it follows that if ∂Ω∗ is sufficiently smooth,the shape Euler condition dJ(Ω∗;V ) = 0 yields the original free boundary problemand the free boundary condition (7.10). The existence of a solution will now followfrom Theorem 6.3 in section 6.1. The case without surface tension is physicallyimportant. It occurs in phenomena with “evaporation ” (cf. M. Souli and J.-P. Zolesio [2]). In that case the previous Bernoulli condition becomes

∣∣∣∣∂ξ

∂n

∣∣∣∣2

= g2 (g ≥ 0)

on the free boundary, so that if we assume (in the channel setting) that there is nocavitation or recirculation in the fluid, then ∂ξ/∂n > 0 on the free boundary and weget the Neumann-like condition ∂ξ/∂n = g together with the Dirichlet condition.In section 7.4 we shall consider the case with surface tension.

7.2 Existence for a Class of Free Boundary ProblemsConsider the following free boundary problem, studied in J.-P. Zolesio [25, 26,29]: find Ω in a fixed holdall D and a function y on Ω such that

−∆y = f in Ω, y = 0 and∂y

∂n= Q2 on ∂Ω, (7.15)

where f and Q are appropriate functions defined in D. To study this type of problemH. W. Alt and L. A. Caffarelli [1] have introduced the following function:

J(ϕ) def=∫

D

12|∇ϕ|2 − fϕ dx +

∫

DQ2χϕ>0 dx, (7.16)

10In Chapter 8 the space H10 (Ω; D) of extensions by zero to D of elements of H1

0 (Ω) is definedfor Ω open. It is then characterized in Lemma 6.1 of Chapter 8 by a capacity condition on thecomplement D\Ω. This characterization extends to quasi-open sets Ω as defined in section 6 ofChapter 8.


which is minimized over

Kdef= u ∈ H1

0 (D) : u(x) ≥ 0 a.e. in D, (7.17)

where χϕ>0 (resp., χϕ=0) is the characteristic function of the set x ∈ D : ϕ(x) >011 (resp., x ∈ D : ϕ(x) = 0). The existence of a solution is based on thefollowing lemma.

Lemma 7.1. Let un and χn be two converging sequences such that un → u inL2(D)-strong, and let the χn’s be characteristic functions, χn(1 − χn) = 0, whichconverge to some function λ in L2(D)-weak. Then

∀n, (1− χn)un = 0 ⇒ λ ≥ χu=0. (7.18)

Proof. We have (1 − χn)un = 0, and in the limit (1 − λ)u = 0. Thus on the setx : u(x) = 0 we have λ = 1; elsewhere λ lies between 0 and 1 as the weak limit ofa sequence of characteristic functions.

Proposition 7.1. Let f and Q be two elements of L2(D) such that f ≥ 0 almosteverywhere. There exists u in K which minimizes the function J over the positivecone K of H1

0 (D).

Proof. Let un ⊂ K be a minimizing sequence for the function J over the convexset K. Denote by χn the characteristic function of the set x ∈ D : un(x) > 0,which is in fact equal to the subset x ∈ D : un(x) = 0. It is easy to verify thatthe sequence un remains bounded in H1

0 (D). Still denote by un a subsequencethat weakly converges in H1

0 (D) to an element u of K. That convergence holdsin L2(D)-strong so that Lemma 7.1 applies and we get λ ≥ χu=0 for any weaklimiting element λ of the sequence χn (which is bounded in L2(D)). Denote byj the minimum of J over K. Then J(un) weakly converges to j. We get

∫

D

12|∇u|2 − fu dx ≤ lim

n→∞inf∫

D

12|∇un|2 − fun dx, (7.19)

∫

Dχu=0Q

2 dx ≤∫

DλQ2dx = lim

n→∞

∫

DχnQ2 dx. (7.20)

Finally, by adding these two estimates we get J(u) ≤ j.

Obviously in the upper bound in (7.20), Q2 is a positive function. Moreover

(i) the set Ω is given by x ∈ D : u(x) > 0, and

(ii) the restriction u|Ω of u to Ω is a weak solution of the free boundary problem

−∆u = f in Ω, u = 0 and∂u

∂n= Q2 on ∂Ω. (7.21)

11This set defined up to a set of zero measure is a quasi-open set in the sense of section 6 inChapter 8.


Effectively, the minimization problem (7.16)–(7.17) can be written as a shape opti-mization problem. First introduce for any measurable Ω ⊂ D the positive cone

H10 (Ω)+

def=u ∈ H1

⋄ (Ω; D) : u(x) ≥ 0 a.e. x ∈ D

in the Hilbert space H1⋄ (Ω; D). Then consider the following shape optimization

problem:

inf E(Ω) : Ω is measurable subset in D , (7.22)

where the energy function E is given by

E(Ω) def= minϕ∈H1

0 (Ω)+

∫

Ω

12|∇ϕ|2 − fϕ dx

+∫

ΩQ2 dx. (7.23)

The necessary condition associated with the minimum could be obtained by thetechniques introduced in section 2.1 of Chapter 10. The important difference withthe previous formulation (7.16)–(7.17) is that the independent variable in the objec-tive function is no longer a function but a domain. The shape formulation (7.22)–(7.23) now makes it possible to handle constraints on the volume or the perimeterof the domain, which were difficult to incorporate in the first formulation.

As we have seen in Theorem 3.6, H1⋄ (Ω; D) endowed with the norm of H1

0 (D)is a Hilbert space, and H1

0 (Ω)+ a closed convex cone in H1⋄ (Ω; D), so that for any

measurable subset Ω in D, problem (7.23) has a unique solution y in H10 (Ω)+.

Thus we have the following equivalence between problems (7.22)–(7.23) and theminimization (7.16)–(7.17) of J over K.

Proposition 7.2. Let u be a minimizing element of J over K. Then

Ω def= x ∈ D : u(x) > 0

is a solution of problem (7.22) and y = u|Ω is a solution of (7.23). Conversely, ifΩ is a measurable subset of D and y is a solution of (7.22)–(7.23) in H1

0 (Ω)+, thenthe element u defined by

u(x) def=

y(x), if x ∈ Ω,

0, if x ∈ D \ Ω,

belongs to K and minimizes J over K.

7.3 Weak Solutions of Some Generic Free Boundary Problems

7.3.1 Problem without Constraint

Problem (7.22)–(7.23) can be relaxed as follows: given any f in L2(D), G in L1(D),

(P0) inf E(Ω) : Ω ⊂ D measurable, (7.24)


where for any measurable subset Ω of D the energy function is now defined by


⋄(Ω;D)ED(ϕ) +

∫

ΩG dx, ED(ϕ) def=

∫

D

12|∇ϕ|2 − fϕ dx, (7.25)

where H1⋄ (Ω; D) is defined in (7.13). By Theorem 3.6 H1

⋄ (Ω; D) is contained inH1

• (Ω; D), and for any element u in H1⋄ (Ω; D) we have ∇u(x) = 0 for almost all x

in D \ Ω so that

∀ϕ ∈ H1⋄ (Ω; D), ED(ϕ) =

∫

Ω

12|∇ϕ|2 − fϕ dx (7.26)

⇒ E(Ω) = minϕ∈H1

⋄(Ω;D)

∫

Ω

12|∇ϕ|2 − fϕ dx +

∫

ΩG dx

. (7.27)

We have the following existence result for problem (P0).

Theorem 7.1. For any f in L2(D), G = Q2 in L1(D), there exists at least onesolution to problem (P0).

Proof. Let Ωn be a minimizing sequence for problem (P0), and for each n letun be the solution to problem (7.25) with Ω = Ωn. If χn is the characteristicfunction of the measurable set Ωn, we have un in H1

⋄ (Ωn;D), which implies that(1−χn)un = 0. On the other hand, the sequence un remains uniformly boundedin H1

0 (D). Taking ϕ = 0 in (7.25) we get∫

D

12

|∇un|2 − fun dx ≤ 0,

and the conclusion follows from the equivalence of norms in H10 (D). We can assume

that the sequence χn weakly converges in L2(D) to an element λ and that thesequence un weakly converges in H1

0 (D) to an element u. From Lemma 7.1 weget λ ≥ χu=0 almost everywhere in D. Define

Ω(u) def= x ∈ D : u(x) = 0.

Then u belongs to H10(Ω(u)

)and we have

m(Ω(u)) = m(x ∈ D : λ(x) = 1) ≤ α,

since we have

α = m(x : λ(x) = 1) + m(x : 0 ≤ λ(x) < 1).

In the limit with Ω = Ω(u) we get∫

Ω

12|∇u|2 − fu dx =

∫

D

12|∇u|2 − fu dx ≤ lim inf

∫

D

12|∇un|2 − fun dx, (7.28)

∫

ΩG dx ≤

∫

DλG dx = lim

n→∞

∫

Ωn

G dx. (7.29)

By adding (7.28) and (7.29) we get that Ω minimizes E and u minimizes E(Ω).


7.3.2 Constraint on the Measure of the Domain Ω

Consider an important variation of problem (7.22)–(7.23). Given any f in L2(D),G in L1(D), and a real number α, 0 < α < m(D),

(Pα0 ) inf E(Ω) : Ω ⊂ D measurable and m(Ω) = α . (7.30)

We have the following existence result for problem (Pα0 ).

Theorem 7.2. For any f in L2(D), G = 0, and any real number α, 0 < α < m(D),there exists at least one solution to problem (Pα

0 ).

Proof. Let Ωn be a minimizing sequence for problem (P0), and for each n let un

be the unique solution to problem (7.25). If χn is the characteristic function of themeasurable set Ωn, we have un in H1

⋄ (Ωn;D), which implies that (1 − χn)un = 0.On the other hand, by picking ϕ = 0 in (7.25), un remains bounded in H1

0 (D),∫

D

12|∇un|2 − fun dx ≤ 0,

and the conclusion follows from the equivalence of norms in H10 (D). We can as-

sume that χn weakly converges in L2(D) to an element λ and that un weaklyconverges in H1

0 (D) to an element u ∈ H10 (D). In the limit we get

∫

Dλ(x) dx = lim

n→∞

∫

Dχn(x) dx = α.

From Lemma 7.1 we get λ ≥ χu=0 almost everywhere in D. Define

Ω(u) def= x ∈ D : u(x) = 0.

Then u belongs to H1⋄ (Ω(u); D) and we have

m(Ω(u)) = m(x ∈ D : λ(x) = 1) ≤ α

(as we have α = m(x : λ(x) = 1) + m(x : 0 ≤ λ(x) < 1)). In the limit we getfor Ω = Ω(u)∫

Ω

12|∇u|2 − f u dx =

∫

D

12|∇u|2 − fu dx ≤ lim inf

n→∞

∫

D

12|∇un|2 − fun dx, (7.31)

∫

DλG dx = lim

n→∞

∫

Ωn

G dx, (7.32)

so that ∫

Ω

12|∇u|2 − fu dx +

∫

DλG dx ≤ inf

Ω′⊂Dm(Ω′)=α

E(Ω′).

If G ≥ 0 almost everywhere in D, we get

E(Ω) ≤ infΩ′⊂D

m(Ω′)=α

E(Ω′) ,


but Ω does not necessarily satisfy the constraint m(Ω) = α. Note that for anymeasurable set Ω′ such that Ω ⊂ Ω′ ⊂ D, we have

∫

Ω′

12|∇u|2 − f u dx =

∫

Ω

12|∇u|2 − fu dx

so that in expression (7.31) Ω can be enlarged to any such Ω′. The inclusion of Ω inΩ′ implies the inclusion of H1

⋄ (Ω; D) in H1⋄ (Ω′;D). So if G ≤ 0 almost everywhere

in D, then it is readily seen that from (7.25) E(Ω′) ≤ E(Ω). In view of the previousassumption on G, now G = 0 almost everywhere in D. To conclude the proof wejust have to select Ω′ with m(Ω′) = α. That measurable set Ω′ is admissible andminimizes the objective function in (7.30), and we have

E(Ω′) = E(Ω) = infΩ′′⊂D

m(Ω′′)=α

E(Ω′′).

Corollary 1. Assume that G = 0 and f in L2(D) and f = Q2 in L1(D). Thenthe following problem has an optimal solution:

(Pα−0 ) inf

Ω⊂Dm(Ω)≤α

E(Ω). (7.33)

Proof. The proof is similar to the proof of the theorem where the minimizingsequence is chosen such that m(Ωn) ≤ α, so that in the weak limit we get

m(Ω) ≤∫

Dλ(x) dx ≤ α.

7.4 Weak Existence with Surface TensionProblems (P0), (Pα

0 ), and (Pα−0 ) have optimal solutions, but as they are associated

with the homogeneous Dirichlet boundary condition, u in H1⋄ (Ω; D), the optimal

domain Ω is in general not allowed to have holes, that is to say, roughly speaking,that the topology of Ω is a priori specified. In many examples it turns out thatthe solution u physically corresponds to a potential and the classical homogeneousDirichlet boundary condition is not the appropriate one. The physical conditionis that the potential u should be constant on each connected component of theboundary ∂Ω in D. When Ω is a simply connected domain in R2, then ∂Ω has asingle connected component so the constant can be taken as zero. In general thisconstant can be fixed only in one connected component; in the others the constantis an unknown of the problem.

The minimization problems (P0), (Pα0 ), and (Pα−

0 ) associated with the Hilbertspace H1

• (Ω; D) fail (in the sense that the previous techniques for existence of anoptimal Ω fail). The main reason is that Lemma 7.1 is no longer true when un

is replaced by ∇un weakly converging in L2(D)N . The key idea is to recover theequivalent of Lemma 7.1 by imposing the strong L2(D)-convergence of the sequenceun. In practice un corresponds to the sequence of characteristic functions


χΩn of a minimizing sequence Ωn. To obtain the strong L2(D)-convergence of asubsequence we add a constraint on the perimeters. Consider the family of finiteperimeter sets in D of Definition 6.2 (ii) in section 6.1,

BPS(D) def= Ω ⊂ D : χΩ ∈ BV(D),

where BV(D) is defined in (6.2). Introduce the following problem indexed by σ > 0:

(Pασ ) inf

Ω⊂Dm(Ω)=α

Eσ(Ω), Eσ(Ω) def= E(Ω) + σPD(Ω), (7.34)


•(Ω;D)

∫

Ω

12|∇ϕ|2 − fϕ dx +

∫

ΩG dx

. (7.35)

Theorem 7.3. Let f in L2(D), G in L1(D), σ > 0, and 0 ≤ α < m(D). Thenproblem (Pα

σ ) has at least one optimal solution Ω in BPS(D).

Proof. Let Ωn be a minimizing sequence for (Pασ ) and let χn be the correspond-

ing sequence of characteristic functions associated with Ωn. By picking ϕ = 0 in(7.35),

PD(Ωn) ≤ σ−1[c∥f∥2L2(D) +

∫

D|G| dx + σPD(D)

].

By Theorem 6.3 there exist a subsequence of Ωn, still indexed by n, and Ω ⊂ Dwith finite perimeter such that χn → χ = χΩ in L1(D)-strong and α = m(Ωn) =m(Ω). For each n, let un in H1

• (Ωn;D) be the unique minimizer of (7.35). Thatsequence remains bounded in H1

0 (D). Pick a subsequence, still indexed by n, suchthat un weakly converges to an element u in H1

0 (D). From the identity (1 −χn)∇un = 0 almost everywhere in D, we get in the limit (1 − χΩ)∇u = 0 almosteverywhere in D, so that the limiting element u belongs to H1

• (Ω; D). Hence wehave

∫

Ω

12|∇u|2 − fu dx =

∫

D

12|∇u|2 dx−

∫

Ωfu dx

≤ lim infn→∞

∫

D

12|∇un|2 dx−

∫

Ωn

fun dx,(7.36)

∫

Ωfu dx = lim

n→∞

∫

Ωn

fun dx,

∫

ΩG dx = lim

n→∞

∫

Ωn

G dx, (7.37)

so that∫

Ω

12|∇u|2 − fu dx +

∫

ΩG dx + σPD(Ω) ≤ inf

Ω⊂Dm(Ω)=α

Eσ(Ω). (7.38)

Chapter 6

Metrics via DistanceFunctions

1 IntroductionIn Chapter 5 the characteristic function was used to embed the equivalence classes ofmeasurable subsets of D into Lp(D) or Lp

loc(D), 1 ≤ p <∞, and induce a completemetric on the equivalence classes of measurable sets. This construction is genericand extends to other set-dependent functions embedded in an appropriate functionspace. The Hausdorff metric is the result of such a construction, where the distancefunction plays the role of the characteristic function. The distance function embedsequivalence classes of subsets A of a closed holdall D with the same closure A intothe space C(D) of continuous functions. When D is bounded the C0-norm of thedifference of the distance functions of two subsets of D is the Hausdorff metric.The Hausdorff topology has many much desired properties. In particular, for Dbounded, the set of equivalence classes of nonempty subsets A of D is compact.

Yet, the volume and perimeter are not continuous with respect to the Haus-dorff topology. This can be fixed by changing the space C(D) to the space W 1,p(D)since distance functions also belong to that space. With that metric the volume isnow continuous. The price to pay is the loss of compactness when D is bounded.But other sequentially compact families can easily be constructed. By analogy withCaccioppoli sets we introduce the sets for which the elements of the Hessian matrixof second-order derivatives of the distance function are bounded measures. Theyare called sets of bounded curvature. Their closure is a Caccioppoli set, and theyenjoy other interesting properties. Convex sets belong to that family. General com-pactness theorems are obtained for such families under global or local conditions.This chapter also deals with the family of open sets characterized by the distancefunction to their complement. They are discussed in parallel along with the distancefunctions to a set.

The properties of distance functions and Hausdorff and Hausdorff complemen-tary metric topologies are studied in section 2. The projections, skeletons, cracks,and differentiability properties of distance functions are discussed in section 3. W 1,p-topologies are introduced in section 4 and related to characteristic functions. The

267

268 Chapter 6. Metrics via Distance Functions

compact families of sets of bounded and locally bounded curvature are characterizedin section 5.

The notion of reach and Federer’s sets of positive reach are studied in sec-tion 6. The smoothness of smooth closed submanifolds is related to the smoothnessof the square of the distance function in a neighborhood. Approximation of distancefunctions by dilated sets/tubular neighborhoods and their critical points are pre-sented in section 7. Convex sets are characterized in section 8 along with the specialfamily of convex sets. Both sets of positive reach and convex sets will be furtherinvestigated in Chapter 7. Finally section 9 gives several compactness theoremsunder global and local conditions on the Hessian matrix of the distance function.

2 Uniform Metric Topologies2.1 Family of Distance Functions Cd(D)In this section we review some properties of distance functions and present thegeneral approach that will be followed in this chapter.

Definition 2.1.Given A ⊂ RN the distance function from a point x to A is defined as

dA(x) def=

infy∈A

|y − x|, A = ∅,

+∞, A = ∅,(2.1)

and the family of all distance functions of nonempty subsets of D as

Cd(D) def=dA : ∀A, ∅ = A ⊂ D

. (2.2)

When D = RN the family Cd(RN) is denoted by Cd.

By definition dA is finite in RN if and only if A = ∅.Recall the following properties of distance functions.

Theorem 2.1. Assume that A and B are nonempty subsets of RN.

(i) The map x '→ dA(x) is uniformly Lipschitz continuous in RN:

∀x, y ∈ RN, |dA(y)− dA(x)| ≤ |y − x| (2.3)

and dA ∈ C0,1loc (RN).1

(ii) There exists p ∈ A such that dA(x) = |p− x| and dA = dA in RN.

(iii) A = x ∈ RN : dA(x) = 0.1A function f belongs to C0,1

loc (RN) if for all bounded open subsets D of RN its restriction toD belongs to C0,1(D).

2. Uniform Metric Topologies 269

(iv) dA = 0 in RN ⇐⇒ A = RN.

(v) A ⊂ B ⇐⇒ dA ≥ dB.

(vi) dA∪B = mindA, dB.

(vii) dA is (Frechet) differentiable almost everywhere and

|∇dA(x)| ≤ 1 a.e. in RN . (2.4)

Proof. (i) For all z ∈ A and x, y ∈ RN

|z − y| ≤ |z − x| + |y − x|,dA(y) = inf

y∈A|z − y| ≤ inf

y∈A|z − x| + |y − x| = dA(x) + |y − x|

and hence the Lipschitz continuity.(ii) As the function y '→ |y − x| is continuous (hence upper semicontinuous),

dA(x) = infy∈A

|y − x| = infy∈A

|y − x| = dA(x).

Since the function |y − x| ≥ 0 is bounded below, its infimum over y ∈ A is finite.Let yn ⊂ A be a minimizing sequence

dA(x) = infy∈A

|y − x| = limn→∞

|yn − x|.

Then yn − x and hence yn are bounded sequences. Hence there exists a subse-quence converging to some y ∈ A and by continuity dA(x) = |y − x|.

(iii) By the continuity of dA, d−1A 0 is closed and

A ⊂ d−1A 0 ⇒ A ⊂ d−1

A 0.

Conversely, if dA(x) = 0, there exists y in A such that 0 = dA(x) = |y − x| andnecessarily x = y ∈ A.

(iv) follows from (iii).(v) From (ii) for x ∈ A, dA(x) = 0 and dB(x) ≤ dA(x) = 0 necessarily implies

dB(x) = 0 and x ∈ B. Conversely, if A ⊂ B, then

dB(x) = dB(x) = infy∈B

|y − x| ≤ infy∈A

|y − x| = dA(x) = dA(x).

(vi) is obvious.(vii) follows from Rademacher’s theorem.

2.2 Pompeiu–Hausdorff Metric on Cd(D)Let D be a nonempty subset of RN and associate with each nonempty subset A ofD the equivalence class

[A]ddef=B : ∀B,B ⊂ D and B = A

,


since from Theorem 2.1 (v) dA = dB if and only if A = B. So A is the invariantclosed representative of the class [A]d. Consider the set

Fd(D) def=[A]d : ∀A, ∅ = A ⊂ D

that can be identified with the setA : ∀A, ∅ = A closed ⊂ D

.

In general, there is no open representative in the class [A]d since

dA = dA ≤ dint A,

where int A = !!A denotes the interior of A. By the definition of [A]d the map

[A]d '→ dA : Fd(D)→ Cd(D) ⊂ C(D)

is injective. So Fd(D) can be identified with the subset of distance functions Cd(D)in C(D). The distance function plays the same role as the set X(D) in Lp(D) ofequivalence classes of characteristic functions of measurable sets.

When D is bounded, C(D) is a Banach space when endowed with the norm

∥f∥C(D) = supx∈D

|f(x)|. (2.5)

As for the characteristic functions of Chapter 5, this induces a complete metric

ρ([A]d, [B]d)def= ∥dA − dB∥C(D) = sup

x∈D|dA(x)− dB(x)| (2.6)

on Fd(D), which turns out to be equal to the classical Pompeiu–Hausdorff metric2

ρH([A]d, [B]d)def= max

supx∈B

dA(x), supy∈A

dB(y)

(cf. J. Dugundji [1, Chap. IX, Prob. 4.8, p. 205] for the definition of ρH). Indeedfor x ∈ D, xA ∈ A, and xB ∈ B

|x− xA| ≤ |x− xB | + |xB − xA|⇒ dA(x) ≤ |x− xB | + dA(xB) ≤ |x− xB | + sup

y∈BdA(y)

⇒ dA(x) ≤ dB(x) + supy∈B

dA(y) ⇒ dA(x)− dB(x) ≤ supy∈B

dA(y).

2The “ecart mutuel” between two sets was introduced by D. Pompeiu [1] in his thesis presentedin Paris in March 1905. This is the first example of a metric between two sets in the literature.It was studied in more detail by F. Hausdorff [2, “Quellenangaben,” p. 280, and Chap. VIII,sect. 6] in 1914.


By interchanging the roles of A and B

∀x ∈ D, |dA(x)− dB(x)| ≤ max

supz∈B

dA(z), supy∈A

dB(y)

.

The converse inequality follows from the fact that A ⊂ D and B ⊂ D.When D is open but not necessarily bounded, we use C0

loc(D) (cf. (2.19) insection 2.6.1 of Chapter 2), the space C(D) of continuous functions on D endowedwith the Frechet topology of uniform convergence on compact subsets K of D, whichis defined by the family of seminorms

∀K compact ⊂ D, qK(f) def= maxx∈K

|f(x)|. (2.7)

The Frechet topology on C(D) is metrizable since the topology induced by thefamily of seminorms qK is equivalent to the one generated by the subfamilyqKkk≥1, where the compact sets Kkk≥1 are chosen as follows:

Kkdef=

x ∈ D : d!D(x) ≥ 1k

and |x| ≤ k

, k ≥ 1. (2.8)

It is equivalent to the topology defined by the metric

δ(f, g) def=∞∑

k=1

12k

qKk(f − g)1 + qKk(f − g)

. (2.9)

It will be shown below that Cd(D) is a closed subset of Cloc(D) and that this willinduce the following complete metric on Fd(D):

ρδ([A]d, [B]d)def= δ(dA, dB) =

∞∑

k=1

12k

qKk(dA − dB)1 + qKk(dA − dB)

, (2.10)

which is a natural extension of the Hausdorff metric to an unbounded domain D.

Theorem 2.2. Let D = ∅ be an open (resp., bounded open) holdall in RN.

(i) The set Cd(D) is closed in Cloc(D) (resp., C(D)) and ρδ (resp., ρ) defines acomplete metric topology on Fd(D).

(ii) When D is a bounded open subset of RN, the set Cd(D) is compact in C(D)and the metrics ρ and ρH are equal.

Proof. (i) We give the proof of C. Dellacherie [1, Thm. 2, p. 42 and Rem. 1, p. 43].The basic constructions will again be used in Chapter 7. Consider a sequence Anof nonempty subsets of D such that dAn converges to some element f of Cloc(D).We wish to prove that f = dA for

Adef= x ∈ D : f(x) = 0


and that the closed subset A of D is nonempty. Fix x in D; then

∀n,∃yn ∈ An, |yn − x| = infz∈An

|z − x| = dAn(x)

⇒ limn→∞

|yn − x| = limn→∞

dAn(x) = f(x).

Hence yn − x and yn ⊂ D are bounded and there exists a subsequence, stillindexed by n, which converges to some y ∈ D

yn → y, and |y − x| = f(x).

In particular, f(y) = 0, since in the inequality

f(y) ≤ f(y)− dAn(y) + dAn(y)− dAn(yn) + dAn(yn),

the last term is zero and dAn is Lipschitz continuous with constant equal to 1:

|f(y)| ≤ |f(y)− dAn(y)| + |y − yn|,

and both terms go to zero. By the definition of A, y ∈ A and A is not empty.Therefore for each x ∈ D, there exists y ∈ A such that

f(x) = |y − x| ≥ infz∈A

|z − x| = dA(x).

Next we prove the inequality in the other direction. By construction for any An ⊂ D

∀x, y ∈ D, |dAn(x)− dAn(y)| ≤ |x− y|

and

|f(x)− f(y)| ≤ |f(x)− dAn(x)| + |dAn(x)− dAn(y)| + |dAn(y)− f(y)|.

By uniform convergence the first and last terms converge to zero, and by Lipschitzcontinuity of dAn ,

∀x, y ∈ D, |f(x)− f(y)| ≤ |x− y|.

Hence for all x ∈ D and y ∈ A, f(y) = 0 and

f(x) ≤ f(y) + |x− y| = |x− y| ⇒ ∀x ∈ D, f(x) ≤ infy∈A

|x− y| = dA(x).

This proves the reverse equality.(ii) Observe that on a compact set D and for any A ⊂ D

dA(x) = infy∈A

|y − x| ≤ supy∈A

|y − x| ≤ cdef= sup

y,x∈D

|y − x| <∞,

∀x, y ∈ D, |dA(y)− dA(x)| ≤ |y − x|.


The compactness of Cd(D) now follows by Ascoli–Arzela Theorem 2.4 of Chapter 2and part (i). By construction, ρ and ρδ are metrics on Fd(D). By definition, for Aand B in the compact D,

ρ(A, B) = maxx∈D

|dA(x)− dB(x)|

≥ max

maxx∈B

|dA(x)− dB(x)|,maxx∈A

|dA(x)− dB(x)|

≥ max

maxx∈B

dA(x), maxx∈A

dB(x)

= ρH(A, B).

Conversely, for any x ∈ D and y in A,

dA(x)− dB(x) ≤ |y − x|− dB(x)

and there exists xB ∈ B such that dB(x) = |x− xB |. Therefore,

∀y ∈ A, dA(x)− dB(x) ≤ |y − x|− |x− xB | ≤ |y − xB |⇒ dA(x)− dB(x) ≤ inf

y∈A|y − xB | = dA(xB) ≤ max

x∈BdA(x).

SimilarlydB(x)− dA(x) ≤ max

x∈AdB(x),

and for all x in D

|dB(x)− dA(x)| ≤ max

maxx∈B

dA(x),maxx∈A

dB(x)⇒ ρ(A, B) ≤ ρH(A, B).

When D is bounded the family Fd(D) enjoys more interesting properties.

Theorem 2.3. Let D be a nonempty open (resp., bounded open) subset of RN.Define for a subset S of RN the sets

H(S) def=dA ∈ Cd(D) : S ⊂ A

,

I(S) def=dA ∈ Cd(D) : A ⊂ S

,

J(S) def=dA ∈ Cd(D) : A ∩ S = ∅

.

(i) Let S be a subset of RN. Then H(S) is closed in Cloc(D) (resp., C(D)).

(ii) Let S be a closed subset of RN. Then I(S) is closed in Cloc(D) (resp., C(D)).If, in addition, S∩D is compact, then J(S) is closed in Cloc(D) (resp., C(D)).

(iii) Let S be an open subset of RN. Then J(S) is open in Cloc(D) (resp., C(D)).If, in addition, !S∩D is compact, then I(S) is open in Cloc(D) (resp., C(D)).

(iv) For a nonempty bounded holdall D, the map

[A]d '→ #(A) : F#(D) def=[A]d : ∀A, ∅ = A ⊂ D and #(A) <∞

→ R


is lower semicontinuous, where #(A) is the number of connected componentsas defined in Definition 2.2 of Chapter 2. For a fixed number c > 0 the subset

dA ∈ Cd(D) : #(A) ≤ c

of Cd(D) is compact in C(D). In particular, the subset

dA ∈ Cd(D) : A is connected

of Cd(D) is compact in C(D).

Proof. (i) If S ⊂ D, then H(S) = ∅ and there is nothing to prove. Assume thatS ⊂ D. From Theorem 2.1 (v), S ⊂ A ⇒ dS ≥ dA. So for any sequence dAn inH(S) converging to dA in Cloc(D),

∀n, dS ≥ dAn ⇒ dS ≥ dA ⇒ S ⊂ S ⊂ A ⇒ dA ∈ H(S).

(ii) We shall use the same technique for I(S) as for H(S), but here we needS = S to conclude. For D ∩ S = ∅, J(S) = ∅ and there is nothing to prove.Assume that D ∩ S = ∅ and consider a sequence dAn in J(S) converging to dA

in Cloc(D). Assume that A∩S = ∅. Then A ⊂ D implies A∩ [S∩D] = A∩S = ∅.By assumption, S ∩D is compact and

∃δ > 0, ∀x ∈ S ∩D, dA(x) ≥ δ,∃N ≥ 1, ∀n ≥ N, ∥dAn − dA∥C(D) ≤ δ/2.

So for all x ∈ D ∩ S

dAn(x) ≥ dA(x)− |dAn(x)− dA(x)| ≥ δ − δ/2 > 0

⇒ An ∩ S = An ∩ [D ∩ S] = ∅,

and this contradicts the fact that An ∈ J(S).(iii) By definition, for any S, !J(S) = I(!S),

!J(S) =dA ∈ Cd(D) : A ∩ S = ∅

=dA ∈ Cd(D) : A ⊂ !S

= I(!S).

Since !S is closed, !J(S) is closed from part (ii) and J(S) is open. Similarly,replacing S by !S in the previous identity, I(S) = !J(!S). So from part (ii) ifD ∩ !S is compact, then !I(S) is closed and I(S) is open. From part (ii) if D ∩ !Sis compact, then !I(S) is closed and I(S) is open.

(iv) Let An and A be nonempty subsets of D such that dAn converges todA in C(D). Assume that #(A) = k is finite. Then there exists a family of disjointopen sets G1,. . . , Gk such that

A ⊂ G = ∪ki=1Gi and ∀i, A ∩Gi = ∅.

In view of the definitions of I(S) and J(S) in part (i)

A ∈ U = ∩ki=1J(Gi) ∩ I(G).


But U is not empty and open as the finite intersection of k + 1 open sets. As aresult there exist ε > 0 and an open neighborhood of [A]d:

Nε([A]d) = [B]d : ∥dB − dA∥ < ε ⊂ U .

Hence since dAn converges to dA, there exists n > 0 such that, for all n ≥ n,[An]d ∈ U , and necessarily

∀n ≥ n, An ⊂ G, An ∩Gi = ∅,∀i ⇒ #(An) ≥ #(A).

Therefore, [A]d '→ #(A) is lower semicontinuous.

This theorem has many interesting corollaries. For instance part (iii) sayssomething about the function that gives the number of connected components of A(cf. T. J. Richardson [1] for an application to image segmentation).

2.3 Uniform Complementary Metric Topology and Ccd(D)

In the previous section we dealt with a theory of closed sets since the equivalenceclass of the subsets of D was completely determined by their unique closure. Inpartial differential equations the underlying domain is usually open. To accommo-date this point of view, consider the set of open subsets Ω of a fixed nonemptyopen holdall D in RN, endowed with the Hausdorff topology generated by thedistance functions d!Ω to the complement of Ω. This approach has been used inseveral contexts, for instance, in J.-P. Zolesio [6, sect. 1.3, p. 405] in the con-text of free boundary problems. Also, V. Sverak [2] uses the family Cc

d(D) foropen sets Ω such that #([!Ω]) ≤ c for some fixed c > 0. His main result is thatin dimension 2 the convergence of a sequence Ωn to Ω of such sets implies theconvergence of the corresponding projection operators PΩn : H1

0 (D) → H10 (Ωn)

to PΩ : H10 (D)→ H1

0 (Ω), where the projection operators are directly related to ho-mogeneous Dirichlet linear boundary value problems on the corresponding domainsΩn and Ω. In dimension 1 the constraint on the number of components can bedropped. This result will be discussed in Theorem 8.2 of section 8 in Chapter 8.

By analogy with the constructions of the previous section, consider for anonempty subset D of RN the family

d!A : ∅ = !A and A ⊂ D

.

By definition,

!A ⊃ !D ⇒ d!A = 0 in !D,

and, by Lipschitz continuity, d!A = 0 in !D and d!A = 0 on ∂D = D ∩ !D. IfintD = ∅, then d!A = 0 in RN and for all sets A in the family intA = ∅. Ifint D = ∅, associate with each A the open set

Ω def= intA = !!A ⇒ !Ω = !A and d!Ω = d!A.


By definition and the previous considerations,

!A ⊃ !D ⇒ Ω = !!A ⊂ !!D = int D.

So finally, for intD = ∅,d!A : ∅ = !A and A ⊂ D

=d!Ω : Ω = RN and Ω open ⊂ intD

.

The invariant set int A is the open representative in the equivalence class

[A]cddef=

B : !B = !A = ∅

,

but, in general, there is no closed representative in that class since

d!int A = d!A= d!A ≤ d!A,

as in the case of dA, where dA = dA ≤ dint A.From this analysis it will be sufficient to consider the family of open subsets

of an open holdall D.

Definition 2.2.Let D be a nonempty open subset of RN. Define the family of functions

Ccd(D) def=

d!Ω : ∀Ω open subset of D and Ω = RN (2.11)

corresponding to the family of open sets

G(D) def=

Ω ⊂ D : ∀Ω open and Ω = RN . (2.12)

For D bounded define the Hausdorff complementary metric ρcH on G(D):

ρcH(Ω2, Ω1)

def= ∥d!Ω2− d!Ω1

∥C(D). (2.13)

Note that for Ω1 ⊂ D and Ω2 ⊂ D, d!Ω1= d!Ω2

= 0 in !D. Therefore,

d!Ω1= d!Ω2

in D ⇐⇒ !Ω1 = !Ω2 ⇐⇒ Ω1 = Ω2.

Theorem 2.4. Let D be a nonempty open subset of RN.

(i) The set Ccd(D) is closed in Cloc(D).

(ii) If, in addition, D is bounded, then Ccd(D) is compact in C0(D) and (G(D), ρc

H)is a compact metric space.

(iii) (Compactivorous property)Let Ωn and Ω be sets in G(D) such as

d!Ωn→ d!Ω in Cloc(D).

Then for any compact subset K ⊂ Ω, there exists an integer N(K) > 0 suchthat

∀n ≥ N(K), K ⊂ Ωn.


Proof. (i) Let Ωn be a sequence of open subsets in D such that d!Ωn is a

Cauchy sequence in Cloc(D). For all n, d!Ωn= 0 in !D and d!Ωn

is also aCauchy sequence in Cloc(RN). By Theorem 2.2 (i) the sequence d!Ωn

convergesin Cloc(RN) to some distance function dA. By construction

Ωn ⊂ D =⇒ !Ωn ⊃ !D =⇒ d!Ωn≤ d!D

=⇒ dA ≤ d!D =⇒ A ⊃ !D =⇒ !A ⊂ D.

By choosing the open set Ω = !A in D we get

d!Ωn→ d!Ω in Cloc(RN) and C0,loc(D).

(ii) To prove the compactness we use the compactness of Cd(D) from The-orem 2.2 (ii) and the fact that Cc

d(D) is closed in C0(D). Observe that since!Ωn ⊃ !D, d!Ωn

= 0 in !D, d!Ωn∈ C0(D), and

!Ωn = [!Ωn ∩D] ∪ [!Ωn ∩ !D] ⊂ [!Ωn ∩D] ∪ !D ⊂ !Ωn

⇒ d!Ωn= d[!Ωn∩D]∪!D = mind!Ωn∩D, d!D.

There exist a subsequence, still indexed by n, and a set A, ∅ = A ⊂ D, such that

d!Ωn∩D → dA in C0(D)

⇒ d!Ωn= mind!Ωn∩D, d!D→ mindA, d!D = dA∪!D in C(D)

⇒ d!Ωn→ dA∪!D in C(RN)

since d!Ωn= 0 = dA∪!D in !D. The result now follows by choosing the open set

Ω def= ![A ∪ !D] = !A ∩D ⊂ D.

(iii) Definem

def= infx∈K

infy∈!Ω

|x− y| = infx∈K

d!Ω(x).

Since K ⊂ Ω ⊂ D, there exist x ∈ K, y ∈ ∂Ω such that

m = infx∈K

infy∈∂Ω

|x− y| = |x− y|.

Necessarily, m > 0 since Ω is open, x ∈ K ⊂ Ω, and y ∈ ∂Ω. Now

∃N > 0,∀n ≥ N, ∥d!Ωn− d!Ω∥C(K) < m/2

⇒ ∀x ∈ K, d!Ωn(x) ≥ d!Ω(x)− |d!Ωn

(x)− d!Ω(x)| ≥ m−m/2 > 0

⇒ x = !Ωn ⇒ x ∈ !!Ωn = int Ωn = Ωn ⇒ K ⊂ Ωn.

Notation 2.1.It will be convenient to write

ΩnHc

→ Ω

for the Hausdorff complementary convergence of open sets of G(D)

d!Ωn→ d!Ω in Cloc(D).


We have the analogue of Theorem 2.3 for the distance function to the com-plement of an open subset of D.

Theorem 2.5. Let D be a nonempty open (resp., bounded open) subset of RN.Define for a subset S of RN and open subsets Ω of D the following families:

Hc(S) def=d!Ω ∈ Cc

d(D) : S ⊂ !Ω

,

Ic(S) def=d!Ω ∈ Cc

d(D) : !Ω ⊂ S

,

Jc(S) def=d!Ω ∈ Cc

d(D) : !Ω ∩ S = ∅

.

(i) Let S be a subset of RN. Then Hc(S) is closed in Cloc(D) (resp., C(D)).

(ii) Let S be a closed subset of RN. Then Ic(S) is closed in Cloc(D) (resp., C(D)).If, in addition, S∩D is compact, then Jc(S) is closed in Cloc(D) (resp., C(D)).

(iii) Let S be an open subset of RN. Then Jc(S) is open in Cloc(D) (resp., C(D)).If, in addition, !S∩D is compact, then Ic(S) is open in Cloc(D) (resp., C(D)).

(iv) For a nonempty bounded open holdall D, the map

!Ω '→ #(!Ω) : G#(D) def=!Ω : Ω open ⊂ D and #

(!Ω)

<∞→ R

is lower semicontinuous, where #(!Ω) is the number of connected componentsof !Ω as defined in Definition 2.2 of Chapter 2. For a fixed number c ≥ 0,the subset

d!Ω ∈ Ccd(D) : #

(!Ω)≤ c

is compact in C(D). In particular, the subset

d!Ω ∈ Ccd(D) : Ω is hole-free

is compact in C(D), where hole-free is in the sense of Definition 2.2 of Chap-ter 2.

2.4 Families Ccd(E; D) and Cc

d,loc(E; D)

Now and then, it is desirable to avoid having the empty set as a solution of a shapeoptimization problem. This can occur for the C(D)-topology on Cc

d(D). One wayto get around this is to assume that each set of the family contains a fixed nonemptyset or, more generally, that each set of the family contains a translated and rotatedimage of a fixed nonempty set.

Theorem 2.6. Let D, ∅ = D ⊂ RN, be bounded open, let E, ∅ = E ⊂ D, and let

Ccd(E;D) def= d!Ω : E ⊂ Ω open ⊂ D , (2.14)

Ccd,loc(E;D) def=

d!Ω :

∃x ∈ RN, ∃A ∈ O(N) such thatx + AE ⊂ Ω open ⊂ D

. (2.15)

3. Projection, Skeleton, Crack, and Differentiability 279

(i) If E is open, Ccd(E;D) and Cc

d,loc(E;D) are compact in C(D).

(ii) If E is closed, Ccd(E;D) and Cc

d,loc(E;D) are open in C(D).

Proof. (a) We first deal with Ccd(E;D) using the fact that

Ccd(E;D) =

d!Ω : !Ω ⊂ !E and Ω open ⊂ D

= Ic(!E)

and Theorem 2.5. If E is open, !E is closed and Ic(!E) is closed and hence compactsince D is bounded. If E is closed, !E is open, E ∩ D is compact, and Ic(!E) isopen.

(b) Assume that E is open. Let d!Ωn be a Cauchy sequence in Cc

D(D)converging to some d!Ω. Denote by xn ∈ RN and An ∈ O(N) the translationand rotation of E such that xn + AnE ⊂ Ωn and d!xn+AnE ≤ d!Ω. Since, forall n, xn + AnE ⊂ D and D is bounded, there exist x ∈ RN and A ∈ O(N) andsubsequences, still indexed by n, such that xn → x and An → A. But

d!xn+AnE(z) = d!E(An(z − xn))

since A−1 = A in O(N) and d!E(An(z − xn))→ d!E(A(z − x)). Finally, d!x+AE =d!E(A(z − x)) ≤ d!Ω and, since !x + AE and !Ω are both closed, !x + AE ⊂ !Ω,which implies x + AE ⊃ Ω and d!Ω ∈ Cc

d,loc(E;D). Therefore Ccd,loc(E;D) as

a closed subset of the compact set Ccd(D) is compact. We leave the proof that

Ccd,loc(E;D) is open when E is closed to the reader.

3 Projection, Skeleton, Crack, and DifferentiabilityIn this section we study the connection between the gradient of dA, the set ofprojections onto A, and the characteristic function of A. We partition the set ofsingularities of the gradient of dA into the skeleton3 and set of cracks of A.

Definition 3.1.Let A ⊂ RN, ∅ = A, and denote by

Sing (∇dA) def=x ∈ RN : ∇dA(x) "

(3.1)

the set of singularities of the gradient of dA.

(i) Given x ∈ RN, a point p ∈ A such that |p− x| = dA(x) is called a projectiononto A. The set of all projections onto A will be denoted by

ΠA(x)def=p ∈ A : |p− x| = dA(x)

. (3.2)

When ΠA(x) is a singleton, its element will be denoted by pA(x).3Our definition of a skeleton does not exactly coincide with the one used in morphological

mathematics, where it is defined as the closure Sk (A) of our skeleton Sk (A) (cf., for instance,G. Matheron [1] or A. Riviere [1], and J. Serra [1] for the pioneering applications in miningengineering in 1968 and later in image processing).


(ii) The skeleton of A is the set of all points of RN whose projection onto A isnot unique. It will be denoted by

Sk (A) def=x ∈ RN : ΠA(x) is not a singleton

. (3.3)

(iii) The set of cracks is defined as

Ck (A) def= Sing∇dA\Sk (A). (3.4)

Note that Sk (A) = Sk (A), Ck (A) = Ck (A), and Sing (A) = Sing (A) since dA = dAby Theorem 2.1 (ii).

Even for a set A with smooth boundary ∂A, the gradient ∇dA(x) may notexist far from the boundary as shown in Figure 6.1, where∇dA(x) exists everywhereoutside A except on Sk (A), a semi-infinite line. Yet, ∇dA(x) exists close to theboundary. An example of the set of cracks is given by the eyes, nose, mouth, andrays of the smiling sun in Figure 5.1 of Chapter 5 that belong to the boundary ofthe set as opposed to the skeleton Sk (A) that lies outside of A.

Sk (A)

Sk (!A)

A

Figure 6.1. Skeletons Sk (A), Sk (!A), and Sk (∂A) = Sk (A) ∪ Sk (!A).

The level sets of dA generate dilations/tubular neighborhoods of A.

Definition 3.2.Let ∅ = A ⊂ RN and let h > 0.

(i) The open h-tubular neighborhood , the closed h-tubular neighborhood , and theh-boundary of A are respectively defined as

Uh(A) def=x ∈ RN : dA(x)| < h

, Ah

def=x ∈ RN : dA(x)| ≤ h

, and d−1

A h.

We shall also use the terminology open or closed dilated sets.

(ii) For 0 < s < h <∞

Us,h(A) def=x ∈ RN : s < dA(x)| < h

and As,h

def=x ∈ RN : s ≤ dA(x)| ≤ h

.

This terminology is justified by the fact that a dilated set is a relatively nice set.


Theorem 3.1. Let ∅ = A ⊂ RN. For r > 0,

Ar = Ur(A), ∂Ar = ∂Ur(A) = d−1A r, !Ar = !Ur(A), intAr = Ur(A). (3.5)

In particular, for 0 < s < r, Us,r(A) = As,r.

Proof. By definition A ⊂ Ur(A) ⊂ Ar and Ur(A) ⊂ Ar. For any x ∈ Ar, dA(x) ≤ r.If dA(x) < r, then x ∈ Ur(A) ⊂ Ur(A). If dA(x) = r > 0, then x /∈ A and thereexists p ∈ ∂A such that dA(x) = |x− p| = r > 0. Consider the points

xndef= p +

(1− 1

n

)r

x− p

|x− p| → p + rx− p

|x− p| = x.

Then p ∈ ΠA(xn), and dA(xn) = r (1 − 1/n) < r, dA(xn) → dA(x) = r, andx ∈ Ur(A). Therefore, since Ur(A) is open, Ar = Ur(A) = Ur(A) ∪ ∂Ur(A) impliesthat ∂Ur(A) = d−1

A r. Moreover, since !Ur(A) is closed, !Ur(A) = int !Ur(A) ∪∂Ur(A) = !Ar ∪ ∂Ur(A) implies that !Ar = !Ur(A). The proof is the same forUs,r(A) = As,r.

We now give several important technical results that will be used later for setsof positive reach and in Chapter 7. We first give a few additional definitions.

Definition 3.3. (i) A function f : U → R defined in a convex subset U of RN

is convex if for all x and y in U and all λ ∈ [0, 1],

f(λx + (1− λ)y) ≤ λf(x) + (1− λ)f(y).

It is concave if the function −f is convex. A function f : U → R defined ina convex subset U of RN is semiconvex (resp., semiconcave) if

∃c ≥ 0, fc(x) = c |x|2 + f(x) (resp., fc(x) = c |x|2 − f(x))

is convex in U .

(ii) A function f : U → R defined in a subset U of RN is locally convex (resp.,locally concave) if it is convex (resp., concave) in every convex subset of U . Afunction f : U → R defined in a subset U of RN is locally semiconvex (resp.,locally semiconcave) if it is semiconvex (resp., semiconcave) in every convexsubset of U .When U is convex the local definitions coincide with the global ones.

The next theorem is a slight extension of a result in J. H. G. Fu [1, Prop. 1.2].

Theorem 3.2. Let A ⊂ RN be such that A = ∅.

(i) Given h > 0, the function

kA,h(x) def=

|x|2 − 2 h dA(x), if dA(x) ≥ h,

|x|2 − d2A(x)− h2, if dA(x) < h,


is convex (and continuous) in RN and the function

x '→ kA,h(x)2h

=|x|2

2 h− dA(x) : RN \Uh(A)→ R

is locally convex (and continuous) in RN \Uh(A).

(ii) The function

x '→ fA(x) def=12(|x|2 − d2

A(x))

: RN → R

is convex and continuous, d2A(x) is the difference of two convex functions

d2A(x) = |x|2 −

(|x|2 − d2

A(x)), (3.6)

∇fA and ∇d2A belong to BVloc(RN)N , and ∇dA ∈ BVloc

(RN \∂A

)N .

Proof. (i) For all p ∈ A define the convex function

ℓp(x) =

|x− p|− h, |x− p| ≥ h,

0, |x− p| < h.

Since ℓp is nonnegative, ℓ2p is convex and

ℓ2p(x) =

|x|2 − 2 h |x− p| + |p|2 + h2 − 2 x · p, |x− p| ≥ h,

0, |x− p| < h.

By subtracting the constant term |p|2 + h2 and the linear term −2x · p from ℓ2p, weget the new convex function

mp(x) =

|x|2 − 2 h |x− p|, |x− p| ≥ h,

|x|2 − |p− x|2 − h2, |x− p| < h.

Then the function

k(x) def= supp∈A

mp(x) =

|x|2 − 2 h dA(x), dA(x) ≥ h,

|x|2 − d2A(x)− h2, dA(x) < h

is convex in RN. Moreover, it is finite for each x ∈ RN and hence continuous in x.Indeed, if x is such that dA(x) ≥ h, by Theorem 2.1 (ii) there exists p ∈ A suchthat |x− p| = dA(x) ≥ h and

|x− p| = dA(x) = infp∈A

|x−p|≥h

|x− p|


and k(x) = |x|2 − 2 h dA(x). If dA(x) < h, then there exists p ∈ A such that|x− p| = dA(x) < h and

|x− p| = dA(x) = infp∈A

|x−p|<h

|x− p|

and k(x) = |x|2 − d2A(x)− h2. We recover the result of J. H. G. Fu [1, Prop. 1.2]

by observing that the restriction of k to RN \Uh(A) is locally convex. In additionthe function |x|2 − d2

A(x) is locally convex in Uh(A).(ii) Since for all h > 0, |x|2 − d2

A(x) is locally convex in Uh(A) it is locallyconvex and continuous in RN = ∪h>0Uh(A) and hence convex in RN. Therefore,from L. C. Evans and R. F. Gariepy [1, Thm. 3, p. 240], ∇fA ∈ BVloc(RN)N

and ∇d2A ∈ BVloc(RN)N . Finally, RN \∂A is the union of the two disjoint open

sets RN \A and intA. On intA, ∇dA = 0. For any point x in RN \A, there existsρ, 0 < 3 ρ < dA(x), such that the open ball B(x, 2 ρ) is contained in RN \Uρ(A),where kA,ρ is locally convex by part (i). Hence kA,ρ is convex in B(x, 2 ρ) and∇kA,ρ, and ∇dA belong to BVloc(B(x, 2ρ))N and, by Definition 6.1 in Chapter 5,∇dA belong to BV(B(x, ρ))N . Finally, by Theorem 6.1 in Chapter 5, ∇dA belongsto BVloc(RN \A)N .

We now give some basic results.

Theorem 3.3. Let A, ∅ = A ⊂ RN, and let x ∈ RN.

(i) The set ΠA(x) is nonempty, compact, and

∀x /∈ A ΠA(x) ⊂ ∂A and ∀x ∈ A ΠA(x) = x.

(ii) For all x and v in RN, the Hadamard semiderivative4 of d2A always exists,

dHd2A(x; v) = lim

t0

d2A(x + tv)− d2

A(x)t

= minp∈ΠA(x)

2(x− p) · v

= 2(x · v − σΠA(x)(v)),

dHfA(x; v) = limt0

fA(x + tv)− fA(x)t

= σΠA(x)(v) = σco ΠA(x)(v),

where σB is the support function of the set B,

σB(v) = supz∈B

z · v,

and co B is the convex hull of B. In particular,

Sk (A) =x ∈ RN : ∇d2

A(x) "⊂ RN \A. (3.7)

4A function f : RN → R has a Hadamard semiderivative in x in the direction v if

dHf(x; v) def= limt0w→v

f(x + tw) − f(x)t

exists

(cf. Chapter 9, Definition 2.1 (ii)).


Given v ∈ RN, for all x ∈ RN \∂A the Hadamard semiderivative of dA exists,

dHdA(x; v) = minp∈ΠA(x)

2x− p

|x− p| · v, (3.8)

and for x ∈ ∂A, dHdA(x; v) exists if and only if

limt0

dA(x + tv)t

exists. (3.9)

(iii) The following statements are equivalent:

(a) d2A(x) is (Frechet) differentiable at x.

(b) d2A(x) is Gateaux differentiable at x.

(c) ΠA(x) is a singleton.

(iv) ∇fA(x) exists if and only if ΠA(x) = pA(x) is a singleton. In that case

pA(x) = ∇fA(x) = x− 12∇d2

A(x).

For all x and y in RN,

∀p(x) ∈ ΠA(x),12(|y|2 − d2

A(y))≥ 1

2(|x|2 − d2

A(x))

+ p(x) · (y − x), (3.10)

or equivalently,

∀p(x) ∈ ΠA(x), d2A(y)− d2

A(x)− 2 (x− p(x)) · (y − x) ≤ |x− y|2. (3.11)

For all x and y in RN

∀p(x) ∈ ΠA(x), ∀p(y) ∈ ΠA(y), (p(y)− p(x)) · (y − x) ≥ 0. (3.12)

(v) The functions

pA : RN \Sk (A)→ RN and ∇d2A : RN \Sk (A)→ RN

are continuous. For all x ∈ RN \Sk (A)

pA(x) = x− 12∇d2

A(x) = ∇fA(x). (3.13)

In particular, for all x ∈ A, ΠA(x) = x and ∇d2A(x) = 0,

Sk (A) =x ∈ RN : ∇d2

A(x) " and ∇dA(x) "⊂ RN \A,

Ck (A) =x ∈ RN : ∇d2

A(x) ∃ and ∇dA(x) "⊂ ∂A,

(3.14)

and Sing (∇dA) = Sk (A) ∪ Ck (A).


(vi) The function

∇dA : RN \(Sk (A) ∪ ∂A

)→ RN

is continuous. For all x ∈ int A, ∇dA(x) = 0 and for all x ∈ RN \(Sk (A) ∪A

)

∇dA(x) =∇d2

A(x)2 dA(x)

=x− pA(x)|x− pA(x)| and |∇dA(x)| = 1. (3.15)

If ∇dA(x) exists for x ∈ A, then ∇dA(x) = 0. In particular, ∇dA = 0 almosteverywhere in A.

(vii) The functions d2A and dA are differentiable almost everywhere and

m(Sk (A)) = m(Ck (A)) = m(Sing (∇dA)) = 0.

For ∂A = ∅

Sk (∂A) = Sk (A) ∪ Sk (!A) ⊂ RN \∂A, (3.16)Ck (∂A) = Ck (A) ∪ Ck (!A) ⊂ ∂(∂A). (3.17)

If ∂A = ∅, then either A = RN or A = ∅.

(viii) Given A ⊂ RN, ∅ = A (resp., ∅ = !A),

χA(x) = 1− |∇dA(x)|, χint !A(x) = |∇dA(x)| in RN \Sing (∇dA)

(resp., χ!A(x) = 1− |∇d!A(x)|, χint A(x) = |∇d!A(x)| in RN \Sing (∇d!A))

and the above identities hold for almost all x in RN.

(ix) Given x ∈ RN, α ∈ [0, 1], p ∈ ΠA(x), and xαdef= p + α (x− p), then

dA(xα) = |xα − p| = α |x− p| = α dA(x) and ∀α ∈ [0, 1], ΠA(xα) ⊂ ΠA(x).

In particular, if ΠA(x) is a singleton, then ΠA(xα) is a singleton and ∇d2A(xα)

exists for all α, 0 ≤ α ≤ 1. If, in addition, x = A, then for all 0 < α ≤ 1∇dA(xα) exists and ∇dA(xα) = ∇dA(x).

Remark 3.1.In general, for each v ∈ RN

∃p = p(v) ∈ ΠA(x), dHd2A(x; v) = 2(x− p) = inf

p∈ΠA(x)2(x− p) · v,

since ΠA(x) is compact. However, when ΠA(x) is not a singleton, p is not necessarilyunique since 2(x − p) depends on the direction v. For points x outside of A andSk (A) the norm of ∇dA(x) is equal to 1. When A is sufficiently smooth, ∇dA(x)coincides with the outward unit normal to A at the point pA(x). When A is notsmooth, this normal is not always unique as shown in Figure 6.2.


∇dA(x)

∇dA(y)

x

y

pA(x) = pA(y)

A

Figure 6.2. Nonuniqueness of the exterior normal.

It is useful to consider examples to illustrate the subtleties of the theorem.The set of cracks Ck (A) of a set A has zero measure, but its boundary ∂A can havea nonempty interior and a nonzero measure.

Example 3.1.Let B1(0) be the open ball of radius 1 at the origin in RN and let

A = x ∈ B1(0) : x has rational coordinates.

Then A = B1(0), !A = RN , ∂A = B1(0), ∂A = ∂B1(0),

∇dA(x) = ∇d2A(x) = 0 in B1(0) and ∇dA(x) =

x

|x| = pA(x) in !B1(0),

Sk (A) = ∅ and Ck (A) = ∂B1(0).

In general ∂A ⊂ ∂A and this inclusion can be strict as illustrated by this example.Also note that the boundary ∂A has a nonempty interior int ∂A = B1(0) anda nonzero Lebesgue measure (nonzero volume in dimension N = 3). Thus theboundary of two members of the equivalence class [A] can be different.

The boundary ∂A of a closed or an open set A has no interior; that is, ∂A isnowhere dense. Yet, its boundary ∂A still does not necessarily have a zero measure.If A is closed, then A = int A∪∂A. If int A = ∅, there exist x ∈ ∂A and r > 0 suchthat Br(x) ⊂ ∂A ⊂ A and x ∈ intA, a contradiction. If A is open, !A is closed,!A = int !A ∪ ∂A, and ∂A has no interior.

Example 3.2 (Modified version of Example 1.10 in E. Giusti [1, p. 7]).Let B1(0) in R2 be the open ball in 0 of radius 1. For i ≥ 1, let xi : i ∈ Nbe the sequence of all points in B1(0) with rational coordinates for some choice ofordering. Associate with each i the open ball

Bi = x ∈ R2 : |x− xi| < ρi, 0 < ρi ≤ min2−i, 1− |xi| ⇒ Bi ⊂ B1(0).


Consider the increasing sequence of open subsets of B1(0)

Ωndef=

n⋃

i=1

Bi Ω def=∞⋃

i=1

Bi

and the closed set Adef= !Ω. We get

A = A = !Ω, !A = Ω, !A = Ω = B1(0), ∂A = B1(0) ∩ !Ω.

As a result, we have the following decreasing sequence of closed subsets of B1(0):

Dndef= B1(0) ∩ !Ωn ∂A = B1(0) ∩ !Ω

⇒ m(Dn) ≥ m(B1(0))−n∑

i=1

m(Bi) = π − πn∑

i=1

(12i

)2

⇒ m(∂A) ≥ π − π∞∑

i=1

(12i

)2

=23π.

Proof of Theorem 3.3. (i) Existence. The function

z '→ |z − x|2 : RN → R (3.18)

is continuous and, a fortiori, upper semicontinuous, and hence

infa∈A

|a− x|2 = infa∈A

|a− x|2.

If A is bounded, there exists p ∈ A such that |p − x|2 = infa∈A |a − x|2. If A isunbounded, the function (3.18) is coercive with respect to A,

lima∈A, |a|→∞

|a− x|2

|a| = +∞,

and we also have the existence of p ∈ A such that |p − x|2 = infa∈A |a − x|2. Bydefinition, ΠA(x) ⊂ A. For x ∈ RN \A, dA(x) > 0. Assume that there existsp ∈ ΠA(x) such that p ∈ int A. Therefore, there exists r, 0 < r < dA(x)/2, suchthat Br(p) ⊂ A. Choose the point

y = p +r

2x− p

|x− p| ⇒ |y − p| = r/2 ⇒ y ∈ Br(p) ⊂ A

⇒ y − x = p− x +r

2x− p

|x− p| =[1− r

2dA(x)

](p− x)

⇒ ∃y ∈ A such that |y − x| =[1− r

2dA(x)

]|p− x| < dA(x);

this contradicts the minimality of dA(x). Hence x ∈ ∂A.


(ii) Semidifferentiability of fA. The semiderivative in a direction v can bereadily computed by using a theorem on the differentiability of the min with respectto a parameter (cf. Chap. 10, sect. 2.3, Thm. 2.1). It is sufficient to prove thesemidifferentiability of d2

A in any direction v, that is, for each x and v the existenceof the limit

dd2A(x; v) def= lim

t0

d2A(x + tv)− d2

A(x)t

.

Recall that for t ≥ 0,

ΠA(x + t v) =pt ∈ A : |x + tv − pt|2 = d2

A(x + tv)

and consider for t > 0 the quotient

qt =d2

A(x + tv)− d2A(x)

t.

For all p ∈ ΠA(x) and pt ∈ ΠA(x + t v)

qt =|x + tv − pt|2 − |x− p|2

t

≤ |x + tv − p|2 − |x− p|2

t= t |v|2 + 2 v · (x− p),

and for all p ∈ ΠA(x)

lim supt0

qt ≤ 2 v · (x− p) ⇒ q = lim supt0

qt ≤ 2 infp∈ΠA(x)

v · (x− p).

In the other direction choose a sequence tk > 0 such that tk → 0 and qtk → q =lim inft0. The corresponding sequence ptk in A is uniformly bounded since

|ptk | ≤ |x + tkv − ptk | + |x + tkv|≤ dA(x + tkv)− dA(x) + dA(x) + |x + tkv|≤ tk|v| + dA(x) + |x + tkv|,

and we can find p0 ∈ A and a subsequence of ptk, still denoted by ptk, suchthat ptk → p0. But by continuity

dA(x + tkv)→ dA(x) and |x + tkv − ptk |→ |x− p0|⇒ ∃p0 ∈ A such that dA(x) = |x− p0| ⇒ p0 ∈ ΠA(x).

Therefore,

qtk =|x + tkv − ptk |2 − |x− p0|2

tk

≥ |x + tkv − ptk |2 − |x− ptk |2

tk= tk|v|2 + 2 v · (x− ptk)

⇒ q ≥ 2 v · (x− p0) ≥ 2 infp∈ΠA(x)

v · (x− p).


Hencedd2

A(x; v) = 2 infp∈ΠA(x)

v · (x− p).

Finally, by Theorem 2.4 (ii), dHd2A(x; v) = dd2

A(x; v) since d2A is locally Lipschitzian.

The same remark applies to ddA(x; v) in RN \∂A.(iii) From part (ii), d2

A(x) is Gateaux differentiable at x if and only if ΠA(x)is a singleton. The (Frechet) differentiability is a standard part of Rademacher’stheorem, but we reproduce it here for completeness as a lemma.

Lemma 3.1. Let f : RN → R be a locally Lipschitzian function. Then f is(Frechet) differentiable at x, that is,

limy→x

f(y)− f(x)−∇f(x) · (y − x)|y − x| = 0

if and only if it is Gateaux differentiable at x, that is,

∀v ∈ RN, df(x; v) = limt0

f(x + tv)− f(x)t

exists

and the map v '→ df(x; v) is linear and continuous.

Proof. It is sufficient to prove that the Gateaux differentiability implies that

limy→x

f(y)− f(x)−∇f(x) · (y − x)|y − x| → 0.

The function f is locally Lipschitzian and there exist δ1 > 0 and c1 > 0 such thatf is uniformly Lipschitzian in the open ball B(x, δ1) with Lipschitz constant c1.Therefore, |∇f(x)| ≤ c1. The unit sphere S(0, 1) = v ∈ RN : |v| = 1 is compactand can be covered by the family of open balls B(v, ε/(4c1)) : v ∈ S(0, 1) for anarbitrary ε > 0. Hence there exists a finite subcover B(vn, ε/(4c1)) : 1 ≤ n ≤ Nof S(0, 1). As a result, given any v ∈ S(0, 1), there exists n, 1 ≤ n ≤ N , such that|v − vn| ≤ ε/(4c1). Define the function

g(x, v, t) =f(x + tv)− f(x)

t−∇f(x) · v

for x ∈ RN, v ∈ S(0, 1), and t > 0. Since f(x) is Gateaux differentiable

∃δ, 0 < δ < δ1, ∀n, 1 ≤ n ≤ N, ∀t < δ, |g(x, vn, t)| < ε/2.

Hence for |y − x| < δ∣∣∣∣g(

x,y − x

|y − x| , |y − x|)∣∣∣∣

≤ |g(x, vn, |y − x|)| +∣∣∣∣g(

x,y − x

|y − x| , |y − x|)− g(x, vn, |y − x|)

∣∣∣∣

≤ |g(x, vn, |y − x|)| + 2 c1

∣∣∣∣y − x

|y − x| − vn

∣∣∣∣ ≤ε

2+ 2 c1

ε

(4c1)= ε,

and we conclude that f(x) is differentiable at x.


(iv) From part (ii), ∇fA(x) exists if and only if ΠA(x) is a singleton. Inequality(3.10) follows directly from the inequality

∀x and y ∈ RN, ∀ p(x) ∈ ΠA(x), d2A(y) ≤ |p(x)− y|2

since

d2A(y)− |y|2 ≤ |p(x)− x + x− y|2 − |y|2

≤ |p(x)− x|2 + |x− y|2 + 2 (p(x)− x) · (x− y)− |y|2

≤ |p(x)− x|2 − |x|2 + 2 (p(x)− x) · (x− y) + |x− y|2 + |x|2 − |y|2

and

− 2fA(y)

≤ −2fA(x) + 2 p(x) · (x− y)− 2 x · (x− y) + |x− y|2 + |x|2 − |y|2

≤ −2fA(x)− 2 p(x) · (y − x) ⇒ fA(y) ≥ fA(x) + 2 p(x) · (y − x).

Inequality (3.11) is (3.10) rewritten. Inequality (3.12) follows by adding inequality(3.10) to the same inequality with x and y permuted.

(v) From the equivalences in (ii), when d2A is differentiable at x, then ΠA(x) =

pA(x) is a singleton, and from part (i)

∇d2A(x) = 2(x− pA(x)),

which yields the expression for pA(x). When x ∈ A, ΠA(x) = x, and by substi-tution ∇d2

A(x) = 0. Finally, it is sufficient to prove the continuity of pA. Givenx ∈ RN \Sk (A), consider a sequence xn ⊂ RN \Sk (A) such that xn → x and theassociated sequence pA(xn) ⊂ A. For all n

|pA(xn)− x| ≤ |pA(xn)− xn| + |xn − x| ≤ |pA(x)− xn| + |xn − x|

≤ |pA(x)− x| + 2|xn − x| < dA(x) + 2r, rdef= sup

n|xn − x| <∞.

So the sequence pA(xn) ⊂ A is bounded and there exists p ∈ A and a subsequence,still denoted by pA(xn), such that pA(xn)→ p. Therefore

dA(x)← dA(xn) = |xn − pA(xn)|→ |x− p|⇒ ∃p ∈ A, dA(x) = |x− p| ⇒ pA(x) = p

and pA is continuous at x since p = pA(x) is independent of the choice of theconverging subsequence.

To complete the proof of identities (3.14), recall that the projection of eachpoint of A onto A being a singleton, Sk (A) ⊂ RN \A. Therefore, for any x ∈Sk (A), we have dA(x) > 0 and if ∇d2

A(x) does not exist, then ∇dA(x) cannotexist. This sharpens the characterization (3.7) of Sk (A). From this we readily getthe characterization of Ck (A) = Sing (∇dA)\Sk (A) and Ck (A) ⊂ A. This last


property can be improved to Ck (A) ⊂ ∂A by noticing that for x ∈ int A, dA(x) = 0and ∇d2

A(x) = ∇dA(x) = 0.(vi) For all x ∈ intA, dA(x) = 0 and hence ∇dA(x) = 0 is constant and,

a fortiori, continuous in intA. For x ∈ RN \(Sk (A) ∪ A), dA(x) > 0 and for allv ∈ RN and t > 0 sufficiently small

dA(x + tv)− dA(x)t

=dA(x + tv)2 − dA(x)2

t

1dA(x + tv) + dA(x)

⇒ ddA(x; v) =dd2

A(x; v)2dA(x)

=1

2dA(x)∇d2

A(x) · v ⇒ ∇dA(x) =1

2dA(x)∇d2

A(x).

By continuity of ∇d2A(x) from part (iv) and continuity of dA, ∇dA(x) is continuous

since dA(x) > 0. Moreover, from part (iv), ∇dA(x) = 2(x − pA(x))/2dA(x) and|∇dA(x)| = 1.

If ∇dA(x) exists in some x ∈ A, then dA(x) = 0 and for all v ∈ RN, thedifferential quotient converges:

0 ≤ limt0

dA(x + tv)t

= limt0


= ∇dA(x) · v

⇒ ∀v, 0 ≤ ±∇dA(x) · v ⇒ ∇dA(x) = 0.

Since ∇dA exists almost everywhere, ∇dA = 0 almost everywhere in A.(vii) The function dA is Lipschitzian and the function d2

A is locally Lipschitz-ian. By Rademacher’s theorem they are both differentiable almost everywhere. Theother properties follow from parts (iii), (iv), and (v).

By definition Sk (∂A) = x ∈ RN : ∇d2∂A(x)" ⊂ RN \∂A = RN \A∪RN \!A.

If x ∈ Sk (∂A) ∩ RN \A, then d∂A(x) = dA(x), ∇d2A(x)", and x ∈ Sk (A); if x ∈

Sk (∂A)∩RN \!A, then d∂A(x) = d!A(x), ∇d2!A

(x)", and x ∈ Sk (!A). Conversely,if x ∈ Sk (A) ⊂ RN \A, then d∂A(x) = dA(x), ∇d2

∂A(x)", and x ∈ Sk (∂A); ifx ∈ Sk (!A) ⊂ RN \!A, then d∂A(x) = d!A(x),∇d2

∂A(x)", and x ∈ Sk (∂A). Finally,Sk (∂A) = Sk (A) ∪ Sk (!A) ⊂ ∂A ∪ ∂!A = ∂(∂A).

The property that Ck (∂A) = Ck (A)∪Ck (!A) is a consequence of the follow-ing property for x ∈ ∂A: ∇d∂A(x) exists if and only if ∇dA(x) and ∇d!A(x) exist.Indeed, for x ∈ ∂A, d∂A(x) = dA(x) = d!A(x) = 0. From part (v), if ∇d∂A(x)exists, it is 0. But ∂A ⊂ A implies d∂A ≥ dA and for all v ∈ RN and t > 0

0 ≤ dA(x + tv)− dA(x)t

≤ d∂A(x + tv)− d∂A(x)t

→ 0 ⇒ ∇dA(x) = 0

and similarly ∇d!A(x) = 0. Conversely, from part (v), if ∇dA(x) and ∇d!A(x)exist, they are 0. Moreover, d∂A(x) = maxdA(x), d!A(x), where

0 ≤ d∂A(x + tv)− d∂A(x)t

= max


,d!A(x + tv)− d!A(x)

t

.

Since both ∇dA(x) = 0 and ∇d!A(x) = 0, then ∇d∂A(x) = 0. Finally, it is easy tocheck that ∂A ∪ ∂!A = ∂(∂A). The other properties for ∂A = ∅ are obvious.


(viii) From part (i), if ∇dA(x) exists for x ∈ A, it is equal to 0. HenceχA(x) = 1 = 1 − |∇dA(x)| on A\Ck (A). From the definition of Sk (A), ∇dA(x)exists and |∇dA(x)| = 1 for all points in int !A\Sk (A). Hence from part (i), χA(x) =0 = 1− |∇dA(x)| = 0 on !A\Sk (A). From this, we get the first line of identities inRN \Sing (∇dA). Since m(Sing (∇dA)) = 0, the above identities are satisfied almosteverywhere in RN. We get the same results with !A in place of A.

(ix) If x ∈ A, ΠA(x) = x and there is nothing to prove. If x /∈ A, thendA(x) = |x− p| > 0 and p = x for all p ∈ ΠA(x). First

dA(xα) ≤ |xα − p| = α |x− p|.

If the inequality is strict, then there exists pα ∈ A such that |xα−pα| = dA(xα) and

|x− pα| ≤ |x− xα| + |xα − pα| = (1− α)|x− p| + dA(xα) < |x− p| = dA(x).

This contradicts the minimality of dA(x) with respect to A. Therefore,

dA(xα) = α |x− p| = |xα − p| ⇒ p ∈ ΠA(xα).

Now for any pα ∈ ΠA(xα), pα ∈ A and

|pα − x| ≤ |pα − xα| + |xα − x| = α |x− p| + (1− α) |p− x| = |p− x| = dA(x)

and pα ∈ ΠA(x). Hence ΠA(xα) ⊂ ΠA(x).

4 W 1,p-Metric Topology and CharacteristicFunctions

4.1 Motivations and Main Properties

Since distance functions are locally Lipschitzian, they belong to C0,1loc (RN) and hence

to W 1,ploc (RN) for all p ≥ 1. Thus the constructions of section 2.1 can be repeated

with W 1,ploc (D) in place of Cloc(D) to generate new W 1,p-metric topologies on the

family Cd(D). One big advantage is that the W 1,p-convergence of sequences willimply the Lp-convergence of the corresponding characteristic functions of the closureof the sets and hence the convergence of volumes (cf. Theorem 3.3 (viii)).

In the uniform Hausdorff topology, the convergence of volumes and perimetersis usually lost. In general, the measure of a closed set is only upper semicontinuouswith respect to that topology. This includes Lebesgue and Hausdorff measures. Thenext example shows that the Hausdorff metric convergence is not sufficient to get theLp-convergence of the characteristic functions of the closure of the correspondingsets in the sequence. The volume function is only upper semicontinuous with respectto the Hausdorff topology. The perimeters do not converge either as illustrated bythe example of the staircase of Example 6.1 and Figure 5.7 in Chapter 5, where thevolumes converge but not the perimeters.

4. W 1,p-Metric Topology and Characteristic Functions 293

Example 4.1.Denote by D = ]− 1, 2[× ]− 1, 2[ the open unit square in R2, p ≥ 1 a real number,and for each n ≥ 1 define the sequence of closed sets

Andef=

(x1, x2) ∈ D :2k

2n≤ x1 ≤

2k + 12n

, 0 ≤ k < n, 0 ≤ x2 ≤ 1

.

set A4 set A8 set S

Figure 6.3. Vertical stripes of Example 4.1.

This defines n vertical stripes of equal width 1/2n each distant of 1/2n (cf.Figure 6.3). Let S = [0, 1]× [0, 1] be the closed unit square. Clearly, for all n ≥ 1,

m(An) =12, PD(An) = 2n + 1,

∀x ∈ S, dAn(x) ≤ 14n

, ∥∇dAn∥Lp(D) ≥ 2−1/p,

where m(An) is the surface and PD(An) the perimeter of An. Hence

dAn → dS in C(D), m(S) = 1, PD(S) = 4.

But

m(An) = m(An) =12# 1 = m(S) ⇒ χAn # χS in Lp(D),

PD(An) # PD(S), χAn 12χS in Lp(D)-weak.

Since the characteristic functions do not converge, the sequence ∇dAn does notconverge in Lp(D)N and dAn does not converge in W 1,p(D).

Theorem 4.1. Let µ be a measure5 and let D ⊂ RN be bounded open such thatµ(D) <∞.

(i) The map

dA '→ µ(A) : Cd(D)→ R (4.1)

is upper semicontinuous with respect to the topology of uniform convergence(Hausdorff topology on Cd(D)).

5A measure in the sense of L. C. Evans and R. F. Gariepy [1, Chap. 1]. It is called an outermeasure in most texts.


(ii) The map

d!A '→ µ(intA) : Ccd(D)→ R (4.2)

is lower semicontinuous with respect to the topology of uniform convergence(Hausdorff complementary topology on Cc

d(D)).

Note that µ can be the Lebesgue measure but also any of the Hausdorffmeasures.

Proof. We prove (ii). The proof of (i) is similar and simpler. It is sufficient to workwith open sets Ω = intA = !!A. Let Ωn be a sequence of open subsets of D suchthat d!Ωn

→ d!Ω in C(D) for some open Ω ⊂ D. Given ε > 0, there exists N suchthat for all n > N

∥d!Ωn− d!ΩC(D) ≤ ε ⇒ !Ωn ⊂ (!Ω)ε.

By definition of a measurable set

µ(Ωn) = µ(Ωn ∩D) = µ(D)− µ(D ∩ !Ωn) ≥ µ(D)− µ(D ∩ (!Ω)ε)⇒ lim inf

n→∞µ(Ωn) ≥ µ(D)− µ(D ∩ (!Ω)ε).

But D ∩ (!Ω)ε is monotonically decreasing to ∩ε>0D ∩ (!Ω)ε = D ∩ !Ω and hence

lim infn→∞

µ(Ωn) ≥ µ(D)− limε→0

µ(D ∩ (!Ω)ε) = µ(D)− µ(D ∩ !Ω) = µ(Ω)

since D ∩ Ω = Ω.

Theorem 4.2. Let D be an open (resp., bounded open) subset of RN.

(i) The topologies induced by W 1,ploc (D) (resp., W 1,p(D)) on Cd(D) and Cc

d(D)are all equivalent for p, 1 ≤ p <∞.

(ii) Cd(D) is closed in W 1,ploc (D) (resp., W 1,p(D)) for p, 1 ≤ p <∞, and

ρD([A2], [A1])def=

∞∑

n=1

12n

∥dA2 − dA1∥W 1.p(B(0,n))

1 + ∥dA2 − dA1∥W 1.p(B(0,n))

(resp., ρD([A2], [A1])def= ∥dA2 − dA1∥W 1,p(D))

defines a complete metric structure on F(D). For p, 1 ≤ p <∞, the map

dA '→ χA = 1− |∇dA| : Cd(D) ⊂W 1,ploc (D)→ Lp

loc(D)

is “Lipschitz continuous”: for all bounded open subsets K of D and nonemptysubsets A1 and A2 of D

∥χA2− χA1

∥Lp(K) ≤ ∥∇dA2 −∇dA1∥Lp(K) ≤ ∥dA2 − dA1∥W 1,p(K).


(iii) Ccd(D) is closed in W 1,p

loc (D) (resp., W 1,p0 (D)) for p, 1 ≤ p <∞, and

ρD(Ω2, Ω1)def=

∞∑

n=1

12n

∥d!Ω2− d!Ω1

∥W 1.p(B(0,n))

1 + ∥d!Ω2− d!Ω1

∥W 1.p(B(0,n))

(resp., ρD(Ω2, Ω1)def= ∥d!Ω2

− d!Ω1∥W 1,p(D))

defines a complete metric structure on the family G(D) of open subsets of D.For p, 1 ≤ p <∞, the map

d!Ω '→ χΩ = |∇d!Ω| : Ccd(D) ⊂W 1,p

loc (D)→ Lploc(D)

is “Lipschitz continuous”: for all bounded open subsets K of D and opensubsets Ω1 = RN and Ω2 = RN of D

∥χΩ2 − χΩ1∥Lp(K) ≤ ∥∇d!Ω2−∇d!Ω1

∥Lp(K) ≤ ∥d!Ω2− d!Ω1

∥W 1,p(K).

Proof. (i) The proof is similar to the proof of Theorem 2.3 in Chapter 5. It issufficient to prove it for D bounded open. Since D is bounded it is contained in asufficiently large ball of radius c. Therefore,

dA(x) = infy∈A

|x− y| ≤ |x| + |y| ≤ 2c

since y ∈ A ⊂ D, and by Theorem 2.1 (vii)

|∇dA(x)| ≤ 1 a.e. in D.

For all p > 1 the injection of W 1,p(D) into W 1,1(D) is continuous since

∥dA∥W 1,1(D) ≤ ∥dA∥W 1,p(D) m(D)1/q

with 1/p + 1/q = 1. Conversely, we have the continuity in the other direction. Forp > 1 and any dA and dB in Cd(D),

∫

D|dA − dB |p + |∇dA −∇dB |p dx

=∫

D|dA − dB | |dA − dB |p−1 + |∇dA −∇dB | |∇dA −∇dB |p−1 dx

≤ max(2c)p−1, 2p−1∫

D|dA − dB | + |∇dA −∇dB | dx

⇒ ∥dA − dB∥pW 1,p(D) ≤ (2 maxc, 1)p−1 ∥dA − dB∥W 1,1(D).

Therefore, for any ε > 0, pick δ = εp/(2 maxc, 1)p−1 and

∥dA − dB∥W 1,1(D) ≤ δ ⇒ ∥dA − dB∥W 1,p(D) ≤ ε.

(ii) It is sufficient to prove it for D bounded open. Let dAn be a Cauchy se-quence in Cd(D) which converges to some f in W 1,p(D)-strong. By Theorem 2.2 (ii),


Cd(D) is compact in C(D) and there exist a subsequence, still denoted by dAn,and ∅ = A ⊂ D such that dAn → dA in C(D) and, a fortiori, in Lp(D)-strongsince D is bounded. By uniqueness of the limit in Lp(D)-strong, f = dA anddAn converges to dA in W 1,p(D)-strong. Therefore Cd(D) is closed in W 1,p(D).For the Lipschitz continuity, recall that the distance function dA is differentiablealmost everywhere in RN for A = ∅. In view of Theorem 3.3 (viii) χA = 1 −|∇dA(x)| almost everywhere in RN. Given two nonempty subsets A1 and A2 of D

|∇dA2 | ≤ |∇dA1 | + |∇dA2 −∇dA1 |

⇒ χA1 ≤ χA2 + |∇dA2 −∇dA1 | ⇒∫

D|χA1 − χA2 |p dx ≤ ∥dA2 − dA1∥

pW 1,p(D)

for 1 ≤ p <∞ and with the ess-sup norm for p =∞.(iii) Again it is sufficient to prove the result for D bounded. In that case

!Ω = ∅ for all open subsets Ω of D. Let Ωn be a sequence of open subsets of Dsuch that d!Ωn

is Cauchy in W 1,p(D). By assumption Ωn ⊂ D, !Ωn ⊃ !D,

∀n ≥ 1, d!Ωn= 0 in !D ⇒ d!Ωn

∈W 1,p0 (D),

and the Cauchy sequence converges to some f ∈ W 1,p0 (D). By Theorem 2.4 (ii),

Ccd(D) is compact in C(D) and there exist a subsequence, still denoted by d!Ωn

,and an open set Ω ⊂ D such that d!Ωn

→ d!Ω in C(D) and hence in Lp(D)-strongsince D is bounded. By uniqueness of the limit in Lp(D), f = d!Ω and the Cauchysequence d!Ωn

converges to d!Ω in W 1,p0 (D). The other part of the proof is similar

to that of part (ii).

4.2 Weak W 1,p-TopologyWe have the following general result.

Theorem 4.3. Let D be a bounded open domain in RN.

(i) If dAn weakly converges in W 1,p(D) for some p, 1 ≤ p <∞, then it weaklyconverges in W 1,p(D) for all p, 1 ≤ p <∞.

(ii) If dAn converges in C(D), then it weakly converges in W 1,p(D) for all p,1 ≤ p < ∞. Conversely if dAn weakly converges in W 1,p(D) for some p,1 ≤ p <∞, it converges in C(D).

(iii) Cd(D) is compact in W 1,p(D)-weak for all p, 1 ≤ p <∞.6

(iv) Parts (i) to (iii) also apply to Ccd(D).

6In a metric space the compactness is equivalent to the sequential compactness. For the weaktopology we use the fact that if E is a separable normed space, then, in its topological dual E′, anyclosed ball is a compact metrizable space for the weak topology. Since Cd(D) is a bounded subsetof the normed reflexive separable Banach space W 1,p(D), 1 ≤ p < ∞, the weak compactness ofCd(D) coincides with the weak sequential compactness (cf. J. Dieudonne [1, Vol. II, Chap. XII,sect. 12.15.9, p. 75]).


Proof. (i) For D bounded there exists a constant c > 0 such that for all dA ∈ Cd(D)

dA(x) ≤ c and |∇dA(x)| ≤ 1 a.e. in D.

If dAn weakly converges in W 1,p(D), then

dAn weakly converges in Lp(D), ∇dAn weakly converges in Lp(D)N .

By Lemma 3.1 (iii) in Chapter 5 both sequences weakly converge for all p ≥ 1, andhence dAn weakly converges in W 1,p(D) for all p ≥ 1.

(ii) If dAn converges in C(D), then by Theorem 2.2 (i) there exists dA ∈Cb(D) such that dAn → dA in C(D) and hence in Lp(D). So for all ϕ ∈ D(D)N ,

∫

D∇dAn · ϕ dx = −

∫

DdAndivϕ dx→ −

∫

DdAdivϕ dx =

∫

D∇dA · ϕ dx.

By density of D(D) in L2(D), ∇dAn → ∇dA in L2(D)N -weak and hence dAn → dA

in W 1,2(D)-weak. From part (i) it converges in W 1,p(D)-weak for all p, 1 ≤ p <∞.Conversely, the weakly convergent sequence converges to some f in W 1,p(D). Bycompactness of Cb(D) there exist a subsequence, still indexed by n, and dA suchthat dAn → dA in C(D) and hence in W 1,p(D)-weak. By uniqueness of the limitdA = f . Therefore, all convergent subsequences in C(D) converge to the same limit,so the whole sequence converges in C(D).

(iii) Consider an arbitrary sequence dAn in Cd(D). From Theorem 2.2 (ii)Cd(D) is compact and there exist a subsequence dAnk

and dA ∈ Cd(D) such thatdAnk

→ dA in C(D). From part (ii) the subsequence weakly converges in W 1,p(D)and hence Cd(D) is compact in W 1,p(D)-weak.

Theorem 4.4. Let D be a closed subset of RN and An, ∅ = An ⊂ D, be asequence such that dAn → dA in Cloc(D) for some A, ∅ = A ⊂ D. Then

∀x ∈ RN \A, limn→∞

χAn(x) = χA(x) = 0 and ∀x ∈ RN, lim sup

n→∞χAn

(x) ≤ χA(x),

and for any compact subset K of D

lim supn→∞

∫

KχAn

dx ≤∫

KχA dx.

Corollary 1. Let D = ∅ be a bounded open subset of RN and Ωn, ∅ = Ωn ⊂ D,be a sequence of open subsets of D converging to an open subset Ω, ∅ = Ω ⊂ D, ofD in the Hausdorff complementary topology: d!Ωn

→ d!Ω in C(D). Then

∀x ∈ Ω, limn→∞

χΩn(x) = χΩ(x) = 1, and ∀x ∈ RN, lim infn→∞

χΩn(x) ≥ χΩ(x),

and

lim infn→∞

∫

DχΩn dx ≥

∫

DχΩ dx.


Proof of Theorem 4.4. For all x /∈ A, dA(x) > 0 and

∃Nx > 0,∀n ≥ Nx, ∥dA − dAn∥ ≤ dA(x)/2⇒ ∃Nx > 0,∀n ≥ Nx, dAn(x) ≥ dA(x)/2 > 0

⇒ ∃Nx > 0,∀n ≥ Nx, x /∈ An and χAn(x) = 0.

So for all x /∈ A

limn→∞

χAn(x) = χA(x) = 0.

Finally, for all x ∈ A and all n ≥ 1

χAn(x) ≤ 1 = χA(x)

and the result follows trivially by taking the limsup of each term.We conclude that

lim supn→∞

χAn≤ χA,

and by using the analogue of Fatou’s lemma for the limsup we get for all compactsubsets K of D

lim supn→∞

∫

KχAn

dx ≤∫

Klim sup

n→∞χAn

dx ≤∫

KχA dx.

In order to get the Lp-convergence of the characteristic functions of the closureof the sets in the sequence, we need the Lp-convergence of the gradients of thedistance functions which are related to the characteristic functions of the closure ofthe sets (cf. Theorem 3.3 (viii)).

We have seen in Example 4.1 that the weak convergence of the characteristicfunctions is not sufficient to obtain the strong convergence of the sequence dAn todA in W 1,2(D). However, if we assume that χAn

is strongly convergent, it con-verges to the characteristic function χB of some measurable subset B of D. Is thissufficient to conclude that χB = χA? The answer is negative. The counterexampleis provided by Example 6.2 in Chapter 5, where

dAn dD in W 1,2(D)-weak and χAn → χB in L2(D)-strong

for some B ⊂ D such that

m(D) = π >π

3≥ m(B) ⇒ χD = χB .

Remark 4.1.In view of part (ii) of Theorem 4.2 an optimization problem with respect to thecharacteristic functions χΩ of open sets Ω in D for which we have the continuitywith respect to χΩ can be transformed into an optimization problem with respect tod!Ω in W 1,1(D) since χΩ = |∇d!Ω|. This would apply to the transmission problem(4.3)–(4.6) in section 4.1 of Chapter 5.

5. Sets of Bounded and Locally Bounded Curvature 299

5 Sets of Bounded and Locally Bounded CurvatureFrom Theorem 6.1 in Chapter 5 and Theorem 3.2, it is readily seen that

∇dA ∈ BVloc(RN)N ⇐⇒ ∀x ∈ ∂A, ∃ρ > 0 such that ∇dA ∈ BV(B(x, ρ))N .

The global properties of ∇dA depend only on its local properties around ∂A. Thislocal property is fairly general and is verified for sets with corners, that is, sets withdiscontinuities in the orientation of the normal along the boundary.

This suggests introducing a new family of sets and motivates the study of theirgeneral properties, We shall later see that convex sets have this property and laterin Chapter 7 that the larger family of sets with positive reach (cf. section 6) alsodoes. One remarkable property is that such sets are Caccioppoli sets (cf. section 6in Chapter 5). We give the definitions for an arbitrary set A, but they can bespecialized to !A and ∂A (associated with the distance functions d!A and d∂A) thatwould require a finer terminology such as exterior, interior, or boundary curvatureto distinguish them.

Definition 5.1. (i) Given a bounded open holdall D ⊂ RN and A, ∅ = A ⊂ D,the set A is said to be of bounded curvature with respect to D if

∇dA ∈ BV(D)N . (5.1)

The family of sets with bounded curvature will be denoted by

BCd(D) def=dA ∈ Cd(D) : ∇dA ∈ BV(D)N

.

(ii) A set A, A = ∅, in RN is said to be of locally bounded curvature if

∇dA ∈ BVloc(RN)N .

The family of sets with locally bounded curvature will be denoted by

BCddef=dA ∈ Cd(RN) : ∇dA ∈ BVloc(RN)N

.

As in Theorem 6.2 of Chapter 5 for Caccioppoli sets, it is sufficient to satisfythe BV property in a neighborhood of each point of the boundary ∂A.

Theorem 5.1. A, ∅ = A ⊂ RN, is of locally bounded curvature if and only if

∀x ∈ ∂A, ∃ρ > 0 such that ∇dA ∈ BV(B(x, ρ))N , (5.2)

where B(x, ρ) is the open ball of radius ρ > 0 in x.

Proof. From Theorem 6.1 and Definition 6.1 in Chapter 5, a function belongs toBVloc(RN) if and only if for each x ∈ RN it belongs to BV(B(x, ρ)) for some ρ > 0.The equivalence now follows from Theorem 3.2.


Recall that the relaxation of the perimeter of a set was obtained from thenorm of the gradient of its characteristic function for Caccioppoli sets. For distancefunctions χA = 1 − |∇dA| almost everywhere and the gradient of dA also has ajump discontinuity along the boundary ∂A of magnitude at most 1. So it is not toosurprising that a set A with bounded curvature is a Caccioppoli set.

Theorem 5.2.

(i) Let D ⊂ RN be a bounded open holdall and let A, ∅ = A ⊂ D. If ∇dA ∈BV(D)N , then

∥∇χA∥M1(D) ≤ 2 ∥D2dA∥M1(D), χA ∈ BV(D),

and A has finite perimeter with respect to D.

(ii) For any subset A of RN, A = ∅,

∇dA ∈ BVloc(RN)N ⇒ χA ∈ BVloc(RN)

and A has locally finite perimeter.

Proof. Given ∇dA in BV(D)N , there exists a sequence uk in C∞(D)N such that

uk → ∇dA in L1(D)N and ∥Duk∥M1(D) → ∥D2dA∥M1(D)

as k goes to infinity, and since |∇dA(x)| ≤ 1, this sequence can be chosen in sucha way that |uk(x)| ≤ 1 for all k ≥ 1. This follows from the use of mollifiers (cf.E. Giusti [1, Thm. 1.17, p. 15]). For all V in D(D)N

−∫

DχAdiv V dx =

∫

D(|∇dA|2 − 1)div V dx =

∫

D|∇dA|2div V dx.

For each uk

∫

D|uk|2div V dx = −2

∫

D

∗[Duk]uk · V dx = −2∫

Duk · [Duk]V dx,

where ∗Duk is the transpose of the Jacobian matrix Duk and∣∣∣∣∫

D|uk|2div V dx

∣∣∣∣ ≤ 2∫

D|uk| |Duk| |V | dx

≤ 2 ∥Duk∥L1∥V ∥C(D) ≤ 2 ∥Duk∥M1∥V ∥C(D)

since for W 1,1(D)-functions ∥∇f∥L1(D)N = ∥∇f∥M1(D)N . Therefore,∣∣∣∣∫

DχAdiv V dx

∣∣∣∣ =∣∣∣∣∫

D|∇dA|2div V dx

∣∣∣∣ ≤ 2∥D2dA∥M1∥V ∥C(D)

as k goes to infinity, where D2dA is the Hessian matrix of second-order partialderivatives of dA. Finally, ∇χA ∈M1(D)N .

5. Sets of Bounded and Locally Bounded Curvature 301

5.1 ExamplesIt is useful to consider the following three simple illustrative examples (cf. Fig-ure 6.4). By convexity, they are all of locally bounded curvature.

Example 5.1 (Half-plane in R2).Consider the domain

A = (x1, x2) : x1 ≤ 0, ∂A = (x1, x2) : x1 = 0.

It is readily seen that

dA(x1, x2) = maxx1, 0, ∇dA(x1, x2) =

(0, 0), x1 < 0,

(1, 0), x1 > 0,

⟨∂11dA,ϕ⟩ =∫

∂Aϕ dH1, ∂12dA = ∂21dA = ∂22dA = 0,

⟨∆dA,ϕ⟩ =∫

∂Aϕ dH1 ⇒ ∆dA = H1.

Thus ∆dA is the one-dimensional Hausdorff measure of ∂A.

Figure 6.4. ∇dA for Examples 5.1, 5.2, and 5.3.

Example 5.2 (Ball of radius R > 0 in R2).Consider the domain

A = x ∈ R2 : |x| ≤ R, ∂A = x ∈ R2 : |x| = R.

Clearly

dA(x) = max0, |x|−R, ∇dA(x) =

x/|x|, |x| > R,

(0, 0), |x| < R,

⟨∂11dA,ϕ⟩ =∫ 2π

0R cos2(θ)ϕ dθ +

∫

!A

x22

(x21 + x2

2)3/2 ϕ dx,

⟨∂22dA,ϕ⟩ =∫ 2π

0R sin2(θ)ϕ dθ +

∫

!A

x21

(x21 + x2

2)3/2 ϕ dx,

⟨∂12dA,ϕ⟩ = ⟨∂21dA,ϕ⟩ =∫ 2π

0R cos(θ) sin(θ)ϕ dθ +

∫

!A

x2x1

(x21 + x2

2)3/2 ϕ dx,

⟨∆dA,ϕ⟩ =∫

∂Aϕ dH1 +

∫

!A

1(x2

1 + x22)1/2 ϕ dx,


where H1 is the one-dimensional Hausdorff measure. ∆dA contains the one-dimensional Hausdorff measure of the boundary ∂A plus a term that correspondsto the volume integral of the mean curvature over the level sets of dA in !A.

Example 5.3 (Unit square in R2).Consider the domain A = x = (x1, x2) : |x1| ≤ 1, |x2| ≤ 1. Since A is symmetricalwith respect to both axes, it is sufficient to specify dA in the first quadrant. Weuse the notation Q1, Q2, Q3, and Q4 for the four quadrants in the counterclockwiseorder and c1, c2, c3, and c4 for the four corners of the square in the same order. Wealso divide the plane into three regions:

D1 = (x1, x2) : |x2| ≤ min1, |x1| ,

D2 = (x1, x2) : |x1| ≤ min1, |x2| ,

D3 = (x1, x2) : |x1| ≥ 1 and |x2| ≥ 1 .

Hence

dA(x) =

⎧⎪⎨

⎪⎩

minx2 − 1, 0, x ∈ D2 ∩Q1,

|x− c1|, x ∈ D3 ∩Q1,

minx1 − 1, 0, x ∈ D1 ∩Q1,

∇dA(x) =

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(0, 1), x ∈ D2 ∩Q1 and x2 > 1,x−c1

|x−c1| , x ∈ D3 ∩Q1,

(1, 0), x ∈ D1 ∩Q1 and x1 > 1,

(0, 0), x ∈ Q1, x1 < 1 and x2 < 1,

⟨∂11dA,ϕ⟩ =4∑

i=1

∫

D3∩Qi

(x2 − ci,2)2

|x− ci|2ϕ dx +

∫

∂A∩Qi∩D1

ϕ dH1,

⟨∂22dA,ϕ⟩4∑

i=1

∫

D3∩Qi

(x1 − ci,1)2

|x− ci|2ϕ dx +

∫

∂A∩Qi∩D2

ϕ dH1,

⟨∂12dA,ϕ⟩ = ⟨∂21dA,ϕ⟩ =4∑

i=1

∫

D3∩Qi

(x2 − ci,2)(x1 − ci,1)|x− ci|2

ϕ dx,

⟨∆dA,ϕ⟩ =4∑

i=1

∫

D3∩Qi

1|x− ci|

ϕ dx +∫

∂Aϕ dH1.

Notice that the structure of the Laplacian is similar to that observed in the previousexamples.

The norm ∥D2dA∥M1(Uh(A)) is decreasing as h goes to zero. The limit isparticularly interesting since it singles out the behavior of the singular part of theHessian matrix in a shrinking neighborhood of the boundary ∂A.

6. Reach and Federer’s Sets of Positive Reach 303

Example 5.4.Let N = 2. For the finite square and the ball of finite radius,

limh0∥∆dA∥M1(Uh(∂A)) = H1(∂A),

where H1 is the one-dimensional Hausdorff measure.

6 Reach and Federer’s Sets of Positive ReachThe material in this section is mostly taken from the pioneering 1959 work ofH. Federer [3, sect. 4], where he introduced the notion of reach and the sets ofpositive reach.

6.1 Definitions and Main PropertiesBy definition, A∩Sk (A) = ∅. For a ∈ int A, there exists r > 0 such that Br(a) ⊂ A.Thus Sk (A) ∩Br(a) = ∅ and

sup r > 0 : Sk (A) ∩Br(a) = ∅ > 0. (6.1)

If a ∈ ∂A, then either a /∈ Sk (A) or a ∈ Sk (A). If a /∈ Sk (A), there exists r > 0such that Br(a) ∩ Sk (A) = ∅. Thus Br(a) ∩ Sk (A) = ∅ and (6.1) is verified. Ifa ∈ Sk (A), then for all r > 0, Br(a) ∩ Sk (A) = ∅. This leads to the followingdefinition.

Definition 6.1.Given A, ∅ = A ⊂ RN, and a point a ∈ A

reach (A, a) def=

0, if a ∈ ∂A ∩ Sk (A),sup r > 0 : Sk (A) ∩Br(a) = ∅ , otherwise,

reach (A) def= inf reach (A, a) : a ∈ A .

The set A is said to have positive reach if reach (A) > 0.

Remark 6.1.(1) Since Sk (A) = Sk (A), reach (A) = reach (A).

(2) For any set A, Sk (A) ⊂ Sk (∂A) and

∀a ∈ ∂A, 0 ≤ reach (∂A, a) ≤ reach (A, a) ≤ +∞.

(3) For all a /∈ ∂A ∩ Sk (A), reach (A, a) > 0.

Lemma 6.1. Let ∅ = A ⊂ RN. The following statements are equivalent:

(i) There exists a ∈ A such that reach (A, a) = +∞.

(ii) Sk (A) = ∅.


(iii) For all a ∈ A, reach (A, a) = +∞.

(iv) reach (A) = +∞.

Proof. (i) ⇒ (ii) If there exists a ∈ A such that reach (A, a) = +∞, then RN =∪r>0Br(a) ⊂ RN \Sk (A) and Sk (A) = ∅. (ii) ⇒ (iii) If Sk (A) = ∅, for alla ∈ A reach (A, a) = +∞. (iii) ⇒ (iv) If for all a ∈ A reach (A, a) = +∞, thenreach (A) = infa∈A reach (A, a) = +∞. (iv) ⇒ (i) is obvious.

Remark 6.2.We shall see later that all nonempty convex sets A have positive reach and thatreach (A) = +∞. Bounded domains and submanifolds of class C2 also have positivereach.

Theorem 6.1. Let A, ∅ = A ⊂ RN, be such that Sk (A) = ∅.

(i) The function a '→ reach (A, a) : A→ R is continuous.

(ii) If reach (A) > 0, then for all h, 0 < h < reach (A), Uh(A) ∩ Sk (A) = ∅.Conversely, if there exists h > 0 such that Uh(A)∩Sk (A) = ∅, reach (A) ≥ h,and A has positive reach.

(iii) If reach (A) > 0, then A ∩ Sk (A) = ∅. If A ∩ Sk (A) = ∅ and A is bounded,then reach (A) > 0.

Remark 6.3.When A is unbounded and A ∩ Sk (A) = ∅, we can possibly have reach (A) = 0even if for all a ∈ A, reach (A, a) > 0 as readily seen from the following example:

Adef= (x1, x2) : |x2| ≥ ex1 , where Sk (A) = (x1, 0) : x1 ∈ R .

Proof. (i) We prove that the function is both lower and upper semicontinuous.For h, reach (A, a) > h, there exists r, h < r < reach (A, a), such that Br(a) ∩Sk (A) = ∅. Let a′ ∈ Bδ(a), δ = (r− h)/2 > 0. Then r− δ = (r + h)/2 > h and forall a′ ∈ Bδ(a)∩A, Br−δ(a′) ⊂ Br(a). Thus Br−δ(a′)∩Sk (A) ⊂ Br(a)∩Sk (A) = ∅,reach (A, a′) ≥ r− δ > h, and reach (A, a) is lower semicontinuous. Similarly, for k,reach (A, a) < k, there exists r, reach (A, a) > r > k, such that Br(a)∩ Sk (A) = ∅.Let a′ ∈ Bδ(a), δ = (k − r)/2 > 0. Then r + δ = (r + k)/2 < k and for alla′ ∈ Bδ(a) ∩ A, Br(a) ⊂ Br+δ(a′). Thus ∅ = Br(a) ∩ Sk (A) ⊂ Br+δ(a′) ∩ Sk (A),reach (A, a′) ≤ r + δ < k, and reach (A, a) is upper semicontinuous.

(ii) By definition of reach (A) as an infimum. Conversely, for all 0 < r < hand all x ∈ A, Br(x) ∩ Sk (x) = ∅, and reach (A, x) ≥ r. Since this is true for all0 < r < h, reach (A, x) ≥ h and reach (A) = infx∈A reach (A, x) ≥ h > 0.

(iii) By separation of a closed set and a compact set.

Theorem 6.2. Let ∅ = A ⊂ RN.

(i) Associate with a ∈ A the sets

P (a) def=v ∈ RN : ΠA(a + v) = a

, Q(a) def=

v ∈ RN : dA(a + v) = |v|

.

Then P (a) and Q(a) are convex and P (a) ⊂ Q(a) ⊂ (−TaA)∗.


(ii) Given a ∈ A and v ∈ RN, assume that

0 < r(a, v) def= sup t > 0 : ΠA(a + tv) = a .

Then for all t, 0 ≤ t < r(a, v), ΠA(a + tv) = a and dA(a + tv) = t |v|.Moreover, if r(a, v) < +∞, then a + r(a, v) v ∈ Sk (A).

(iii) If there exists h such that 0 < h ≤ reach (A), then for all x ∈ Uh(A)\A and0 < t < h, ΠA(x(t)) = pA(x) and dA(x(t)) = t, where

x(t) def= pA(x) + t∇dA(x) = pA(x) + tx− pA(x)

dA(x). (6.2)

(iv) Given b ∈ A, x ∈ RN \Sk (A), a = pA(x), such that reach (A, a) > 0, then

(x− a) · (a− b) ≥ − |a− b|2|x− a|2 reach (A, a)

. (6.3)

(v) Given 0 < r < q <∞ and

x ∈ RN \Sk (A) such that dA(x) ≤ r and reach (A, pA(x)) ≥ q,

y ∈ RN \Sk (A) such that dA(y) ≤ r and reach (A, pA(y)) ≥ q,

then

|pA(y)− pA(x)| ≤ q

q − r|y − x|. (6.4)

(vi) If 0 < reach (A) < +∞, then for all h, 0 < h < reach (A),

|pA(y)− pA(x)| ≤ reach (A)reach (A)− h

|y − x|, ∀x, y ∈ Ah = x ∈ RN : dA(x) ≤ h,

∣∣∇d2A(y)−∇d2

A(x)∣∣ ≤ 2

(1 +

reach (A)reach (A)− h

)|y − x|, ∀x, y ∈ Ah; (6.5)

for all 0 < s < h < reach (A) and all x, y ∈ As,h =x ∈ RN : s ≤ dA(x) ≤ h

,

|∇dA(y)−∇dA(x)| ≤ 1s

(2 +


)|y − x| .

If reach (A) = +∞,

∀x, y ∈ RN, |pA(y)− pA(x)| ≤ |y − x|, (6.6)∣∣∇d2

A(y)−∇d2A(x)

∣∣ ≤ 4 |y − x|;

for all 0 < s and all x, y ∈ x ∈ RN : s ≤ dA(x),

|∇dA(y)−∇dA(x)| ≤ 3s

|y − x| .


(vii) If reach (A) > 0, then, for all 0 < r < reach (A), Ur(A) is a set of class C1,1,Ur(A) = Ar, ∂Ur(A) = x ∈ RN : dA(x) = r, and !Ar = !Ur(A) = x ∈RN : dA(x) ≥ r.

(viii) For all a ∈ A,

TaA =

h ∈ RN : lim inft0

dA(a + th)t

= 0

.

In particular, if a ∈ intA, TaA = RN.

Remark 6.4.For part (viii), more properties of TaA and the normal cone (−TaA)∗ (cf. Defini-tions 2.3 and 2.4 in Chapter 2) can be found in H. Federer [3, sect. 4].

Proof. (i) By definition for all a ∈ A,

a ∈ P (A) ⇐⇒ ∀b ∈ A\a, |a + v − b| > |v|,a ∈ Q(A) ⇐⇒ ∀b ∈ A\a, |a + v − b| ≥ |v|

and P (a) ⊂ Q(a). To prove the convexity, we need the identity

|a + v − b|2 − |v|2 = (a− b) · (2v + a− b).

The convexity of P (a) and Q(a) now follow from the following identities: for allλ ∈ [0, 1] and v, w ∈ RN,

|a + (λv + (1− λ)w)− b|2 − |(λv + (1− λ)w|2

= (a− b) · (2((λv + (1− λ)w) + a− b)= λ (a− b) · (2v + a− b) + (1− λ) (a− b) · (2w + a− b)

= λ(|a + v − b|2 − |v|2

)+ (1− λ)

(|a + w − b|2 − |v|2

).

Finally, for h ∈ TaA, there exist sequences bn ⊂ A and εn > 0, εn 0, suchthat (bn − a)/εn → h and bn → a. For all v ∈ Q(a),

2bn − a

εn· v ≤ |bn − a|2

εn|bn − a| ≤ 0 ⇒ h · v ≤ 0 ⇒ h ∈ [−TaA]∗.

(ii) Let Sdef= t > 0 : ΠA(a + tv) = a. Since, by assumption, r(a, v) =

supS > 0, then S = ∅ and v = 0. For each t ∈ S, ΠA(a + tv) = a anddA(a + tv) = t |v|. In particular for all 0 < s < t, dA(a + sv) ≤ s |v|. For anyp ∈ ΠA(a + sv) ⊂ A, dA(a + sv) = |a + sv − p| ≤ s |v| and

dA(a + tv) ≤ |a + tv − p| = |a + sv − p + (t− s)v|≤ |a + sv − p| + (t− s)|v| ≤ s |v| + (t− s)v| = t |v| = dA(a + tv).

As a result p ∈ ΠA(a + tv) = a, p = a, and necessarily ΠA(a + sv) = a anddA(a + sv) = s |v|. This proves that for all t ∈ S, s : 0 < s ≤ t ⊂ S and hence forall 0 ≤ t < r(a, v).


Assume that when r = r(a, v) <∞, a + rv /∈ Sk (A). Then there exists δ > 0such that Bδ(a+rv)∩(Sk (A)∪A) = ∅. By continuity of the projection, ΠA(a+rv) =a and dA(a+rv) = r |v| > 0 so that a+rv /∈ A. By Theorem 3.3 (vi), the function∇dA : RN \

(Sk (A) ∪ ∂A

)→ R is continuous and for all x ∈ RN \

(Sk (A) ∪A

),

∇dA(x) = (x−pA(x))/|x−pA(x)|. For 0 < t < r, pA(a+ tv) = a, dA(a+ tv) = t|v|,and ∇dA(a + tv) = v/|v|. By continuity, ∇dA(a + rv) = v/|v|.

Since∇dA is continuous in Bδ(a+rv), the Caratheodory conditions are verifiedand there exists ε, 0 < ε < δ, such that the differential equation

x′(t) = ∇dA(x(t)), x(0) = a + rv = a + r |v| v

|v|

has a solution in (−ε, ε). This yields

d

dtdA(x(t)) = ∇dA(x(t)) · x′(t) = ∇dA(x(t)) ·∇dA(x(t)) = 1

⇒ dA(x(t)) = r|v| + t in (−ε, ε).

For −ε < p < q < ε, the length of the curve x(t) is given by∫ q

p|x′(t)| dt =

∫ q

p

d

dtdA(x(t)) dt = dA(x(q))− dA(x(p)) ≤ |x(q)− x(p)|

so that the length of the curve x(t) between the points x(p) and x(q) is less than orequal to the distance |x(q) − x(p)| between x(p) and x(q). Therefore x(t) is a linebetween x(p) and x(q):

x(t) = x(p) + (t− p)x(q)− x(p)

q − p, x′(t) =

x(q)− x(p)q − p

⇒ x(q)− x(p)q − p

= x′(0) = ∇dA(x(0)) = ∇dA(a + rv) =v

|v|

⇒ ∀t, 0 < t < ε, x(t) = x(0) + (t− 0)v

|v| = a + rv + tv

|v|⇒ |x(t)− a| = r|v| + t = dA(x(t)) ⇒ pA(x(t)) = a and ΠA(a + r|v| + t) = a

and we get the contradiction supS ≥ r + t/|v| > r = supS. Therefore, a + rv /∈Sk (A).

(iii) If there exists 0 < h ≤ reach (A), then for all a ∈ A and all 0 < s < h,Bs(a) ∩ Sk (A) = ∅. For x ∈ Uh(A)\A, 0 < dA(x) < h and for all p ∈ ΠA(x)

(dA(x) + h)/2 < h ≤ reach (A) ⇒ B(dA(x)+h)/2(p) ∩ Sk (A) = ∅.

Since x ∈ B(dA(x)+h)/2(p), ΠA(x) is a singleton and p = pA(x). From part (ii), Sdef=

t > 0 : ΠA(pA(x) + t∇dA(x)) = pA(x) = ∅ and rAdef= r(pA(x),∇dA(x)) ≥

dA(x) > 0. If rA = +∞, then for all t ≥ 0 we have ΠA(x(t)) = pA(x) anddA(x(t)) = t. If rA < +∞, then x(rA) ∈ Sk (A) and for all ρ > 0, BrA+ρ(pA(x)) ∩Sk (A) = ∅. This implies that, for all ρ > 0, reach (pA(x), A) ≤ rA + ρ and


reach (A) ≤ reach (pA(x), A) ≤ rA. Thus, for all 0 < t < h, t < h ≤ reach (A) ≤ rA

and we have ΠA(x(t)) = pA(x) and dA(x(t)) = t.(iv) If b = a, the inequality is verified. For b = a, let

vdef=

x− a

|x− a| , Sdef= t > 0 : ΠA(a + tv) = a .

By assumption pA(x) = a and

x = a + |x− a| v ⇒ |x− a| ∈ S, supS ≥ |x− a| > 0,

and, from part (ii), a + supS v ∈ Sk (A).If reach (A, a) > supS, then for ρ, reach (A, a) > ρ > supS,

Bρ(a) ∩ Sk (A) = ∅ ⇒ Bsup S(a) ∩ Sk (A) = ∅

and this contradicts the fact that a + supS v ∈ Sk (A). Therefore, reach (A, a) ≤supS. Given t, 0 < t < reach (A, a), define xt = a + tv. Then t ∈ S and t < supS.Thus ΠA(xt) = a, pA(xt) = a, and dA(xt) = |xt − a| = t. For b ∈ A, |xt − b| ≥|xt − a| = t > 0 and

∣∣∣∣a + tx− a

|x− a| − b

∣∣∣∣2

≥ t2 ⇒ (a− b) · (x− a) ≥ −|a− b|2 |x− a|2t

.

To conclude, choose a sequence tn, 0 < tn < reach (A, a), such that tn →reach (A, a) in the above inequality.

(v) Apply the inequality of part (iii) twice to the triplets (a, x, b) and (b, y, a):

(a− b) · (x− a) ≥ −|a− b|2 |x− a|2 reach (A, a)

≥ −|a− b|2 r

2 q,

(b− a) · (y − b) ≥ −|a− b|2 |y − b|2 reach (A, b)

≥ −|a− b|2 r

2 q.

By adding the two inequalities

(x− a + b− y) · (a− b) ≥ −r

q|a− b|2 ⇒ (x− y) · (a− b) ≥

(1− r

q

)|a− b|2

⇒ |pA(x)− pA(y)| = |a− b| ≤ q

q − r|x− y|.

(vi) Given 0 < h < r < reach (A), for all a ∈ A,

0 < h < r < reach (A) ≤ reach (A, a) and Br(a) ∩ Sk (A) = ∅.

Therefore Ah ⊂ Ur(A) = ∪a∈ABr(a) ⊂ RN \Sk (A) and from (iv)

∀x, y ∈ Ah, ∀r, h < r < reach (A), |pA(x)− pA(y)| ≤ r

r − h|x− y|.


The function r '→ r/(r − h) : (h,∞)→ R is strictly decreasing to 1. If reach (A) =+∞, then for all a ∈ A, reach (A, a) = +∞ and by letting r go to infinity

∀x, y ∈ Ah, |pA(x)− pA(y)| ≤ |x− y|.

If reach (A) is finite, then for all a ∈ A, reach (A, a) ≥ reach (A) > r and by lettingr go to reach (A)

∀x, y ∈ Ah, |pA(x)− pA(y)| ≤ reach (A)reach (A)− h

|x− y|.

The inequalities for ∇d2A follow from identity (3.13) in Theorem 3.3 (v).

Finally, the Lipschitz continuity of ∇dA follows from identity (3.15) in The-orem 3.3 (vi). For all pairs 0 < s < h < reach (A) and all x, y ∈ x ∈ RN : s ≤dA(x) ≤ h

|∇dA(y)−∇dA(x)| =∣∣∣∣y − pA(y)

dA(y)− x− pA(x)

dA(x)

∣∣∣∣

=|(y − pA(y)) dA(x)− (x− pA(x)) dA(y)|

dA(x) dA(y)

≤ |(y − pA(y)) (dA(x)− dA(y))|dA(x) dA(y)

+|[(y − pA(y))− (x− pA(x))] dA(y)|

dA(x) dA(y)

≤ 1dA(x)

(|dA(x)− dA(y)| + |y − x| + |pA(y)− pA(x)|)

≤ 1s

(2 +


)|y − x| .

When reach (A) = +∞, for all s > 0 and all x, y ∈ x ∈ RN : s ≤ dA(x)

|∇dA(y)−∇dA(x)| ≤ 3s

|y − x| .

(vii) For 0 < r < reach (A), there exists h such that 0 < r < h < reach (A).The open set Ur(A) is defined via the r-level set d−1

A r = x ∈ RN : dA(x) = r ofthe function dA. From part (vi), for each x ∈ d−1

A r, there exists a neighborhoodV (x) = Bρ(x), ρ = minr, h − r/2, of x in Uh(A)\A such that dA ∈ C1,1(V (x)).Therefore, ∇dA exists and is not zero on the level set ∂Ar = d−1

A r since |∇dA| = 1.From Theorem 4.2 in Chapter 2, the set Ur(A) is of class C1,1, Ur(A) = Ar, and∂Ur(A) = d−1

A r, and !Ar = !Ur(A).(viii) Given h ∈ TaA (cf. Definition 2.3 in Chapter 2), there exist sequences

an ⊂ A and εn, εn 0, such that (an − a)/εn → h. For each n,

0 ≤ dA(a + εnh)εn

≤ |a + εnh− an|εn

=∣∣∣∣h−

an − a

εn

∣∣∣∣→ 0 ⇒ lim inft0

dA(a + th)t

= 0.


Conversely, there exists tn > 0, tn 0, such that dA(a + tnh)/tn → 0. Thismeans that there exists a sequence of projections an ⊂ A such that

∣∣∣∣h−an − a

tn

∣∣∣∣ =|a + tnh− an|

tn=

dA(a + tnh)tn

→ 0

and h ∈ TaA.

The next theorem summarizes several characterizations of sets of positivereach. Note that condition (ii) is a global condition on the smoothness of d2

A inthe tubular neighborhood Uh(A).

Theorem 6.3. Given ∅ = A ⊂ RN, the following conditions are equivalent:

(i) A has positive reach, that is, reach (A) > 0.

(ii) There exists h > 0 such that d2A ∈ C1,1(Uh(A)).

(iii) There exists h > 0 such that for all s, 0 < s < h, dA ∈ C1,1(Us,h(A)).

(iv) There exists h > 0 such that dA ∈ C1(Uh(A)\A).

(v) There exists h > 0 such that for all x ∈ Uh(A), ΠA(x) is a singleton.

Proof. (i) ⇒ (ii) For all h, 0 < h < reach (A), and all a ∈ A, Bh(a) ∩ Sk (A) = ∅.Hence for all x ∈ Uh(A) ΠA(x) is a singleton and ∇d2

A(x) exists. The functions d2A

and ∇d2A are bounded and uniformly continuous in Uh(A):

∀x ∈ Uh(A), d2A(x) ≤ h2, ∇d2

A(x) = |2(x− pA(x)| = 2 dA(x) ≤ 2h,

∀x, y ∈ Uh(A), |d2A(y)− d2

A(x)| = (|dA(y)− dA(x)) |dA(y)− dA(x)| ≤ 2h |y − x|,

∀x, y ∈ Uh(A),∣∣∇d2

A(y)−∇d2A(x)

∣∣ ≤ 2(

1 +reach (A)

reach (A)− h

)|y − x|

from inequality (6.5) of Theorem 6.2 (vi). By definition, d2A ∈ C1,1(Uh(A)).

(ii) ⇒ (iii) From the proof of Theorem 6.2 (vi).(iii) ⇒ (iv) By assumption, for all 0 < s < h, dA ∈ C1(Us,h(A)). Since

Uh(A)\A = ∪0<s<hUs,h(A), dA ∈ C1(Uh(A)\A).(iv) ⇒ (v) For all x ∈ Uh(A)\A, dA(x) > 0 and ∇d2

A(x) = 2 dA(x)∇dA(x)exists; from Theorem 3.3 (v), for all x ∈ A, ∇d2

A(x) = 0. This means that ΠA(x) isa singleton in Uh(A) = Uh(A)\A ∪A.

(v) ⇒ (i) By assumption, for all a ∈ A and all x ∈ Bh(a), Bh(a) ∩Sk (A) = ∅. This implies that for all a ∈ A, reach (A, a) ≥ h, which finally impliesthat reach (A) ≥ h > 0.

6.2 Ck-SubmanifoldsFor arbitrary closed submanifolds M of RN of codimension greater than or equalto one, ∇dM has a discontinuity along M and generally does not exist. In thatcase the smoothness of M is related to the existence and the smoothness of ∇d2

Min a neighborhood of M that is, a fortiori, locally of positive reach. The followinganalysis of the smoothness of M was given by J.-B. Poly and G. Raby [1] in 1984.


Theorem 6.4. Let A be a closed nonempty subset of RN and k ≥ 2 be an integer(k =∞ and ω included7). Then

singkA = A ∩ sing

kd2

A,

where singkd2

A = RN \regkd2

A, singkA = RN \reg

kA, and

regkd2

Adef=x ∈ RN : d2

A is Ck in a neighborhood of x

,

regkA

def=x ∈ A : A is a Ck-submanifold of RN in a neighborhood of x

.

It is more interesting to rephrase their theorem in terms of regularities thansingularities.

Theorem 6.5. Let ∅ = A ⊂ RN and let k ≥ 2 be an integer (k = ∞ andω included), and x ∈ A.8

(i) d2A is Ck in a neighborhood of a point x ∈ A if and only if A is a Ck-

submanifold in a neighborhood of x.

(ii) Under conditions (i), the dimension of A in x is equal to the rank of DpA(x)(dimension of the subspace DpA(RN)) and DpA(x) is the orthogonal projectoronto the tangent space to A in x.

(iii) If ∂A = ∅, then A = RN. If ∂A = ∅ and d2A is C2 in a neighborhood of

x ∈ A, then there exists an open neighborhood V (x) of x such that

(a) if x ∈ intA, then dA = dRN = 0 in V (x);(b) if x ∈ ∂A, then dA = d∂A in V (x), intA ∩ V (x) = ∅, d!A = 0 in V (x),

A is a C2-submanifold of dimension less than or equal to N − 1 in V (x),and m(∂A ∩ V (x)) = 0.

Proof. (i) This follows from Theorem 6.4.(ii) For y ∈ V (x)

fA(y) def=12(|y|2 − d2

A(y))⇒

∣∣∣∣∣∣∣

∇fA(y) = y − 12∇d2

A(y) = pA(y),

D2fA(y) = I − 12D2d2

A(y) = DpA(y).

Since pA is a projection, pA(pA(y)) = pA(y) implies that DpA pA DpA = DpA and(DpA)2 = DpA on A. Therefore, if R(x) denotes the image of DpA(x), DpA(x) isan orthogonal projector onto R(x) and I −DpA(x) is an orthogonal projector ontoR(x)⊥ and

∀v ∈ RN, v = DpA(x)v + [I −DpA(x)]v ∈ R(x)⊕R(x)⊥.

7ω indicates the analytical case.8Note that A can have several connected components with the same smoothness k but different

dimension. When A = RN, dA is identically zero. Another set of results for Holderian sets wasobtained by G. M. Lieberman [1] (see also V. I. Burenkov [1]) by introducing a regularizeddistance function.


Consider the C1-mapping

Tx(y) def= (DpA(x)(pA(y)− x), [I −DpA(x)](y − pA(y))) : W (x)→ R(x)⊕R(x)⊥.

Tx is a C1-diffeomorphism from a neighborhood U(x) ⊂ V (x) of x onto a neighbor-hood W (0) of 0 since DTx is continuous in V (x) and

DTx(y) def= (DpA(x)DpA(y), [I −DpA(x)](I −DpA(y))),

DTx(x) def= (DpA(x)DpA(x), [I −DpA(x)](I −DpA(x)))= (DpA(x)DpA(x), [I −DpA(x)](I −DpA(x))= (DpA(x), I −DpA(x)) = I.

By construction

Tx(U(x) ∩A) = V (0) ∩R(x),

V (0) ∩R(x)⊥ = (0, [I −DpA(x)](y − pA(y))) : ∀y ∈ U(x) ,

and U(x)∩A is a C1-submanifold in RN of codimension r, where N − r is the rankof DpA(x).

(iii) If ∂A = ∅, !A ⊂ !A and !A is both open and closed. Since A = ∅,then A = RN. Let ∂A = ∅. If x ∈ ∂A, then !A = ∅. We claim that int A = ∅.If this is not true, there exists a neighborhood V (x) of x where d2

A is C2 andintA ∩ V (x) = ∅ where dA = 0 and D2d2

A = 0. This implies that DpA = I inintA ∩ V (x). In !A ∩ V (x), D2d2

A is continuous and dA > 0. Therefore, ∇dA andD2dA exist, |∇dA| = 1, and

12D2d2

A = ∇dA∗∇dA + dA D2dA

⇒ DpA = I −∇dA∗∇dA − dA D2dA and DpA∇dA = 0

since |∇dA|2 = 1 implies that D2dA∇dA = 0. As a result the rank of DpA isless than or equal to N − 1 in !A ∩ V (x) and equal to N in intA ∩ V (x). Thiscontradicts the continuity of D2d2

A and DpA in x. Therefore, A∩V (x) = ∂A∩V (x),and d∂A = dA in V (x) and, by the previous argument in !A ∩ V (x), A is a C2-submanifold of dimension less than or equal to N − 1 in a neighborhood of x.

At this juncture, several interesting remarks can be made.

(a) The equivalence in (i) fails in the direction (⇒) for d2A ∈ C1,1 as seen from

Example 6.1 of J.-B. Poly and G. Raby [1].

(b) The equivalence in (i) also fails in the direction (⇐) for a C1,ℓ submanifold,0 ≤ ℓ < 1, as seen from Example 6.2 of S. G. Krantz and H. R. Parks [1].

(c) The square d2A is really pertinent for Ck-submanifolds of codimension greater

than or equal to one and not for sets of class Ck. Indeed, when ∂A = ∅ andd2

A is Ck, k ≥ 2, in a neighborhood of each point of A, then A is necessarilya Ck-submanifold of codimension greater than or equal to one. In particular,!A = RN, A = ∂A, and m(A) = 0.


Example 6.1.Consider the half-space in RN

Adef=x ∈ RN : x · eN ≤ 0

,

dA(x) =

0, x · eN ≤ 0,

x · eN , x · eN > 0,∇d2

A(x) =

0, x · eN ≤ 0,

2 (x · eN ) eN , x · eN > 0,

and d2A is locally C1,1. For any point x ∈ ∂A = x ∈ RN : x · eN = 0, A ∩ V (x) is

not even a manifold in any neighborhood V (x) of x.

The equivalence in (i) also fails in the direction (⇐) for C1,ℓ, 0 ≤ ℓ < 1, asseen from Example 6.2, that shows that for a set of class C1,ℓ, 0 ≤ ℓ < 1, we do notget bA ∈ C1,λ in a neighborhood of the boundary ∂A.

Example 6.2.Consider the two-dimensional domain Ω defined as the epigraph of the function f :

Ω def= (x, z) : f(x) > z, ∀x ∈ R , f(x) def= |x|2− 1n

for some arbitrary integer n ≥ 1. Ω is a set of class C1,1−1/n and ∂Ω is a C1,1−1/n-submanifold of dimension 1. In view of the presence of the absolute value in (0, 0),it is the point where the smoothness of ∂Ω is minimum. We claim that

Sk (∂Ω) = Sk (Ω) = (0, y) : y > 0 and ∂Ω ∩ Sk (∂Ω) = (0, 0) = ∅.

Because the skeleton is a line touching ∂Ω in (0, 0), d2∂Ω and d2

Ω cannot be C1 inany neighborhood of (0, 0).

Indeed, for each (0, y)

d2∂Ω(0, y) = inf

x∈R

x2 + |f(x)− y|2

.

The point (0, y), y > 0, belongs to Sk (∂Ω) since there exist two different points xthat minimize the function

F (x, y) def= x2 + +|f(x)− y|2. (6.7)

Since f is symmetric with respect to the y-axis it is sufficient to show that thereexists a strictly positive minimizer x > 0. A locally minimizing point x ≥ 0 mustsatisfy the conditions

F ′x(x, y) = 2x + 2

(|x|2− 1

n − y)(

2− 1n

)x1− 1

n = 0, (6.8)

12F ′′

x (x, y) = 1 +(

2− 1n

)(3− 2

n

)x2− 2

n −(2− 1

n

)(1− 1

n

)y x− 1

n ≥ 0. (6.9)

Equation (6.8) can be rewritten as

2x1− 1n

[x

1n +

(x2− 1

n − y)

(2− 1/n)]

= 0,


and x = 0 is a solution. The second factor can be written as an equation in thenew variable X

def= x1/n and

g(X) def=n

2n− 1X + X2n−1 − y = 0.

It has exactly one solution X since

∀X,dg

dX(X) =

n

2n− 1+ (2n− 1)X2n−2 > 0,

g(0) = −y, and g(X) goes to infinity as X goes to infinity. In particular, X > 0 fory > 0 and X = 0 for y = 0. For y > 0 and x = Xn

12F ′

x(x, y) = 0 ⇒ y = x2− 1n +

n

2n− 1x

1n (6.10)

and

12F ′′

x (x, y) =1 +(

2− 1n

)(3− 2

n

)x2− 2

n

−(

2− 1n

)(1− 1

n

) [x2− 2

n +n

2n− 1

]

=1−(

n− 1n

)+(

2n− 1n

)2

x2− 2n =

1n

+(

2n− 1n

x1− 1n

)2

> 0.

Therefore, x is a local minimum. To complete the proof for y > 0 we must compareF (0, y) = y2 for the solution corresponding to x = 0 with

F (x, y) = x2 + |x2− 1n − y|2

for the solution x > 0. Again using identity (6.10)

F (x, y) = x2 +(

n

2n− 1

)2

x2n ,

F (0, y) =∣∣∣∣x

2− 1n +

n

2n− 1x

1n

∣∣∣∣2

=(x2− 1

n

)2+(

n

2n− 1

)2

x2n + 2

n

2n− 1x2

⇒ F (0, y)− F (x, y) =(

2n

2n− 1− 1)

x2 +(x2− 1

n

)2=

12n− 1

x2 +(x2− 1

n

)2> 0.

This proves that for y > 0, x > 0 is the minimizing positive solution. Therefore, forall n ≥ 1, f(x) = |x|2−1/n yields a domain Ω for which the strictly positive part ofthe y-axis is the skeleton. It is interesting to note that, for each point x, the points(x, f(x)) ∈ ∂Ω are the projections of the point

(0, y) =(

0, |x|2− 1n +

n

2n− 1|x| 1

n

)


and that the square of the distance function is equal to

b2Ω(0, y) = x2 +

(n

2n− 1

)2

x2n ,

∇bΩ(x, f(x)) =1√

1 + f ′(x)2(f ′(x),−1)

=1√

1 +( 2n−1

n

)2 |x|2− 2n

((2− 1/n) x |x|− 1n ,−1).

Finally for all n ≥ 1, f /∈ C1,1. For n = 1, f ∈ C0,1, and for n > 1, f ∈ C1,1−1/n.This gives an example in dimension 2 of a domain Ω of class C1,λ, 0 ≤ λ < 1, withskeleton converging to the boundary.

6.3 A Compact Family of Sets with Uniform Positive ReachThe following family of sets of positive reach and the associated compactness theo-rem were first given in H. Federer [3, Thm. 4.13, Rem. 4.14].

Theorem 6.6. Given an open (resp., bounded open) holdall D in RN and h > 0,the family

Cd,h(D) def=dA : ∅ = A ⊂ D, reach (A) ≥ h

(6.11)

is closed in Cloc(D) (resp., compact in C(D)).

Proof. It is sufficient to prove the theorem for a bounded open D. Given any C(D)-Cauchy sequence dAn ⊂ Cd,h(D), there exists dA ∈ Cd(D) such that dAn → dA

in C(D). Choose W open such that W ⊂ Uh(A). Then supx∈W dA(x) < h. Chooser, supx∈W dA(x) < r < h. Then W ⊂ Ur(A) ⊂ Ar ⊂ Uh(A). Choose N such that

∀n > N, supx∈W

dAn(x) < r < h.

Then for all n > N , W ⊂ Ur(An).From Theorem 6.2 (v), for all n > N and

∀x, y ∈ (An)r,∣∣∇d2

An(y)−∇d2

An(x)∣∣ ≤ 2

(1 +

h

h− r

)|y − x|. (6.12)

But W ⊂ Ur(A) ⊂ Ar ⊂ Uh(A), d2An→ d2

A in C(D), and

∀n > N, d2An∈ C1,1(W ) ⊂ C1(W ).

From Theorem 2.5 in Chapter 2, the embedding C1,1(W ) ⊂ C1(W ) is compact.Therefore, for all open subsets such that W ⊂ Uh(A), d2

An→ d2

A ∈ C1(W ), d2A ∈

C1(Ur(A)), and reach (A) ≥ r. Since this is true for all r < h, we finally getreach (A) ≥ h and dA ∈ Cd,h(D).


The compactness of Theorem 6.7 can also be expressed as follows.

Theorem 6.7. Let D be a fixed bounded open subset of RN. Let An, An = ∅,be a sequence of subsets of D. Assume that there exists h > 0 such that

∀n, d2An∈ C1,1(Uh(An)). (6.13)

Then there exist a subsequence Ank and A ⊂ D, A = ∅, such that reach (A) ≥ hand for all r, 0 < r < h, d2

A ∈ C1,1(Ur(A)),

d2Ank→ d2

A in C1(Ur(A)) and dAnk→ dA in C(D). (6.14)

This condition in an h-tubular neighborhood is generic of other such conditionsthat will be expressed in different function spaces.

7 Approximation by Dilated Sets/TubularNeighborhoods and Critical Points

The W 1,p-topology turns out to be an appropriate setting for the approximation ofthe closure of a set by its dilated sets as defined in Definition 3.2.

Theorem 7.1. Let ∅ = A ⊂ RN. For r > 0,

Ar = Ur(A), ∂Ar = ∂Ur(A) = d−1A r, !Ar = !Ur(A), intAr = Ur(A), (7.1)

dUr(A) = dAr (x) =

dA(x)− r, if dA(x) ≥ r,

0, if dA(x) < r,(7.2)

∀x ∈ RN, 0 ≤ dA(x)− dAr (x) ≤ r, (7.3)

dUr(A) = dAr (x)→ dA(x) uniformly in RN as r → 0,

dUr(A) = dAr → dA in W 1,ploc (RN) and χAr → χA in Lp

loc(RN) (7.4)

for all p, 1 ≤ p <∞.

Proof. The first line of properties follows from Theorem 3.1. For r > 0, A ⊂Ur(A) ⊂ Ar and dA(x) ≥ dAr (x) for all x ∈ RN. For x ∈ RN such that dA(x) < r,dAr (x) = 0 and 0 ≤ dA(x) − dAr (x) < r. For x ∈ RN such that dA(x) ≥ r and alla ∈ A and ar ∈ Ar,

|x− a| ≤ |x− ar| + |x− ar| ⇒ dA(x) ≤ |x− ar| + dA(ar) ≤ |x− ar| + r

⇒ dA(x) ≤ dAr (x) + r ⇒ 0 ≤ dA(x)− dAr (x) ≤ r.

We show that dA(x)− dAr (x) = r. Assume that dA(x)− dAr (x) < r. There existsa unique p ∈ A such that dA(x) = |x− p| ≥ r > 0. Consider the point

xrdef= p + r

x− p

|x− p| ⇒ |x− xr| = dA(x)− r and |xr − p| = r.

7. Approximation by Dilated Sets/Tubular Neighborhoods and Critical Points 317

By definition, dA(xr) ≤ |xr − p| = r and xr ∈ Ar. As a result,

dA(x) < dAr (x) + r ≤ |x− xr| + r = dA(x)− r + r = dA(x)

yields a contradiction. This establishes identity (7.2) and for all x ∈ RN, |dAr (x)−dA(x)| ≤ r → 0 as r goes to zero.

When ∇dAr (x) exists at a point ∈ Ar, ∇dAr (x) = 0 (cf. Theorem 3.3 (vi)).This means that ∇dAr = 0 almost everywhere in Ar and that the gradient of dAr

is almost everywhere equal to

∇dAr (x) =

∇dA(x), dA(x) > r,

0, dA(x) ≤ r.

For any compact K ⊂ RN

∫

K|∇dAr −∇dA| dx =

∫

K|∇dA|χAr dx =

∫

K(1− χA)χAr dx =

∫

K

(χAr − χA

)dx.

Since K ∩ Ar is a decreasing family of closed sets as r → 0 and ∩r>0Ar = A,m(K ∩ Ar) → m(K ∩ A) (cf. W. Rudin [1, Thm. 1.1.9 (e), p. 16]). Finally,∥∇dAr −∇dA∥L1 → 0 and dAr → dA in W 1,1

loc (RN).

S. Ferry [1] proved in 1975 that if N = 2 or 3, then d−1A r is an (N − 1)-

manifold for almost all r, and that if A is a finite polyhedron in RN, then d−1A r

is an (N − 1)-manifold for all sufficiently small r. On the other hand, there is aCantor set K in R4 such that d−1

K r is not a 3-manifold for any r between 0 and1. The set K generalizes to higher dimensions.

Generalizing those results, J. H. G. Fu [1] investigated the regularity prop-erties of ∂Ar and proved the following theorem involving the notion of critical pointof dA and of Lipschitz manifold. In order to make things simple we use a specificcharacterization of the critical points of dA as a definition (cf. J. H. G. Fu [1, 2]and Figure 6.5).

Definition 7.1.Given ∅ = A ⊂ RN, x ∈ RN \A is said to be a critical point of dA if x ∈ co ΠA(x).The set of all critical points of dA will be denoted by crit (dA).

Definition 7.2.An m-dimensional Lipschitz manifold is a paracompact metric space M such thatthere is a system of open sets Uα covering M and, for each α, a bi-Lipschitzianhomeomorphism ϕα of Uα onto an open subset of Rm.

Theorem 7.2. Let A ⊂ RN be compact. The subset C(A) = dA(crit (dA)) ⊂ R iscompact:

C(A) ⊂[0,

(N

2N + 2

)1/2

diam (A)

]and H(N−1)/2(C(A)) = 0

(the assertion about the measure of C(A) is nontrivial only for N ≤ 3). For allr /∈ C(A), !Ar has positive reach and ∂Ar is a Lipschitz manifold.


critical points of A

Figure 6.5. Set of critical points of A.

8 Characterization of Convex Sets8.1 Convex Sets and Properties of dA

In the convex case the squared distance function is differentiable everywhere andthis property can be used to characterize the convexity of a set.

Theorem 8.1. Let ∅ = A ⊂ RN. The following statements are equivalent:

(i) There exists a ∈ A such that reach (A, a) = +∞.

(ii) Sk (A) = ∅.

(iii) For all a ∈ A, reach (A, a) = +∞.

(iv) reach (A) = +∞.

(v) For all x ∈ RN, ΠA(x) is a singleton.9

(vi) A is convex.

(vii) dA is convex.

In particular,

∀x,∀y ∈ RN, |pA(y)− pA(x)| ≤ |y − x| (8.1)

and d2A belongs to C1,1

loc (RN) (and, a fortiori, to W 2,∞loc (RN)). Moreover, if A is

convex, then !A = !A and d!A = d!A.

Remark 8.1.The convexity of dA implies the convexity of A but not of A. The set A of allpoints of the unit open ball with rational coordinates has a convex closure but isnot convex.

9This part of the theorem is related to deeper results on the convexity of Chebyshev sets inmetric spaces. A subset A of a metric space X is called a Chebyshev set provided that every pointx of X has a unique projection pA(x) in A. The reader is referred to V. L. Klee [1] for detailsand background material.

8. Characterization of Convex Sets 319

Proof. The equivalence of properties (i) to (v) follows from Lemma 6.1.(v)⇒ (vi) Given x, y ∈ A and λ ∈ [0, 1], consider the point xλ = λx+(1−λ)y

and the continuous function λ '→ dA(xλ) : [0, 1] → R is continuous. There existsµ ∈ [0, 1] such that dA(xµ) = supλ∈[0,1] dA(xλ). If supλ∈[0,1] dA(xλ) = 0, thenxλ ∈ A. If supλ∈[0,1] dA(xλ) > 0, then x = y, dA(xµ) > 0, µ ∈ (0, 1), and xµ /∈ A.Since Sk (A) = ∅, ∇dA(xµ) exists and the maximum is characterized by

d

dλdA(xλ)

∣∣∣∣λ=µ

= 0 ⇒ ∇dA(xµ) · (x− y) = 0.

By assumption there exists a unique projection pµ = pA(xµ) ∈ A such that dA(xµ) =|xµ − pµ| and by Theorem 3.3 (vi)

∇dA(xµ) =xµ − pµ

|xµ − pµ| .

From inequality (6.6) in Theorem 6.2 (vi)

|pµ − x| = |pA(xµ)− pA(x)| ≤ |xµ − x| = (1− µ) |x− y|,|pµ − y| = |pA(xµ)− pA(y)| ≤ |xµ − y| = µ |x− y|⇒ |x− y| ≤ |pµ − x| + |pµ − y| ≤ |x− y|.

Therefore the point pµ lies on the segment between x and y: there exists µ′ ∈ [0, 1],µ′ = µ, such that pµ = x′

µ and xµ − pµ = xµ − xµ′ = (µ− µ′) (x− y). Finally,

µ− µ′

|µ− µ′| |x− y| =xµ − xµ′

|xµ − xµ′ | · (x− y) =xµ − pµ

|xµ − pµ| · (x− y) = ∇dA(xµ) · (x− y) = 0

and we get the contradiction x = y.(vi) ⇒ (v) By definition for each x ∈ RN

d2A(x) = inf

z∈A|z − x|2 = inf

z∈A|z − x|2,

and since A is closed and convex and z '→ |z − x|2 is strictly convex and coercive,there exists a unique minimizing point pA(x) in A and ΠA(x) is a singleton.

(vi) ⇒ (vii) Given x and y in RN, there exist x and y in A such that dA(x) =|x−x| and dA(y) = |y−y|. By convexity of A, for all λ, 0 ≤ λ ≤ 1, λx+(1−λ) y ∈ Aand

dA(λx + (1− λ) y) ≤|λx + (1− λ) y − (λx + (1− λ) y)|≤λ |x− x| + (1− λ) |y − y| = λ dA(x) + (1− λ) dA(y)

and dA is convex in RN.(vii) ⇒ (vi) If dA is convex, then

∀λ ∈ [0, 1],∀x, y ∈ A, dA(λx + (1− λ)y) ≤ λdA(x) + (1− λ)dA(y).


But x and y in A imply that dA(x) = dA(y) = 0 and hence

∀λ ∈ [0, 1], dA(λx + (1− λ)y) = 0.

Thus λx + (1− λ)y ∈ A and A is convex.Properties (8.1) follow from Theorem 6.2 and the identity on the complements

follows from Theorem 3.5 (ii) in Chapter 5.

The next theorem is a consequence of the fact that the gradient of a convexfunction is locally of bounded variations.

Theorem 8.2. For all subsets A of RN such that ∂A = ∅ and A is convex,

(i) ∇dA belongs to BVloc(RN)N ,

(ii) the Hessian matrix D2dA of second-order derivatives is a matrix of signedRadon measures that are nonnegative on the diagonal, and

(iii) dA has a second-order derivative almost everywhere and for almost all x andy in RN,∣∣∣∣dA(y)− dA(x)−∇dA(x) · (y − x)− 1

2(y − x) · D2dA(x) (y − x)

∣∣∣∣ = o(|y − x|2)

as y → x.

Proof. From the equivalence of (vi) and (vii) in Theorem 8.1 dA is convex andcontinuous and the result follows from L. C. Evans and R. F. Gariepy [1, Thm. 3,p. 240, Thm. 2, p. 239, and Aleksandrov’s Thm., p. 242].

8.2 Semiconvexity and BV Character of dA

Since the BV properties in Theorem 8.2 arise solely from the convexity of the dis-tance function dA, the BV property will extend to sets A such that dA is semiconvex.

Theorem 8.3. Let A be a nonempty subset of RN such that

∀x ∈ ∂A, ∃ρ > 0 such that dA is semiconvex in B(x, ρ).

Then ∇dA ∈ BVloc(RN)N .

Proof. For all x ∈ intA, dA = 0 and there exists ρ > 0 such that B(x, ρ) ⊂ intAand the result is trivial. For each x ∈ RN \A, dA(x) > 0. Pick h = dA(x)/2. Thenfor all y ∈ B(x, h) and all a ∈ A,

|y − a| ≥ |x− a|− |y − x| ≥ dA(x)− |y − x| > 2h− h = h

and B(x, h) ⊂ RN \Ah. By Theorem 3.2 (i) the function |x|2/(2h) − dA(x) islocally convex in RN \Ah and, a fortiori, convex in B(x, h). Finally, by assumptionfor all x ∈ ∂A, there exists ρ > 0 such that dA is semiconvex. Hence for eachx ∈ RN, ∇dA belongs to BV(B(x, ρ/2)) for some ρ > 0. Therefore, ∇dA belongsto BVloc(RN)N .


However, it is never semiconcave in RN for nontrivial sets.

Theorem 8.4.

(i) Given a subset A of RN, A = ∅,

∃c ≥ 0, fc(x) = c |x|2 − dA(x) is convex in RN

if and only if∃h > 0, ∃c ≥ 0, fc(x) = c |x|2 − dA(x)

is locally convex in Uh(A).

(ii) Given a subset A of RN, ∅ = A = RN,

"c ≥ 0, fc(x) = c |x|2 − dA(x) is convex in RN .

Proof. (i) If fc is convex in RN, it is locally convex in Uh(A). Conversely, assumethat there exist h > 0 and c ≥ 0 such that fc is locally convex in Uh(A). FromTheorem 3.2 (i),

1h

|x|2 − dA(x)

is locally convex in RN \Ah/2. Therefore, for c = maxc, 1/h, the function fc islocally convex in RN and hence convex on RN.

(ii) Assume the existence of a c ≥ 0 for which fc is convex in RN. For eachy ∈ A, the function

x '→ Fc(x, y) = c |x− y|2 − dA(x)

is also convex since it differs from fc(x) by the linear term

x '→ c (|y|2 − 2 x · y).

Since ∅ = A = RN, there exist x ∈ !A and p ∈ A such that

0 < dA(x) = |x− p| ≤ 12c

.

For any t > 0, define xt = p− t (x− p) and λ = t/(1 + t) ∈ ]0, 1[ and observe that

xλdef= λx + (1− λ) xt = p and Fc(xλ, p) = 0.

But

λFc(x, p) + (1− λ) Fc(xt, p)

=t

1 + t[c |x− p|2 − dA(x)] +

11 + t

[c |xt − p|2 − dA(xt)]

≤ t

1 + t[c dA(x)2 − dA(x) + c tdA(x)2]

≤ t

1 + tdA(x) [(1 + t) c dA(x)− 1]

≤ t

1 + tdA(x)

[(1 + t)

12− 1]

=dA(x) t

1 + t

t− 12

,


since by construction dA(x) > 0 and c dA(x) < 1/2. Therefore for t, 0 < t < 1,the above quantity is strictly negative and we have constructed two points x andxt and a λ, 0 < λ < 1, such that

Fc(λx + (1− λ) xt, p) = 0 > λFc(x, p) + (1− λ) Fc(xt, p).

This contradicts the convexity of the function x '→ Fc(x, p) and, a fortiori, of fc.

8.3 Closed Convex Hull of A and Fenchel Transform of dA

Another interesting property is that the closed convex hull of A is completely char-acterized by the convex envelope of dA that can be obtained from the double Fencheltransform of dA.Theorem 8.5. Let A be a nonempty subset of RN:

(i) The Fenchel transform

d∗A(x∗) = sup

x∈RNx∗ · x− dA(x)

of dA is given by

d∗A(x∗) = σA(x∗) + I

B(0,1)(x∗), (8.2)

where σA(x∗) is the support function of A and

IB(0,1)(x

∗) =

+∞, x /∈ B(0, 1),0, x ∈ B(0, 1),

is the indicator function of the closed unit ball B(0, 1) at the origin.

(ii) The Fenchel transform of d∗A is given by

d∗∗A = dco A = dco A. (8.3)

In particular, dco A is the convex envelope of dA.The function dA and the indicator function IA are both zero on A and the

Fenchel transform of IA,

I∗A

= σA = σA,

coincides with d∗A on B(0, 1).

Proof. (i) From the definition,

d∗A(x∗) = sup

x∈RNx∗ · x− dA(x) = sup

x∈RN

[x∗ · x− inf

p∈A|x− p|

]

= supx∈RN

supp∈A

x∗ · x− |x− p| = supp∈A

supx∈RN

x∗ · x− |x− p|

= supp∈A

supx∈RN

x∗ · p + x∗ · (x− p)− |x− p|

= supp∈A

x∗ · p + supx∈RN

[x∗ · (x− p)− |x− p| ]

= supp∈A

x∗ · p + supx∈RN

[x∗ · x− |x| ] = σA(x∗) + IB(0,1)(x

∗).


(ii) We now use the property that σA = σco A:

d∗∗A (x∗∗) = sup

x∗∈RNx∗∗ · x∗ − d∗

A(x∗) = supx∗∈RN

x∗∗ · x∗ − σA(x∗)− IB(0,1)(x

∗)

= sup|x∗|≤1

x∗∗ · x∗ − σA(x∗) = sup|x∗|≤1

[x∗∗ · x∗ − sup

x∈co Ax∗ · x

]

= sup|x∗|≤1

infx∈co A

(x∗∗ − x) · x∗

≤ sup|x∗|≤1

(x∗∗ − p(x∗∗)) · x∗ ≤ |x∗∗ − p(x∗∗)| = dco A(x∗∗).

But in fact we have a saddle point. If x∗∗ ∈ co A, pick x∗ = 0 and

d∗∗A (x∗∗) = sup

|x∗|≤1inf

x∈co A(x∗∗ − x) · x∗ ≥ 0 = dco A(x∗∗).

For x∗∗ /∈ co A rewrite the function as

(x∗∗ − x) · x∗ = (x∗∗ − p(x∗∗)) · x∗ + (p(x∗∗)− x) · x∗.

Since p(x∗∗) is the minimizing point of the function |x−x∗∗|2 over the closed convexset co A it is completely characterized by the variational inequality

∀x ∈ co A, (p(x∗∗)− x∗∗) · (x− p(x∗∗)) ≥ 0. (8.4)

By choosing the following special x∗,

x∗ =x∗∗ − p(x∗∗)|x∗∗ − p(x∗∗)| ,

we get

d∗∗A (x∗∗) = sup

|x∗|≤1inf

x∈co A(x∗∗ − x) · x∗

≥ |x∗∗ − p(x∗∗)| + (p(x∗∗)− x) · x∗∗ − p(x∗∗)|x∗∗ − p(x∗∗)|

≥ |x∗∗ − p(x∗∗)| = dco A(x∗∗)

in view of (8.4). Therefore

sup|x∗|≤1

infx∈co A

(x∗∗ − x) · x∗ = dco A(x∗∗) = infx∈co A

sup|x∗|≤1

(x∗∗ − x) · x∗.

Finally from the previous theorem dco A is a convex function that coincides with theconvex envelope since it is the bidual of dA.

8.4 Families of Convex Sets Cd(D), Ccd(D), Cc

d(E; D), andCc

d,loc(E; D)

Theorem 8.6. Let D be a nonempty open subset of RN. The subfamily

Cd(D) def= dA ∈ Cd(D) : A = ∅ convex (8.5)


of Cd(D) is closed in Cloc(D). It is compact in C(D) when D is bounded. Thesubfamily

Ccd(D) def= d!Ω ∈ Cc

d(D) : Ω open and convex (8.6)

and for E open, ∅ = E ⊂ D, the subfamilies

Ccd(E;D) def= d!Ω : E ⊂ Ω open and convex ⊂ D , (8.7)

Ccd,loc(E;D) def=

d!Ω :

∃x ∈ RN, ∃A ∈ O(N) such thatx + AE ⊂ Ω open and convex ⊂ D

(8.8)

of Ccd(D) are closed in Cloc(D). If D is bounded, they are compact in C(D).

Proof. The set of all convex functions in Cloc(D) is a closed convex cone with vertexat 0. Hence its intersection with Cd(D) is closed. Any Cauchy sequence in Cd(D)converges to some convex dA. From part (i) A is convex. But dA = dA and thenonempty convex set A can be chosen as the limit set. For the complementarydistance function, it is sufficient to prove the compactness for D bounded. Givenany sequence d!Ωn

, there exist an open subset Ω of D and a subsequence, stillindexed by n, such that d!Ωn

→ d!Ω in C(D). If Ω is empty, there is nothing toprove. If Ω is not empty, consider two points x and y in Ω and λ ∈ [0, 1]. Thereexists r > 0 such that

B(x, r) ⊂ Ω and B(y, r) ⊂ Ω ⇒ d!Ω(x) ≥ r and d!Ω(y) ≥ r.

There exists N > 0 such that for all n > N , ∥d!Ωn− d!Ω∥C(D) < r/2 and

d!Ωn(x) ≥ d!Ω(x)− r/2 = r/2⇒ B(x, r/2) ⊂ Ωn,

d!Ωn(y) ≥ d!Ω(y)− r/2 = r/2⇒ B(y, r/2) ⊂ Ωn.

By convexity of Ωn, xλ = λx + (1− λ)y ∈ Ωn,

B(xλ, r/2) ⊂ λB(x, r/2) + (1− λ)B(y, r/2) ⊂ Ωn

⇒ ∀n > N, d!Ωn(xλ) ≥ r/2 ⇒ d!Ω(xλ) ≥ r/2 ⇒ B(xλ, r/2) ⊂ Ω.

Therefore xλ ∈ Ω, Ω is convex, and Ccd(D) is compact. Finally, Cc

d(E;D) =Cc

d(E;D)∩ Ccd(D) and Cc

d,loc(E;D) = Ccd,loc(E;D)∩ Cc

d(D) are closed since Ccd(D) is

closed and Ccd(E;D) and Cc

d,loc(E;D) are closed by Theorem 2.6.

9 Compactness Theorems for Sets of BoundedCurvature

We have seen two compactness conditions in Theorems 6.6 and 6.7 of section 6.3 forsets of positive reach from a condition in a tubular neighborhood. In this section wegive compactness theorems for families of sets of global and local bounded curvature.

9. Compactness Theorems for Sets of Bounded Curvature 325

For the family of sets with bounded curvature, the key result is the compact-ness of the embeddings

BCd(D) =dA ∈ Cd(D) : ∇dA ∈ BV(D)N

→W 1,p(D), (9.1)

BCcd(D) =

d!Ω ∈ Cc

d(D) : ∇d!Ω ∈ BV(D)N→W 1,p(D) (9.2)

for bounded open Lipschitzian subsets D of RN and p, 1 ≤ p < ∞. It is theanalogue of the compactness Theorem 6.3 of Chapter 5 for Caccioppoli sets

BX(D) = χ ∈ X(D) : χ ∈ BV(D)→ Lp(D), (9.3)

which is a consequence of the compactness of the embedding

BV(D)→ L1(D) (9.4)

for bounded open Lipschitzian subsets D of RN (cf. C. B. Morrey, Jr. [1,Def. 3.4.1, p. 72, Thm. 3.4.4, p. 75] and L. C. Evans and R. F. Gariepy [1,Thm. 4, p. 176]).

As for characteristic functions in Chapter 5, we give a first version involvingglobal conditions on a fixed bounded open Lipschitzian holdall D. In the secondversion, the sets are contained in a bounded open holdall D with local conditionsin their tubular neighborhood or the tubular neighborhood of their boundary.

9.1 Global Conditions in D

Theorem 9.1. Let D be a nonempty bounded open Lipschitzian holdall in RN. Theembedding (9.1) is compact. Thus for any sequence An, ∅ = An, of subsets of Dsuch that

∃c > 0,∀n ≥ 1, ∥D2dAn∥M1(D) ≤ c, (9.5)

there exist a subsequence Ank and A = ∅, such that ∇dA ∈ BV(D)N and

dAnk→ dA in W 1,p(D)-strong

for all p, 1 ≤ p <∞. Moreover, for all ϕ ∈ D0(D),

limn→∞

⟨∂ijdAnk,ϕ⟩ = ⟨∂ijdA,ϕ⟩, 1 ≤ i, j ≤ N, and ∥D2dA∥M1(D) ≤ c. (9.6)

Proof. Given c > 0 consider the set

Scdef=dA ∈ Cd(D) : ∥D2dA∥M1(D) ≤ c

.

By compactness of the embedding (9.4), given any sequence dAn, there exist asubsequence, still denoted by dAn, and f ∈ BV(D)N such that ∇dAn → f inL1(D)N . But by Theorem 2.2 (ii), Cd(D) is compact in C(D) for bounded D andthere exist another subsequence dAnk

and dA ∈ Cd(D) such that dAnk→ dA in

C(D) and, a fortiori, in L1(D). Therefore, dAnkconverges in W 1,1(D) and also in


L1(D). By uniqueness of the limit, f = ∇dA and dAnkconverges in W 1,1(D) to

dA. For Φ ∈ D1(D)N×N as k goes to infinity∫

D∇dAnk

·−→div Φ dx→

∫

D∇dA ·

−→div Φ dx

⇒∣∣∣∣∫

D∇dA ·

−→div Φ dx

∣∣∣∣ = limk→∞

∣∣∣∣∫

D∇dAnk

·−→div Φ dx

∣∣∣∣ ≤ c∥Φ∥C(D),

∥D2dA∥M1(D) ≤ c, and ∇dA ∈ BV(D)N . This proves the compactness of theembedding for p = 1 and properties (9.6). The conclusions remain true for p ≥ 1by the equivalence of the W 1,p-topologies on Cd(D) in Theorem 4.2 (i).

When D is bounded open, Ccd(D) is compact in C(D) and closed in W 1,p

0 (D),1 ≤ p <∞, and we have the analogue of the previous compactness theorem.

Theorem 9.2. Let D, ∅ = D ⊂ RN, be bounded open Lipschitzian. The embedding(9.2) is compact. Thus for any sequence Ωn of open subsets of D such that

∃h > 0, ∃c > 0, ∀n, ∥D2d!Ωn∥M1(D) ≤ c, (9.7)

there exist a subsequence Ωnk and an open subset Ω of D such that ∇d!Ω ∈BV(D)N and

d!Ωnk→ d!Ω in W 1,p

0 (D) (9.8)

for all p, 1 ≤ p <∞. Moreover, for all ϕ ∈ D0(D)N×N ,

⟨D2d!Ωn,ϕ⟩ → ⟨D2d!Ω,ϕ⟩, ∥D2d!Ω∥M1(D) ≤ c, and χΩ ∈ BV(D). (9.9)

Proof. Given c > 0 consider the set

Scc

def=d!Ω ∈ Cc

d(D) : ∥D2d!Ω∥M1(D) ≤ c

.

By compactness of the embedding (9.4), given any sequence d!Ωn there exist a

subsequence, still denoted by d!Ωn, and f ∈ BV(D)N such that ∇d!Ωn

→ f inL1(D)N . But by Theorem 2.4 (ii), Cc

d(D) is compact in C0(D) for bounded D andthere exist another subsequence d!Ωnk

and d!Ω ∈ Ccd(D) such that d!Ωnk

→ d!Ω

in C0(D) and, a fortiori, in L1(D). Therefore, d!Ωnkconverges in W 1,1

0 (D) and alsoin L1(D). By uniqueness of the limit, f = ∇d!Ω and d!Ωnk

converges in W 1,10 (D)

to d!Ω. For all Φ ∈ D1(D)N×N as k goes to infinity∫

D∇d!Ωnk

·−→divΦ dx→

∫

D∇d!Ω ·

−→divΦ dx

⇒∣∣∣∣∫

D∇d!Ω ·

−→divΦ dx

∣∣∣∣ = limk→∞

∣∣∣∣∫

D∇d!Ωnk

·−→divΦ dx

∣∣∣∣ ≤ c∥Φ∥C(D),

∥D2d!Ω∥M1(D) ≤ c, ∇d!Ω ∈ BV(D)N , and χΩ ∈ BV(D). This proves the compact-ness of the embedding for p = 1 and properties (9.9). The conclusions remain truefor p ≥ 1 by the equivalence of the W 1,p-topologies on Cc

d(D) in Theorem 4.2 (i).


9.2 Local Conditions in Tubular NeighborhoodsThe global conditions (9.5) and (9.7) can be weakened to local ones in a neigh-borhood of each set of the sequence, and the Lipschitzian condition on D can beremoved since only the uniform boundedness of the sets of the sequence is required.

Theorem 9.3. Let D, ∅ = D ⊂ RN, be a bounded open holdall and An, ∅ = An,be a sequence of subsets of D. Assume that there exist h > 0 and c > 0 such that

∀n, ∥D2dAn∥M1(Uh(∂An)) ≤ c. (9.10)

Then there exist a subsequence Ank and a subset A, ∅ = A, of D such that∇dA ∈ BVloc(RN)N , and for all p, 1 ≤ p <∞,

dAnk→ dA in W 1,p(Uh(D))-strong. (9.11)

Moreover, for all ϕ ∈ D0(Uh(A))

limk→∞

⟨∂ijdAnk,ϕ⟩ = ⟨∂ijdA,ϕ⟩, 1 ≤ i, j ≤ N, ∥D2dA∥M1(Uh(A)) ≤ c, (9.12)

and χA belongs to BVloc(RN).

Proof. First notice that, since An is bounded and nonempty, ∂An = ∅. FurtherUh(∂An) can be replaced by Uh(An) in condition (9.10) since dAn = 0 in An. So itis sufficient to prove the theorem for that case.

Lemma 9.1. For any A, ∂A = ∅ and h > 0,

∥D2dA∥M1(Uh(A)) = ∥D2dA∥M1(Uh(∂A)).

The proof of the lemma will be given after the proof of the theorem.(i) The assumption An ⊂ D implies Uh(An) ⊂ Uh(D). Since Uh(D) is

bounded, there exist a subsequence, still indexed by n, and a set A, ∅ = A ⊂ D,such that dAn → dA in C(Uh(D)) and another subsequence, still denoted by dAn,such that

dAn → dA in H1(Uh(D))-weak.

For all ε > 0, 0 < 3ε < h, there exists N > 0 such that for all n ≥ N and x inUh(D)

dAn(x) ≤ dA(x) + ε, dA(x) ≤ dAn(x) + ε.

Therefore,

An ⊂ Uh−2 ε(An) ⊂ Uh−ε(A) ⊂ Uh(An), (9.13)

!Uh−ε(A) ⊂ !Uh−2 ε(An) ⊂ !An. (9.14)

From (9.10) and (9.13)

∀n ≥ N, ∥D2dAn∥M1(Uh−ε(A)) ≤ c.


In order to use the compactness of the embedding (9.4) as in the proof of The-orem 9.1, we would need Uh−ε(A) to be Lipschitzian. To get around this, weconstruct a bounded Lipschitzian set between Uh−2ε(A) and Uh−ε(A). Indeed, bydefinition,

Uh−ε(A) = ∪x∈AB(x, h− ε) and Uh−2ε(A) ⊂ Uh−ε(A),

and by compactness, there exists a finite sequence of points xini=1 in A such that

|xi − xj | < (h− ε)/2 for all i = j so that no two balls are tangent, and

Uh−2ε(A) ⊂ UBdef= ∪n

i=1B(xi, h− ε) ⊂ Uh(D).

Since UB is Lipschitzian as the union of a finite number of nontangent balls, itnow follows by compactness of the embedding (9.4) for UB that there exist a sub-sequence, still denoted by dAn, and f ∈ BV(UB)N such that ∇dAn → f inL1(UB)N . Since Uh(D) is bounded, Cd(Uh(D)) is compact in C(Uh(D)) and thereexist another subsequence, still denoted by dAn, and ∅ = A ⊂ D such thatdAn → dA in C(Uh(D)) and, a fortiori, in L1(Uh(D)). Therefore, dAn converges inW 1,1(UB) and also in L1(UB). By uniqueness of the limit, f = ∇dA on UB anddAn converges to dA in W 1,1(UB). By Definition 3.3 and Theorem 5.1, ∇dA and∇dAn all belong to BVloc(RN)N since they are BV in tubular neighborhoods oftheir respective boundaries. Moreover, by Theorem 5.2 (ii), χA ∈ BVloc(RN). Theabove conclusions also hold for the subset Uh−2ε(A) of UB .

(ii) Convergence in W 1,p(Uh(D)). Consider the integral∫

Uh(D)|∇dAn −∇dA|2 dx

=∫

Uh−2ε(A)|∇dAn −∇dA|2 dx +

∫

Uh(D)\Uh−2ε(A)|∇dAn −∇dA|2 dx.

From part (i) the first integral on the right-hand side converges to zero as n goesto infinity. The second integral is on a subset of !Uh−2ε(A). From (9.14) for alln ≥ N ,

|∇dAn(x)| = 1 a.e. in !An ⊃ !Uh−3ε(An) ⊃ !Uh−2ε(A),

|∇dA(x)| = 1 a.e. in !A ⊃ !Uh−2ε(A).

The second integral reduces to∫

Uh(D)\Uh−2ε(A)|∇dAn −∇dA|2 dx =

∫

Uh(D)\Uh−2ε(A)2 (1−∇dAn ·∇dA) dx,

which converges to zero since ∇dAn ∇dA in L2(Uh(D))N -weak in part (i) andthe fact that |∇dA| = 1 almost everywhere in Uh(D)\Uh−2ε(A). Therefore, sincedAn → dA in C(Uh(D)),

dAn → dA in H1(Uh(D))-strong,

and by Theorem 4.2 (i) the convergence is true in W 1,p(Uh(D)) for all p ≥ 1.


(iii) Properties (9.12). Consider the initial subsequence dAn which convergesto dA in H1(Uh(D))-weak constructed at the beginning of part (i). This sequenceis independent of ε and the subsequent constructions of other subsequences. Byconvergence of dAn to dA in H1(Uh(D))-weak for each Φ ∈ D1(Uh(A))N×N ,

limn→∞

∫

Uh(A)∇dAn ·

−→div Φ dx =

∫

Uh(A)∇dA ·

−→div Φ dx.

Each such Φ has compact support in Uh(A), and there exists ε = ε(Φ) > 0, 0 <3ε < h, such that

supp Φ ⊂ Uh−2ε(A).

From part (ii) there exists N(ε) > 0 such that

∀n ≥ N(ε), Uh−2 ε(An) ⊂ Uh−ε(A) ⊂ Uh(An).

For n ≥ N(ε) consider the integral∫

Uh(A)∇dAn ·

−→div Φ dx =

∫

Uh−2ε(A)∇dAn ·

−→div Φ dx =

∫

Uh(An)∇dAn ·

−→div Φ dx

⇒∣∣∣∣∫

Uh(A)∇dAn ·

−→div Φ dx

∣∣∣∣ ≤ ∥D2dAn∥M1(Uh(An)) ∥Φ∥C(Uh(An))

≤ c ∥Φ∥C(Uh−2ε(A)) = c ∥Φ∥C(Uh(A)).

By convergence of ∇dAn to ∇dA in L2(D∪Uh(A))-weak, for all Φ ∈ D1(Uh(A))N×N

∣∣∣∣∫

Uh(A)∇dA ·

−→div Φ dx

∣∣∣∣ ≤ c ∥Φ∥C(Uh(A)) ⇒ ∥D2dA∥M1(Uh(A)) ≤ c.

Finally, the convergence remains true for all subsequences constructed in parts (i)and (ii). This completes the proof.

Proof of Lemma 9.1. First check that

Uh(A) = Uh(∂A)∪x∈A−hB(x, h), A−hdef=x ∈ RN : B(x, h) ⊂ A

.

It is sufficient to show that all points x of intA are contained in the right-handside of the above expression. If d∂A(x) < h, then x ∈ Uh(∂A); if d∂A(x) ≥ h, thenB(x, h) ⊂ A and x ∈ A−h. For all Φ ∈ D1(Uh(A))N×N , Φ has compact supportand there exists ε > 0, 0 < 2ε < h, such that

supp Φ ⊂ Uh−2ε(A).

But Uh−ε(A) ⊂ Uh(A) and there exists xj ∈ A−h : 1 ≤ j ≤ m such that

Uh−ε(A) ⊂ Uh(∂A)∪mj=1B(xj , h).

Let ψ0 ∈ D(Uh(∂A)) and let ψj ∈ D(B(xj , h)) be a partition of unity such that

0 ≤ ψj ≤ 1,m∑

j=0

ψj = 1 on Uh−2ε(A).


Consider the integral∫

Uh(A)∇dA ·

−→div Φ dx =

∫

Uh−2ε(A)∇dA ·

−→div Φ dx =

∫

Uh−2ε(A)∇dA ·

−→div

⎛

⎝m∑

j=0

ψjΦ

⎞

⎠ dx

=∫

Uh−2ε(A)∇dA ·

−→div(ψ0Φ

)dx +

m∑

j=1

∫

Uh−2ε(A)∇dA ·

−→div(ψjΦ

)dx.

By construction for ∥Φ∥C(Uh(A)) ≤ 1

ψ0Φ ∈ D1(Uh−2ε(A) ∩ Uh(∂A))N×N , ∥ψ0Φ∥C(Uh(∂A)) ≤ 1,

ψjΦ ∈ D1(Uh−2ε(A) ∩B(xj , h))N×N , ∥ψjΦ∥C(B(xj ,h)) ≤ 1, 1 ≤ j ≤ m.

But for j, 1 ≤ j ≤ m, B(xj , h) ⊂ intA, where both dA and ∇dA are identicallyzero. As a result the above integral reduces to∫

Uh(A)∇dA ·

−→div Φ dx =

∫

Uh−2ε(A)∇dA ·

−→div (ψ0Φ) dx =

∫

Uh(∂A)∇dA·

−→div (ψ0Φ) dx

⇒∣∣∣∣∫

Uh(A)∇dA ·

−→div Φ dx

∣∣∣∣ ≤ ∥D2dA∥M1(Uh(∂A)) ∥ψ0Φ∥C(Uh(∂A))

⇒ ∥D2dA∥M1(Uh(A)) ≤ ∥D2dA∥M1(Uh(∂A)),

and this completes the proof.

Theorem 9.4. Let D, ∅ = D ⊂ RN, be a bounded open holdall. Let Ωn,∅ = Ωn ⊂ D, be a sequence of open subsets and assume that

∃h > 0 and ∃c > 0 such that ∀n, ∥D2d!Ωn∥M1(Uh(∂Ωn)) ≤ c. (9.15)

Then there exist a subsequence Ωnk and an open subset Ω of D such that ∇d!Ω ∈BVloc(RN)N and for all p, 1 ≤ p <∞,

d!Ωnk→ d!Ω in W 1,p

0 (D)-strong. (9.16)

Moreover for all ϕ ∈ D0(Uh(!Ω))

limk→∞

⟨∂ijd!Ωnk,ϕ⟩ = ⟨∂ijd!Ω,ϕ⟩, 1 ≤ i, j ≤ N, ∥D2d!Ω∥M1(Uh(!Ω)) ≤ c, (9.17)

and χΩ belongs to BVloc(RN).

Proof. First note that ∂Ωn = ∅ since Ωn is bounded and nonempty. By Lemma 9.1,Uh(∂Ωn) can be replaced by Uh(!Ωn) in condition (9.15). Moreover since !Ωn canbe unbounded, we shall work with the bounded neighborhoods Uh(!DΩn) of !DΩn

rather than with Uh(!Ωn).

Lemma 9.2. Let h > 0 and let ∅ = Ω ⊂ D ⊂ RN be bounded open. Then

∥D2d!Ω∥M1(Uh(!D

Ω)) = ∥D2d!Ω∥M1(Uh(!Ω)).

If ∂Ω = ∅,

∥D2d!Ω∥M1(Uh(∂Ω)) = ∥D2d!Ω∥M1(Uh(!Ω)).


The proof of this lemma will be given after the proof of the theorem.(i) By Theorem 2.4 (ii) since D is bounded, there exist a subsequence, still

denoted by d!Ωn, and an open subset Ω of D such that

d!Ωn→ d!Ω in C0(D) and Cc(D ∪ Uh(!Ω))

and another subsequence, still denoted by d!Ωn, such that

d!Ωn→ d!Ω in H1

0 (D)-weak and H10 (D ∪ Uh(!Ω))-weak.

For all ε > 0, 0 < 3ε < h, there exists N > 0 such that for all n ≥ N

d!Ωn(x) ≤ d!Ω(x) + ε, d!Ω(x) ≤ d!Ωn

(x) + ε.

Clearly, since Ω ⊂ D and Ωn ⊂ D and D is open, !DΩn = ∅ and !DΩ = ∅.Furthermore,

!DΩn ⊂ Uh−2 ε(!DΩn) ⊂ Uh−ε(!DΩ) ⊂ Uh(!DΩn), (9.18)!DUh−ε(!DΩ) ⊂ !DUh−2 ε(!DΩn) ⊂ Ωn. (9.19)

From (9.15) and (9.18)

∀n ≥ N, ∥D2d!Ωn∥M1(Uh−ε(!

DΩ)) ≤ c. (9.20)

As in part (i) of the proof of Theorem 9.3, we can construct a bounded openLipschitzian set UB between Uh−2ε(!DΩ) and Uh−ε(!DΩ) such that

Uh−2ε(!DΩ) ⊂ UB ⊂ Uh−ε(!DΩ).

Since UB is bounded and Lipschitzian, it now follows by compactness of the em-bedding (9.4) for UB that there exist a subsequence, still denoted by d!Ωn

, andf ∈ BV(UB)N such that ∇d!Ωn

→ f in L1(UB)N . Since D ∪ UB is bounded,Cc

d(D ∪ UB) is compact in C0(D ∪ UB), and there exist another subsequence, stilldenoted by d!Ωn

, and an open subset Ω of D such that d!Ωn→ d!Ω in C0(D ∪ UB)

and, a fortiori, in L1(D ∪ UB). Therefore d!Ωnconverges in W 1,1(UB) and also in

L1(UB). By uniqueness of the limit, f = ∇d!Ω on BD and d!Ωnconverges to d!Ω

in W 1,1(UB). By Definition 3.3 and Theorem 5.1, ∇d!Ω and ∇d!Ωnall belong to

BVloc(RN)N since they are BV in tubular neighborhoods Uh(!DΩ) and Uh(!DΩn)of their respective boundaries ∂Ω and ∂Ωn. Moreover, by Theorem 5.2 (ii), χ!Ω ∈BVloc(RN) and, a fortiori, χΩ since Ω is open. The above conclusions also hold forthe subset Uh−2ε(!DΩ) of UB .

(ii) Convergence in W 1,p0 (D). Consider the integral

∫

D|∇d!Ωn

−∇d!Ω|2 dx

=∫

D∩Uh−2ε(!D

Ω)|∇d!Ωn

−∇d!Ω|2 dx +∫

D\Uh−2ε(!D

Ω)|∇d!Ωn

−∇d!Ω|2 dx.


From part (i) the first integral on the right-hand side converges to zero as n goes toinfinity. The second integral is on a subset of !DUh−2ε(!DΩ). From the relations(9.19) for all n ≥ N ,

|∇d!Ωn(x)| = 1 a.e. in Ωn ⊃ !DUh−3ε(!DΩn) ⊃ !DUh−2ε(!DΩ),

|∇dΩ(x)| = 1 a.e. in Ω ⊃ !DUh−2ε(!DΩ).


D\Uh−2ε(!D

Ω)|∇d!Ωn

−∇d!Ω|2 dx =∫

D\Uh−2ε(!D

Ω)2 (1−∇d!Ωn

·∇d!Ω) dx,

which converges to zero by weak convergence of ∇d!Ωnto ∇d!Ω in L2(D)N and

the fact that |∇d!Ω| = 1 almost everywhere in D\Uh−2ε(!DΩ). Therefore, sinced!Ωn

→ d!Ω in C0(D)d!Ωn

→ d!Ω in H10 (D)-strong,

and by Theorem 4.2 (i) the convergence is true in W 1,p0 (D) for all p ≥ 1.

(iii) Properties (9.17). Consider the initial subsequence d!Ωn, which con-

verges to d!Ω in H10 (D ∪ Uh(!Ω))-weak constructed at the beginning of part (i).

This sequence is independent of ε and the subsequent constructions of other sub-sequences. By convergence of d!Ωn

to d!Ω in H10 (D ∪ Uh(!Ω))-weak for each

Φ ∈ D1(Uh(!Ω))N×N ,

limn→∞

∫

Uh(!Ω)∇d!Ωn

·−→div Φ dx =

∫

Uh(!Ω)∇d!Ω ·

−→div Φ dx.

Now each Φ ∈ D1(Uh(!Ω))N×N has compact support in Uh(!Ω) and there existsε = ε(Φ) > 0, 0 < 3ε < h, such that

supp Φ ⊂ Uh−2ε(!Ω).


∀n ≥ N(ε), Uh−2 ε(!Ωn) ⊂ Uh−ε(!Ω) ⊂ Uh(!Ωn).


Uh(!Ω)∇d!Ωn

·−→div Φ dx =

∫

Uh−2ε(!Ω)∇d!Ωn

·−→div Φ dx =

∫

Uh(!Ωn)∇dΩn ·

−→div Φ dx

⇒∣∣∣∣∫

Uh(!Ω)∇d!Ωn

·−→div Φ dx

∣∣∣∣ ≤ ∥D2d!Ωn

∥M1(Uh(!DΩn)) ∥Φ∥C(Uh(!DΩn))

≤ c ∥Φ∥C(Uh−2ε(!DΩ)) = c ∥Φ∥C(Uh(!Ω)).

By convergence of∇d!Ωnto∇d!Ω in L2(D∪Uh(!Ω))-weak, for all Φ∈D1(Uh(!Ω))N×N

∣∣∣∣∫

Uh(!Ω)∇d!Ω ·

−→div Φ dx

∣∣∣∣ ≤ c ∥Φ∥C(Uh(!Ω)) ⇒ ∥D2d!Ω∥M1(Uh(!Ω)) ≤ c.

Finally the convergence remains true for all subsequences constructed in parts (i)and (ii). This completes the proof.


Proof of Lemma 9.2. The proof is similar to that of Lemma 9.1 with

Uh(!Ω) = Uh(!DΩ) ∪ int !D = Uh(!DΩ)∪x∈(!D)−hB(x, h),

(!D)−hdef=x ∈ RN : B(x, h) ⊂ !D

,

since both d!Ω and ∇d!Ω are identically zero on !D.

Chapter 7

Metrics via OrientedDistance Functions

1 IntroductionIn this chapter we study oriented distance functions and their role in the descriptionof the geometric properties and smoothness of domains and their boundary. Theyconstitute a special family of the so-called algebraic or signed distance functions.Signed distance functions make sense only when some appropriate orientation ofthe underlying set is available to assign a sign, while oriented distance functionsare always well-defined. Our choice of terminology emphasizes the fact that, fora smooth open domain, the associated oriented distance function specifies the ori-entation of the normal to the boundary of the set. They enjoy many interestingproperties. For instance, they retain the nice properties of the distance functionsbut also generate the classical geometric properties associated with sets and theirboundary. The smoothness of the oriented distance function in a neighborhood ofthe boundary of the set is equivalent to the local smoothness of its boundary. Simi-larly, the convexity of the function is equivalent to the convexity of the closure of theset. In addition, their respective gradient and Hessian matrix respectively coincidewith the unit outward normal and the second fundamental form on the boundaryof the set. Finally, they provide a framework for the classification of domains andsets according to their degree of smoothness, much like Sobolev spaces and spacesof continuous and Holderian functions do for functions.

The first part of the chapter deals with the basic definitions and construc-tions and the main results. The second part specializes to specific subfamilies oforiented distance functions. The last part concentrates on compact families of sub-sets of oriented distance functions. In the first part, section 2 presents the basicproperties, introduces the uniform metric topology, and shows its connection withthe Hausdorff and complementary Hausdorff topologies of Chapter 6. Section 3 isdevoted to the differentiability properties and the associated set of projections ontothe boundary, and it completes the treatment of skeletons and cracks. Section 8gives the equivalence of the smoothness of a set and the smoothness of its orienteddistance function in a neighborhood of its boundary for sets of class C1,1 or better.

335

336 Chapter 7. Metrics via Oriented Distance Functions

When the domain is sufficiently smooth the trace of the Hessian matrix of second-order partial derivatives on the boundary is the classical second fundamental formof geometry. Section 4 deals with the W 1,p-topology on the set of oriented distancefunctions and the closed subfamily of sets for which the volume of the boundaryis zero.

In the second part of the chapter, section 5 studies the subfamily of sets forwhich the gradient of the oriented distance function is a vector of functions ofbounded variation: the sets with global or local bounded curvature. There are largeclasses for which general compactness theorems will be proved in section 11. Exam-ples are given to illustrate the behavior of the norms in tubular neighborhoods asthe thickness of the neighborhood goes to zero in section 5.1. Section 9 introducesSobolev or W s,p-domains which provide a framework for the classification of setsaccording to their degree of smoothness. For smooth sets this classification inter-twines with the classical classification of Ck- and Holderian domains. Section 10extends the characterization of closed convex sets by the distance function to theoriented distance function and introduces the notion of semiconvex sets. The set ofequivalence classes of convex subsets of a compact holdall D is again compact, asit was for all the topologies considered in Chapters 5 and 6. Section 7 shows thatsets of positive reach introduced in Chapter 6 are sets of locally bounded curvatureand that the boundary of their closure has zero volume. A compactness theoremis also given. Section 12 introduces the notion of γ-density perimeter, which is arelaxation of the (N − 1)-dimensional upper Minkowski content, and a compactnesstheorem for a family of subsets of a bounded holdall that have a uniformly boundedγ-density perimeter.

In the last part of the chapter, section 11 gives compactness theorems for sets ofglobal and local bounded curvatures from a uniform bound in tubular neighborhoodsof their boundary. They are the analogues of the theorems of Chapter 6. Section 13gives the compactness theorem for the family of subsets of a bounded holdall thathave the uniform fat segment property. Equivalent conditions are given on the localgraph functions of the family in section 13.2. The results are specialized to thecompactness under the uniform cusp/cone property. In particular we recover thecompactness of Theorem 6.11 of section 6.4 in Chapter 5 for the associated familyof characteristic functions in Lp(D). Section 14 combines the compactness underthe uniform fat segment property with a bound on the De Giorgi perimeter ofCaccioppoli sets in section 14.1 and the γ-density perimeter in section 14.2.

Section 15 introduces the families of cracked sets and uses them in section 16to provide an original solution to a variation of the image segmentation problemof D. Mumford and J. Shah [2]. Cracked sets are more general than sets whichare locally the epigraph of a continuous function in the sense that they includedomains with cracks, and sets that can be made up of components of different co-dimensions. The Hausdorff (N − 1) measure of their boundary is not necessarilyfinite. Yet compact families (in the W 1,p-topology) of such sets can be constructed.

2. Uniform Metric Topology 337

2 Uniform Metric Topology2.1 The Family of Oriented Distance Functions Cb(D)The distance function dA provides a good description of a domain A from the“outside.” However, its gradient undergoes a jump discontinuity at the boundaryof A which prevents the extraction of information about the smoothness of A fromthe smoothness of dA in a neighborhood of ∂A. To get around this, it is natural totake into account the negative of the distance function of its complement !A thatcancels the jump discontinuity at the boundary of smooth sets.

Definition 2.1.Given A ⊂ RN, the oriented distance function from x ∈ RN to A is defined as

bA(x) def= dA(x)− d!A(x). (2.1)

This function gives a level set description of a set whose boundary coincideswith the zero level set. From Chapter 6 the function bA is finite in RN if and onlyif ∅ = A = RN, since bA = dA = +∞ when A = ∅ and bA = −d!A = −∞ when!A = ∅. This condition is completely equivalent to ∂A = ∅, in which case bA

coincides with the algebraic distance function to the boundary of A:

bA(x) =

⎧⎪⎨

⎪⎩

dA(x) = d∂A(x), x ∈ int !A,

0, x ∈ ∂A,

−d!A(x) = −d∂A(x), x ∈ intA.

(2.2)

Noting that b!A = −bA, it means that we have implicitly chosen the negative signfor the interior of A and the positive sign for the interior of its complement. As bA

is an increasing function from its interior to its exterior, we shall see later that forsets with a smooth boundary, the restriction of the gradient of bA to ∂A coincideswith the outward unit normal to ∂A. Changing the sign of bA gives the inwardorientation to the gradient and the normal.

Associate with a nonempty subset D of RN the family

Cb(D) def=bA : A ⊂ D and ∂A = ∅

. (2.3)

Theorem 2.1. Let A be a subset of RN. Then the following hold:

(i) A = ∅ and !A = ∅ ⇐⇒ ∂A = ∅.

(ii) Given bA and bB in Cb(D),

A ⊃ B ⇒ bA ≤ bB and A = B ⇒ bA = bB ,

bA ≤ bB in D ⇐⇒ B ⊂ A and !A ⊂ !B,

bA = bB in D ⇐⇒ B = A and !A = !B ⇐⇒ B = A and ∂A = ∂B.


In particular, bint A ≤ bA ≤ bA and

bA = bA ⇐⇒ ∂A = ∂A and bint A = bA ⇐⇒ ∂ intA = ∂A. (2.4)

(iii) |bA| = dA + d!A = maxdA, d!A = d∂A and ∂A = x ∈ RN : bA(x) = 0.

(iv) bA ≥ 0 ⇐⇒ !A ⊃ ∂A ⊃ A ⇐⇒ ∂A = A.

(v) bA = 0 ⇐⇒ !A = ∂A = A ⇐⇒ ∂A = RN.

(vi) If ∂A = ∅, the function bA is uniformly Lipschitz continuous in RN and

∀x, y ∈ RN, |bA(y)− bA(x)| ≤ |y − x|. (2.5)

Moreover, bA is (Frechet) differentiable almost everywhere and

|∇bA(x)| ≤ 1 a.e. in RN . (2.6)

Proof. (i) The proof is obvious.(ii) By assumption

dA = b+A ≤ b+

B = dB and d!A = b−A ≥ b−

B = d!B in D.

Since A ⊂ D and B ⊂ D, then !A ⊃ !D and !B ⊃ !D and

d!A = d!B = 0 in !D ⇒ d!A ≥ d!B in RN ⇒ !A ⊂ !B.

Also since B ⊂ D for all x ∈ B, dA(x) ≤ dB(x) = 0 and x ∈ A. Therefore, !A ⊂ !Band B ⊂ A. Conversely,

B ⊂ A ⇒ dA ≤ dB in RN and !A ⊂ !B ⇒ d!B ≤ d!A in RN

and, a fortiori, in D. The equality case follows from the fact that bA = bB if and onlyif bA ≥ bB and bA ≤ bB . By definition intA = !!A ⊂ A ⊂ A and bint A ≤ bA ≤ bA.

(iii) For x in A

bA(x) = −d!A(x) =⇒ |bA(x)| = d!A(x) ≤ d∂A(x)

since !A ⊃ ∂A and the inf over !A is smaller than the inf over its subset ∂A.Similarly, for x in !A

bA(x) = dA(x) =⇒ |bA(x)| = dA(x) ≤ d∂A(x),

and finally|bA(x)| ≤ maxd!A(x), dA(x) ≤ d∂A(x).

Conversely, for each x in !A, the set of projections ΠA(x) ⊂ A ∩ !A = ∂A is notempty. Hence

|bA(x)| = dA(x) = miny∈ΠA(x)

|x− y| ≥ infy∈∂A

|x− y| = d∂A(x),


and similarly for all x in A,

|bA(x)| = d!A(x) ≥ d∂A(x).

Therefore |bA(x)| ≥ d∂A(x).(iv) If bA = dA−d!A ≥ 0, then dA ≥ d!A and A ⊂ !A, and necessarily A ⊂ ∂A

and A = ∂A. Conversely, if x ∈ ∂A, then by definition bA(x) = 0. If x /∈ ∂A, thenx ∈ !∂A = !A = int !A and bA(x) = dA(x) ≥ 0.

(v) bA = 0 is equivalent to bA ≥ 0 and b!A = −bA ≥ 0. Then we apply (v)twice. But !A = ∂A = A ⇐⇒ int !A = ∅ = int A ⇐⇒ ∂A = RN.

(vi) Clearly,

∀x, y ∈ A, |bA(y)− bA(x)| = |d!A(y)− d!A(x)| ≤ |y − x|,

∀x, y ∈ !A, |bA(y)− bA(x)| = |dA(y)− dA(x)| ≤ |y − x|.

For x ∈ A and y ∈ int !A = !A, dA(y) > 0 and

bA(y)− bA(x) = dA(y) + d!A(x) > 0 ⇒ |bA(y)− bA(x)| = dA(y) + d!A(x).

By assumption B(y, dA(y)) ⊂ int !A. Define the point

x = y +dA(y)|x− y| (x− y) ∈ !A

⇒ d!A(x) ≤ |x− x| =∣∣∣∣

(1− dA(y)

|x− y|

)(y − x)

∣∣∣∣ = |x− y|− dA(y)

⇒ |bA(y)− bA(x)| = dA(y) + d!A(x) ≤ |x− y|.

The argument is similar for x ∈ intA and y ∈ !A. The differentiability follows fromTheorem 2.1 (vii) in Chapter 6.

2.2 Uniform Metric Topology

From Theorem 2.1 (i) the function bA is finite at each point when ∂A = ∅. Thisexcludes A = ∅ and A = RN. The zero function bA(x) = 0 for all x in RN

corresponds to the equivalence class of sets A such that

A = ∂A = !A or ∂A = RN .

This class of sets is not empty. For instance, choose the subset of points of RN withrational coordinates or the set of all lines parallel to one of the coordinate axes withrational coordinates.

Let D be a nonempty subset of RN and associate with each subset A of D,∂A = ∅, the equivalence class

[A]bdef=B : ∀B, B ⊂ D, B = A and ∂A = ∂B


and the family of equivalence classes

Fb(D) def=[A]b : ∀A, A ⊂ D and ∂A = ∅

.

The equivalence classes induced by bA are finer than those induced by dA since boththe closures and the boundaries of the respective sets must coincide.

For Cd(D), we have an invariant closed representative A in the equivalenceclass [A]d and for Cc

d(D) an invariant open representative intA for [!A]d. For Cb(D),the deficiencies of Cd(D) and Cc

d(D) combine

dA = dA ≤ dint A and d!int A = d!A= d!A ≤ d!A,

and, in general, there is neither a closed nor an open representative, since

bA ≤ bA ≤ bint A.

One invariant in the class is ∂A, but it does not completely describes the classwithout one of the other two A or !A. We shall see that for sets verifying a uniformsegment property bA = bA = bint A and for convex sets that bA = bA.

As in the case of dA we identify Fb(D) with the family Cb(D) through theembedding

[A]b .→ bA : Fb(D)→ Cb(D) ⊂ C(D).

When D is bounded, the space C(D) endowed with the norm ∥f∥C(D) is a Banachspace. Moreover, for each A ⊂ D, bA is bounded, uniformly continuous on D, andbA ∈ C(D). This will induce the following complete metric:

ρ([A]b, [B]b)def= ∥bA − bB∥C(D) (2.7)

on Fb(D).When D is open but not necessarily bounded, we use the space Cloc(D) defined

in section 2.2 of Chapter 6 endowed with the complete metric ρδ defined in (2.9) forthe family of seminorms qK defined in (2.8). It will be shown below that Cb(D)is a closed subset of Cloc(D) and that this induces the following complete metric onFb(D):

ρδ([A]b, [B]b)def=

∞∑

k=1

12k

qKk(bA − bB)1 + qKk(bA − bB)

. (2.8)

The subfamilies

F0b (D) def= [A]b : ∀A, A ⊂ D, ∂A = ∅, m(∂A) = 0 ,

C0b (D) def= bA ∈ Cb(D) : m(∂A) = 0

of Fb(D) and Cb(D) will be important. We now have the equivalent of Theorem 2.2in Chapter 6.


Theorem 2.2. Let D = ∅ be an open (resp., bounded open) holdall in RN.

(i) The set Cb(D) is closed in Cloc(D) (resp., C(D)) and ρδ (resp., ρ) defines acomplete metric topology on Fb(D).

(ii) For a bounded open subset D of RN, the set Cb(D) is compact in C(D).

(iii) For a bounded open subset D of RN, the map

bA .→ (b+A, b−

A, |bA|) = (dA, d!A, d∂A) : Cb(D) ⊂ C(D)→ C(D)3

is continuous: for all bA and bB in Cb(D),

max∥dB − dA∥C(D), ∥d!B − d!A∥C(D), ∥d∂B − d∂A∥C(D)

≤ ∥bB − bA∥C(D).(2.9)

Proof. (i) It is sufficient to consider the case for which D is bounded. Consider asequence An ⊂ D, ∂An = ∅, such that bAn → f in C(D) for some f in C(D).Associate with each g in C(D) its positive and negative parts

g+(x) = maxg(x), 0, g−(x) = max−g(x), 0.

Then by continuity of this operation

dAn = b+An→ f+ and d!An

= b−An→ f− in C(D),

d∂An = |bAn |→ |f | in C(D).

By Theorem 2.2 (i) of Chapter 6, there exists a closed subset F , ∅ = F ⊂ D,such that

f+ = dF in RN and f+ > 0 in !D.

Moreover F = RN since D is bounded. By Theorem 2.4 and the remark at thebeginning of section 2.3 of Chapter 6, there exists an open subset G ⊂ D, G = RN,such that

f− = d!G in RN and d!G ∈ C0(D).

Therefore,

f = f+ − f− = dF − d!G,

f = f+ − f− = dF in !D and f > 0 in !D.

Define the sets

A+ def=x ∈ RN : f(x) > 0

=x ∈ D : f(x) > 0

∪ !D,

A− def=x ∈ RN : f(x) < 0

=x ∈ D : f(x) < 0

,

A0 def=x ∈ RN : f(x) = 0

=x ∈ D : f(x) = 0

.


They form a partition of RN, RN = A− ∪A0 ∪A+, and

RN = A0 ∪A− = F = ∅, A0 ∪A+ = !G = ∅ ⇒ A0 = F ∩ !G = ∅,

!D ⊂ A+ and F = A0 ∪A− ⊂ D,

f = dA0∪A− − dA0∪A+ .

If A0 were empty, RN could be partitioned into two disjoint nonempty closed sub-sets. Since A0 is closed

∂A0 = A0 ∩ !A0 = A0 ∩A+ ∪A− = A0 ∩ [A+ ∪A−]

= A0 ∩ [A+ ∪A− ∪ ∂A− ∪ ∂A+] = A0 ∩ [∂A− ∪ ∂A+].

Moreover since A− and A+ are open, A− ⊂ A− ∪A0, A+ ⊂ A+ ∪A0,

∂A− = !A− ∩A− ⊂ [A0 ∪A+] ∩ [A0 ∪A−] = A0,

∂A+ = !A+ ∩A+ ⊂ [A0 ∪A−] ∩ [A0 ∪A+] = A0

⇒ ∂A− ∪ ∂A+ = [∂A− ∪ ∂A+] ∩A0 = ∂A0

⇒ ∂A− ∪ ∂A+ = ∂A0 ⇒ ∂A0 = ∂A− ∪ (∂A+\∂A−).

Let Q be the subset of points in RN with rational coordinates. Define

B0 def= intA0, B0+

def= B0 ∩Q, B0−

def= B0 ∩ !Q,

and notice that by density of Q and !Q in RN,

B0+ = B0 = B0

−.

Consider the following new partition of RN:

RN = A+ ∪ (∂A+\∂A−) ∪A− ∪ ∂A− ∪B0+ ∪B0

−.

Define

Adef= A− ∪ (∂A+\∂A−) ∪B0

+ ⇒ !A = A+ ∪ ∂A− ∪B0−

and

A = A− ∪ (∂A+\∂A−) ∪B0+ = A− ∪ ∂A− ∪ (∂A+\∂A−) ∪B0,

!A = A+ ∪ ∂A− ∪B0− = A+ ∪ ∂A+ ∪ ∂A− ∪B0.

But

∂A− ∪ ∂A+ = ∂A− ∪ (∂A+\∂A−) ⊂ ∂A− ∪ (∂A+\∂A−) ⊂ ∂A− ∪ ∂A+,


and since ∂A− ∪ ∂A+ = ∂A0

A = A− ∪ ∂A0 ∪ intA0 = A− ∪A0

!A = A+ ∪ ∂A0 ∪ intA0 = A+ ∪A0

∣∣∣∣∣ ⇒ ∂A = A0 = ∅

⇒ bA = dA − d!A = bA − d!A= dF − d!G = f.

(ii) For the compactness, consider any sequence bAn ⊂ Cb(D) ⊂ C0,1(D).Since D is bounded, bAn and ∇bAn are both pointwise uniformly bounded in D.From Theorem 2.5 in Chapter 2, the injection of C0,1(D) into C(D) is compact andthere exist f ∈ C(D) and a subsequence bAnk

such that bAnk→ f in C(D). From

the proof of the closure in part (i), there exists A ⊂ D, ∂A = ∅, such that f = bA.(iii) For all bA and bB in Cb(D) and x in D,

|bB(x)| ≤ |bA(x)| + |bB(x)− bA(x)|⇒ | |bB(x)|− |bA(x)| | ≤ |bB(x)− bA(x)|

⇒ ∥d∂B − d∂A∥C(D) = ∥ |bB |− |bA| ∥C(D) ≤ ∥bB − bA∥C(D).

Moreover, dA = b+A = (|bA| + bA)/2 and d!A = b−

A = (|bA|− bA)/2, and necessarily

∥b±B − b±

A∥C(D) ≤ ∥bB − bA∥C(D).

By combining the above three inequalities we get (2.9).

We have the analogue of Theorem 2.3 of Chapter 6 and its corollary for A,!A, and ∂A. In the last case it takes the following form (to be compared withT. J. Richardson [1, Lem. 3.2, p. 44] and S. R. Kulkarni, S. K. Mitter, andT. J. Richardson [1] for an application to image segmentation).

Corollary 1. Let D be a nonempty open (resp., bounded open) holdall in RN.Define for a subset S of RN the sets

Hb(S) def= bA ∈ Cb(D) : S ⊂ ∂A ,

Ib(S) def= bA ∈ Cb(D) : ∂A ⊂ S ,

Jb(S) def= bA ∈ Cb(D) : ∂A ∩ S = ∅ .

(i) Let S be a subset of RN. Then Hb(S) is closed in Cloc(D) (resp., C(D)).

(ii) Let S be a closed subset of RN. Then Ib(S) is closed in Cloc(D) (resp., C(D)).If, in addition, S∩D is compact, then Jb(S) is closed in Cloc(D) (resp., C(D)).

(iii) Let S be an open subset of RN. Then Jb(S) is open in Cloc(D) (resp., C(D)).If, in addition, !S∩D is compact, then Ib(S) is open in Cloc(D) (resp., C(D)).

(iv) For D bounded, associate with an equivalent class [A]b the number

# ∂A = number of connected components of ∂A.


Then the map

[A]b .→ # ∂A : F#b(D) def=[A]b : ∀A ⊂ D, ∂A = ∅, and # ∂A <∞

→ R

is lower semicontinuous. For a fixed number c > 0, the subset

bA ∈ Cb(D) : # ∂A ≤ c

of Cb(D) is compact in C(D). In particular, the subset

bA ∈ Cb(D) : ∂A is connected

of Cb(D) is compact in C(D).

Proof. The proof is the same as for Theorem 2.3 of Chapter 6 using the Lipschitzcontinuity of the map bA .→ d∂A = |bA| from C(D) to C(D).

3 Projection, Skeleton, Crack, and DifferentiabilityThis section is the analogue of section 3 in Chapter 6: connection between thegradient of bA and the projection onto ∂A and the characteristic functions associatedwith ∂A, singularities of the gradients, and the notions of skeleton1 and cracks.

Let A ⊂ RN, ∅ = ∂A, and recall the notation

Sing (∇bA) def=x ∈ RN : ∇bA(x) "

(3.1)

for the set of singularities of the gradient of bA,

Π∂A(x) def= z ∈ ∂A : |z − x| = d∂A(x) (3.2)

for the set of projections of x onto ∂A (when Π∂A(x) is a singleton, the uniqueelement is denoted by p∂A(x)), and

Sk (∂A) def=x ∈ RN : Π∂A(x) is not a singleton

=x ∈ RN : ∇d2

∂A(x) "

(3.3)

for the skeleton of ∂A. Since Π∂A(x) is a singleton for all x ∈ ∂A we have Sk (∂A) ⊂RN \∂A.

In general, the functions bA and d∂A are different and Sing∇bA is usuallysmaller than Sing (∇d∂A) = Sk (∂A) ∪ Ck (∂A). As a result, using the orienteddistance function bA rather than d∂A requires a new definition of the set of crackswith respect to bA that will be justified in Theorem 3.1 (iv).

Definition 3.1.Given A ⊂ RN, ∅ = ∂A, the set of b-cracks of A is defined as

Ck b(A) def= Sing∇bA\Sk (∂A). (3.4)

Sk (∂A), Ck b(A), and Sing (∇bA) have zero N -dimensional Lebesgue measure, sincebA is Lipschitz continuous and hence differentiable almost everywhere.

1Our definition of a skeleton does not exactly coincide with the one used in morphologicalmathematics (cf., for instance, G. Matheron [1] or A. Riviere [1]).


Remark 3.1.(1) In general Ck b(A) ⊂ Ck (∂A) = Ck (A) ∪ Ck (!A), but Ck b(A) can be strictlysmaller than Ck (∂A). Consider the open ball of unit radius A = B1(0):

bA(x) = |x|− 1, ∇bA(x) =x

|x| , D2bA(x)ij =1|x|

(δij −

xi

|x|xj

|x|

),

Sing (∇bA) = 0, Sk (∂A) = 0, and Ck b(A) = ∅.

It is easy to check that for all x ∈ ∂A, ∇bA(x) is the outward unit normal, andD2bA(x) is the second fundamental form with eigenvalues 0 for the space spannedby the normal x and 1 for the tangent space to ∂A in x orthogonal to x. But

d∂A(x) = ||x|− 1| ,Sing (∇d∂A) = 0 ∪ ∂B1(0), Sk (∂A) = 0, and Ck (∂A) = ∂B1(0).

Using bA rather than d∂A or dA removes artificial singularities.(2) In general Sk (∂A) ⊂ RN \∂A, but if ∂A has positive reach h > 0 in

the sense of Definition 6.1 in section 6 of Chapter 6 (there exists h > 0 such thatb2A ∈ C1,1(Uh(∂A)), the skeleton will remain at a distance h from ∂A.

The function bA enjoys properties similar to the ones of dA and d!A since|bA| = d∂A = maxdA, d!A and we have the analogues of Theorems 3.2 and 3.3 inChapter 6. Recall the definition of

f∂A(x) def=12(|x|2 − d2

∂A(x)).

Theorem 3.1. Let A be a subset of RN such that ∅ = ∂A, and let x ∈ RN.

(i) For all x ∈ RN, d2∂A(x) = b2

A(x),

f∂A(x) =12(|x|2 − b2

A(x))

is convex and continuous, the function

x .→ k∂A,h(x)2h

=|x|2

2 h− |bA|(x) : RN \Uh(∂A)→ R

is locally convex (and continuous) in RN \Uh(∂A), ∇f∂A and ∇b2A belong to

BVloc(RN)N , and ∇d∂A ∈ BVloc(RN \∂(∂A)

)N .

(ii) The set Π∂A(x) is nonempty and compact, and

∀x /∈ ∂A Π∂A(x) ⊂ ∂(∂A) and ∀x ∈ ∂A Π∂A(x) = x.

For x /∈ A, Π∂A(x) = ΠA(x) ⊂ ∂A and for x /∈ !A, Π∂A(x) = Π!A(x) ⊂ ∂!A.Moreover, ∂(∂A) = ∂A ∪ ∂!A.


(iii) For all x and v in RN, the Hadamard semiderivative2 of d2A always exists,

dHb2A(x; v) = lim

t0

b2A(x + tv)− b2

A(x)t

= minz∈Π∂A(x)

2(x− z) · v,

dHf∂A(x; v) = limt0

f∂A(x + tv)− f∂A(x)t

= σΠ∂A(x)(v) = σco Π∂A(x)(v),

where σB is the support function of the set B,

σB(v) = supz∈B

z · v,

and co B is the convex hull of B. In particular,

Sk (∂A) =x ∈ RN : ∇b2

A(x) "⊂ RN \∂A. (3.5)

Given v ∈ RN, for all x ∈ RN \∂(∂A) the Hadamard semiderivative of bA

exists,

dHbA(x; v) =1

bA(x)min

p∈ΠA(x)(x− p) · v, (3.6)

and for x ∈ ∂(∂A), dHbA(x; v) exists if and only if

limt0

bA(x + tv)t

exists. (3.7)

(iv) The following statements are equivalent:

(a) b2A(x) is (Frechet) differentiable at x.

(b) b2A(x) is Gateaux differentiable at x.

(c) Π∂A(x) is a singleton.

(v) ∇f∂A(x) exists if and only if Π∂A(x) = p∂A(x) is a singleton. In that case

p∂A(x) = ∇f∂A(x) = x− 12∇b2

A(x). (3.8)


∀p(x) ∈ Π∂A(x),12(|y|2 − b2

A(y))≥ 1

2(|x|2 − b2

A(x))

+ p(x) · (y − x), (3.9)

or equivalently,

∀p(x) ∈ Π∂A(x), b2A(y)− b2

A(x)− 2 (x− p(x)) · (y − x) ≤ |x− y|2. (3.10)


∀p(x) ∈ Π∂A(x), ∀p(y) ∈ Π∂A(y), (p(y)− p(x)) · (y − x) ≥ 0. (3.11)2A function f : RN → R has a Hadamard semiderivative in x in the direction v if

dHf(x; v) def= limt0w→v

f(x + tw) − f(x)t

exists

(cf. Chapter 9, Definition 2.1 (ii)).


(vi) The functions

p∂A : RN \Sk (∂A)→ RN and ∇b2A : RN \Sk (∂A)→ RN

are continuous. For all x ∈ RN \Sk (∂A)

p∂A(x) = x− 12∇b2

A(x) = ∇f∂A(x). (3.12)

In particular, for all x ∈ ∂A, Π∂A(x) = x, ∇b2A(x) = 0,

Sk (∂A) =x ∈ RN : ∇b2

A(x) " and ∇bA(x) "⊂ RN \∂A,

Ck b(A) =x ∈ RN : ∇b2

A(x) ∃ and ∇bA(x) "⊂ ∂(∂A),

(3.13)

and Sing (∇bA) = Sk (∂A) ∪ Ck b(A).

(vii) The function

∇bA : RN \ (Sk (∂A) ∪ ∂(∂A))→ RN

is continuous. For all x ∈ int ∂A, ∇bA(x) = 0; for all x ∈ RN \ (Sk (∂A) ∪ ∂A),

∇bA(x) =∇b2

A(x)2 bA(x)

=

⎧⎪⎪⎨

⎪⎪⎩

x− p∂A(x)|x− p∂A(x)| , x ∈ RN \A,

− x− p∂A(x)|x− p∂A(x)| , x ∈ RN \!A,

⎫⎪⎪⎬

⎪⎪⎭, |∇bA(x)| = 1.

(3.14)

If ∇d∂A(x) exists for some x ∈ ∂A, then ∇bA(x) = ∇d∂A(x) = 0. In partic-ular, ∇bA(x) = 0 almost everywhere in ∂A.

(viii) The functions b2A and bA are differentiable almost everywhere and

m(Sk (∂A)) = m(Ck b(A)) = m(Sing (∇bA)) = 0.

If ∂A = ∅,

Sk (∂A) = Sk (A) ∪ Sk (!A) ⊂ RN \∂A,

Ck b(A) ⊂ Ck (∂A) = Ck (A) ∪ Ck (!A) ⊂ ∂(∂A).(3.15)

If ∂A = ∅, then either A = RN or A = ∅.

(ix) Given A ⊂ RN, ∂A = ∅,

χ∂A(x) = 1− |∇bA(x)| in RN \Sing (∇bA), (3.16)

the identity holds for almost all x in RN, and the set

∂bAdef= x ∈ ∂A : ∇bA(x) exists and ∇bA(x) = 0 ⊂ ∂(∂A) (3.17)

has zero N -dimensional Lebesgue measure.


(x) Let µ be a measure in the sense of L. C. Evans and R. F. Gariepy [1,Chap. 1] and D ⊂ RN be bounded open such that µ(D) <∞. The map

bA .→ µ(∂A) : Cb(D)→ R (3.18)

is upper semicontinuous with respect to the topology of uniform convergenceon Cb(D). Moreover, if bAn → bA in C(D) for bAn and bA in Cb(D), then

µ(intA) ≤ lim inf µ(intAn) ≤ lim supµ(An) ≤ µ(A). (3.19)

If µ(∂A) = 0, the map bA .→ µ(A) : Cb(D)→ R is continuous in bA.

(xi) Given x ∈ RN, α ∈ [0, 1], p ∈ Π∂A(x), and xαdef= p + α (x− p), then

bA(xα) = α bA(x) and ∀α ∈ [0, 1] , Π∂A(xα) ⊂ Π∂A(x).

In particular, if Π∂A(x) is a singleton, then Π∂A(xα) is a singleton and∇b2

A(xα) exists for all α, 0 ≤ α ≤ 1. If, in addition, x = ∂A, then forall 0 < α ≤ 1 ∇bA(xα) exists and ∇bA(xα) = ∇bA(x).

Proof. (i) to (v) follow from Theorems 3.2 and 3.3 of Chapter 6 using the identityb2A = d2

∂A.(v) To complete the proof of identities (3.13), recall that the projection of

all points of ∂A onto ∂A being a singleton, Sk (∂A ⊂ RN \∂A. Therefore, for anyx ∈ Sk (∂A), we have bA(x) = 0 and if ∇b2

A(x) does not exist, then ∇bA(x) cannotexist. This sharpens the characterization (3.5) of Sk (∂A). From this we readilyget the characterization of Ck b(A) = Sing (∇bA)\Sk (∂A) and Ck b(A) ⊂ ∂A. Thislast property can be improved to Ck b(A) ⊂ ∂(∂A) by noticing that for x ∈ int ∂A,bA(x) = 0 and ∇b2

A(x) = ∇bA(x) = 0.(vi) From Theorem 3.3 (iv) of Chapter 6.(vii) In RN \A, bA = dA and ∇bA is continuous in RN \(Sk (A) ∪ A); in

RN \!A, bA = −d!A and ∇bA is continuous in RN \(Sk (!A) ∪ !A); in int ∂A,bA = dA = d!A = 0 and ∇bA = 0. So, in view of part (v) it is continuous in

RN \(Sk (A) ∪A) ∪RN \(Sk (!A) ∪ !A) ∪ int ∂A = RN \ (Sk (∂A) ∪ ∂(∂A)) .

If x ∈ RN \!A, then bA = −d!A and repeat the proof of Theorem 3.3 (vi) inChapter 6 to obtain

−∇bA(x) = ∇d!A(x) =x− p∂A(x)|x− p∂A(x)| =

x− p∂A(x)−bA(x)

and p∂A(x) ∈ ∂!A is unique. Similarly, for x ∈ RN \A, bA = dA,

∇bA(x) = ∇dA(x) =x− p∂A(x)|x− p∂A(x)| =

x− p∂A(x)bA(x)

and p∂A(x) ∈ ∂A is unique. In both cases |∇bA(x) = 1.

4. W 1,p(D)-Metric Topology and the Family C0b (D) 349

Finally, from Theorem 3.3 (i) in Chapter 6, whenever d∂A is differentiable ata point x ∈ ∂A, ∇d∂A(x) = 0. But for all v in RN and t > 0

∣∣∣∣bA(x + tv)− bA(x)

t

∣∣∣∣ =|bA(x + tv)|

t=

d∂A(x + tv)− d∂A(x)t

since |bA| = d∂A. Therefore, if ∇d∂A(x) exists, ∇bA(x) exists and ∇bA(x) =∇d∂A(x) = 0. Finally, since ∇d∂A(x) exists almost everywhere in RN, then foralmost all x ∈ ∂A, ∇bA(x) = 0.

(viii) From Theorem 3.3 (vii) in Chapter 6 and the fact that Ck b(A) ⊂ C(∂A).Indeed, for x ∈ ∂A, if ∇dA(x) and ∇d!A(x) exist, they are 0 and for all t > 0and v ∈ RN

∣∣∣∣bA(x + tv)− bA(x)

t

∣∣∣∣ =∣∣∣∣dA(x + tv)− dA(x)

t− d!A(x + tv)− d!A(x)

t

∣∣∣∣→ 0

as t 0. Hence∇bA(x) = 0. As a consequence, if∇bA(x) does not exist, then either∇dA(x) or ∇d!A(x) does not exist and Ck b(A) ⊂ C(A) ∪ C(!A) = C(∂A) ⊂ ∂(∂A)from Theorem 3.3 (vii) in Chapter 6.

(ix) Since bA is a Lipschitzian function, it is differentiable almost everywherein RN, and in view of the previous considerations, when it is differentiable

|∇bA(x)| =

1, x /∈ ∂A,

0 a.e. in ∂A⇒ χ∂A(x) = 1− |∇bA(x)| a.e. in RN .

Therefore, the set ∂bA has at most a zero measure.(x) From Theorem 4.1 in Chapter 6 applied to A, !A, and ∂A.(xi) Same proof as the one of Theorem 3.3 (ix) in Chapter 6.

Remark 3.2.In general ∇dA(x) and ∇d!A(x) do not exist for x ∈ ∂A. This is readily seen byconstructing the directional derivatives for the half-space

A = (x1, x2) ∈ R2 : x1 ≤ 0 (3.20)

at the point (0, 0). Nevertheless ∇bA(0, 0) exists and is equal to (1, 0), which isthe outward unit normal at (0, 0) ∈ ∂A to A. Note also that for all x ∈ ∂A,|∇bA(x)| = 1. This is possible since mN (∂A) = 0. If ∇bA(x) exists, then it is easyto check that |∇bA(x)| ≤ 1. For sufficiently smooth domains, the subset ∂bA of theboundary ∂A coincides with the reduced boundary of finite perimeter sets.

4 W 1,p(D)-Metric Topology and the Family C0b (D)

4.1 Motivations and Main Properties

From Theorem 2.1 (vi) bA is locally Lipschitzian and belongs to W 1,ploc (RN) for all

p ≥ 1. So the previous constructions for dA and d!A can be repeated with the space


W 1,ploc (RN) in place of Cloc(D) to generate new W 1,p metric topologies on the family

Cb(D). Moreover the other distance functions can be recovered from the map

bA .→ (b+A, b−

A, |bA|) = (dA, d!A, d∂A)

and the characteristic functions from the maps

bA .→ b−A = d!A .→ χint A = |∇d!A|, (4.1)

bA .→ b+A = dA .→ χint !A = |∇dA|, (4.2)

bA .→ χ∂A = 1− |∇bA|. (4.3)

One of the advantages of the function bA is that the W 1,p-convergence of sequenceswill imply the Lp-convergence of the corresponding characteristic functions of intA,int !A, and ∂A, that is, continuity of the volume of these sets.

It will be useful to introduce the following set and terminology.

Definition 4.1. (i) A set A ⊂ RN, ∂A = ∅, is said to have a thin3 boundary ifmN (∂A) = 0.

(ii) Denote by

C0b (D) def= bA : ∀A, A ⊂ D, ∂A = ∅ and m(∂A) = 0 (4.4)

the subset of oriented distance functions with thin boundaries.

In this section we give the analogues of Theorems 4.2 and 4.3 in Chapter 6.

Theorem 4.1. Let D be an open (resp., bounded open) subset of RN.

(i) The topologies induced by W 1,ploc (D) (resp., W 1,p(D)) on Cb(D) are all equiv-

alent for p, 1 ≤ p <∞.

(ii) Cb(D) is closed in W 1,ploc (D) (resp., W 1,p(D)) for p, 1 ≤ p <∞, and

ρD([A2]b, [A1]b)def=

∞∑

n=1

12n

∥bA2 − bA1∥W 1.p(B(0,n))

1 + ∥bA2 − bA1∥W 1.p(B(0,n))

(resp., ρD([A2]b, [A1]b)def= ∥bA2 − bA1∥W 1,p(D))

defines a complete metric structure on Fb(D).

(iii) For p, 1 ≤ p <∞, the map

bA .→ χ∂A = 1− |∇bA| : Cb(D) ⊂W 1,ploc (D)→ Lp

loc(D)

is “Lipschitz continuous”: for all bounded open subsets K of D and bA1 andbA2 in Cb(D),

∥χ∂A2 − χ∂A1∥Lp(K) ≤ ∥∇bA2 −∇bA1∥Lp(K) ≤ ∥bA2 − bA1∥W 1,p(K).

In particular, C0b (D) is a closed subset of Cb(D) for the W 1,p-topology.

3This terminology should not be confused with the notion of thin set in the capacity sense.


(iv) Let D be a bounded open subset of RN. For each bA ∈ Cb(D),

∥bA∥W 1,p(D) = ∥ |bA| ∥W 1,p(D) = ∥d∂A∥W 1,p(D),

∥dA∥W 1,p(D) ≤ ∥bA∥W 1,p(D), ∥d!A∥W 1,p(D) ≤ ∥bA∥W 1,p(D).(4.5)

The map

bA .→ (b+A, b−

A, |bA|) = (dA, d!A, d∂A) : Cb(D) ⊂W 1.p(D)→W 1.p(D)3 (4.6)

is continuous.

(v) Let D be an open (resp., bounded open) subset of RN. For all p, 1 ≤ p <∞,the map

bA .→ (χ∂A,χint A,χint !A)

: W 1,ploc (D)→ Lp

loc(D)3(resp.,W 1,p(D)→ Lp(D)3)(4.7)

is continuous.

Proof. The proofs of (i) and (ii) are essentially the same as the proof of Theorem 4.2of Chapter 6 using the properties established for Cb(D) in C(D) of Theorem 2.2.

(iii) From Theorem 3.1 (ix) for any two subsets A1 and A2 of D with nonemptyboundaries and for any open U in D

|∇bA2 | ≤ |∇bA1 | + |∇bA2 −∇bA1 |⇒ χ∂A1 ≤ χ∂A2 + |∇bA2 −∇bA1 | ⇒ |χ∂A1 − χ∂A2 | ≤ |∇bA2 −∇bA1 |

⇒∫

U|χ∂A2 − χ∂A1 |p dx ≤ ∥∇bA2 −∇bA1∥

pLp(U) ≤ ∥bA2 − bA1∥

pW 1,p(U)

for p, 1 ≤ p < ∞, and with the ess sup norm for p = ∞. The closure of C0b (D)

follows from the continuity of the maps

bA .→ m(∂A ∩ U) =∫

∂A∩Uχ∂A dx : W 1,p(U)→ R

for all bounded open subsets U of D.(iv) First observe that

|bA(x)| = dA(x) + d!A(x) and ∇|bA(x)| = ∇dA(x) +∇d!A(x),bA(x) = dA(x)− d!A(x) and ∇bA(x) = ∇dA(x)−∇d!A(x)

and since ∇bA = ∇dA = ∇d!A = 0 almost everywhere on ∂A = A ∩ !A, then

|∇bA(x)| = |∇dA(x)| + |∇d!A(x)| = |∇|bA(x)| | for almost all x in RN .

From this we readily get (4.5). However, this is not sufficient to prove the continuitysince the map (4.6) is not linear. To get around this consider a sequence bAn ⊂Cb(D) converging to bA in W 1,p(D). From Theorem 2.2 (iii), |bAn | = d∂An , b+

An=

dAn , and b−An

= d!Anconverge to |bA| = d∂A, b+

A = dA, and b−A = d!A in C(D)


and hence in W 1,p(D)-weak. To prove that the convergence is strong, consider theL2-norm

∫

D|∇dAn −∇dA|2 + |∇d!An

−∇d!A|2 dx

=∫

D|∇dAn |2 + |∇dA|2 + |∇d!An

|2 + |∇d!A|2

− 2∇dAn ·∇dA − 2∇d!An·∇d!A dx

=∫

D|∇bAn |2 + |∇bA|2 − 2∇dAn ·∇dA − 2∇d!An

·∇d!A dx

→∫

D2 |∇bA|2 − 2 |∇dA|2 − 2 |∇d!A|2 dx =

∫

D2 |∇bA|2 − 2 |∇bA|2 dx = 0

by weak L2-convergence of ∇dAn and ∇d!Anto ∇dA and ∇d!A. So both se-

quences dAn and d!Anconverge in W 1,2(D)-strong, and hence from part (i) in

W 1,p(D)-strong for all p, 1 ≤ p <∞.(v) It is sufficient to prove the result for D bounded open. From part (iii) it

is true for χ∂A, and from part (iv) and Theorem 4.2 (ii) and (iii) of Chapter 6 forthe other two.

4.2 Weak W 1,p-TopologyTheorem 4.2. Let D be a bounded open domain in RN.

(i) If bAn weakly converges in W 1,p(D) for some p, 1 ≤ p <∞, then it weaklyconverges in W 1,p(D) for all p, 1 ≤ p <∞.

(ii) If bAn converges in C(D), then it weakly converges in W 1,p(D) for all p,1 ≤ p < ∞. Conversely, if bAn weakly converges in W 1,p(D) for some p,1 ≤ p <∞, it converges in C(D).

(iii) Cb(D) is compact in W 1,p(D)-weak for all p, 1 ≤ p <∞.

Proof. (i) Recall that for D bounded there exists a constant c > 0 such that for allbA ∈ Cb(D)

|bA(x)| ≤ c and |∇bA(x)| ≤ 1 a.e. in D.

If bAn weakly converges in W 1,p(D) for some p ≥ 1, then

bAn weakly converges in Lp(D),

∇bAn weakly converges in Lp(D)N .

By Lemma 3.1 (i) in Chapter 5 both sequences weakly converge for all p ≥ 1 andhence bAn weakly converges in W 1,p(D) for all p ≥ 1.

(ii) If bAn converges in C(D), then by Theorem 2.2 (ii) there exists bA ∈Cb(D) such that bAn → bA in C(D) and hence in Lp(D). So for all ϕ ∈ D(D)N

∫

D∇bAn · ϕ dx = −

∫

DbAndivϕ dx→ −

∫

DbAdivϕ dx =

∫

D∇bA · ϕ dx.


By density of D(D) in L2(D), ∇bAn → ∇bA in L2(D)N -weak and hence bAn → bA

in W 1,2(D)-weak. From part (i) it converges in W 1,p(D)-weak for all p, 1 ≤ p <∞.Conversely, the weakly convergent sequence converges to some f in W 1,p(D). Bycompactness of Cb(D), there exist a subsequence, still indexed by n, and bA suchthat bAn → bA in C(D) and hence in W 1,p(D)-weak. By uniqueness of the limit,bA = f . Therefore, all convergent subsequences in C(D) converge to the same limit.So the whole sequence converges in C(D). This concludes the proof.

(iii) Consider an arbitrary sequence bAn in Cd(D). From Theorem 2.2 (ii)Cb(D) is compact and there exist a subsequence bAnk

and bA ∈ Cb(D) such thatbAnk

→ bA in C(D). From part (ii) the subsequence weakly converges in W 1,p(D)and hence Cb(D) is compact in W 1,p(D)-weak.

For sets with thin boundaries the strong and weak W 1,p(D)-convergences ofelements of C0

b (D) to an element of C0b (D) are equivalent as in the case of Theo-

rem 3.2 in section 3.2 of Chapter 5 for characteristic functions in Lp(D)-strong.

Theorem 4.3. Let ∅ = D ⊂ RN be bounded open.

(i) Let An be a sequence of subsets of D such that ∂An = ∅ and m(∂An) = 0and A ⊂ D be such that ∂A = ∅ and m(∂A) = 0. Then

bAn bA in W 1,2(D)-weak ⇒ bAn → bA in W 1,2(D)-strong,

and hence in W 1,p(D)-strong for all p, 1 ≤ p <∞.

(ii) The map

bA .→ mN (intA) = mN (A) = mN (A) : C0b (D)→ R (4.8)

is continuous with respect to the W 1,p-topology on C0b (D) for all p, 1 ≤ p <∞.

Proof. (i) From Theorem 4.1 (i) on the equivalence of the W 1,p-topologies, it issufficient to prove the result for p = 2. By Theorem 4.2 (ii), the weak L2-convergenceof bAn to bA implies the strong convergence in C(D) and hence in L2(D)-strong.Since, for all n ≥ 1, m(∂An) = 0 = m(∂A), |∇bA| = 1 = |∇bAn | almost everywherein D by Theorem 3.1 (vii). As a result

∫

D|∇bAn −∇bA|2 dx =

∫

D|∇bAn |2 + |∇bA|2 − 2∇bAn ·∇bA dx

= 2∫

D(1−∇bAn ·∇bA) dx→ 2

∫

D(1− |∇bA|2) dx = 2

∫

Dχ∂A dx = 0.

Therefore ∇bAn → ∇bA in L2(D)N -strong and bAn → bA in W 1,2(D)-strong, sincethe convergence bAn → bA in L2(D)-strong follows from the weak convergence inW 1,2(D). The convergence in W 1,p(D)-strong follows from the equivalence of thetopologies on Cb(D) (cf. Theorem 4.1 (i)).

(ii) From Theorem 3.1 (x) since mN (∂A) = 0 for bA ∈ C0b (D) and C0

b (D) isclosed in the W 1,p-topology.


5 Boundary of Bounded and Locally BoundedCurvature

We introduce the family of sets with bounded or locally bounded curvature which arethe analogues for bA of the sets of exterior and interior bounded curvature associatedwith dA and d!A in section 5 of Chapter 6. They include C1,1-domains, convexsets, and the sets of positive reach of H. Federer [3]. They lead to compactnesstheorems for subfamilies of Cb(D) in W 1,p(D).

Definition 5.1. (i) Given a bounded open nonempty holdall D in RN and asubset A of D, ∂A = ∅, its boundary ∂A is said to be of bounded curvaturewith respect to D if

∇bA ∈ BV(D)N . (5.1)

This family of sets will be denoted as follows:

BCb(D) def=bA ∈ Cb(D) : ∇bA ∈ BV(D)N

.

(ii) Given a subset A of RN, ∂A = ∅, its boundary ∂A is said to be of locallybounded curvature if ∇bA belongs to BVloc(RN)N . The family of sets whoseboundary is of locally bounded curvature will be denoted by

BCbdef=bA ∈ Cb(RN) : ∇bA ∈ BVloc(RN)N

.

As in Theorem 6.2 of Chapter 5 for Caccioppoli sets, and in Theorem 5.1 ofChapter 6 for sets with a locally bounded curvature, it is sufficient to satisfy theBV property in a neighborhood of each point of the boundary ∂A.

Theorem 5.1. Let A, ∂A = ∅, be a subset of RN. Then ∂A is of locally boundedcurvature if and only

∀x ∈ ∂(∂A), ∃ρ > 0 such that ∇bA ∈ BV(B(x, ρ))N , (5.2)

where B(x, ρ) is the open ball of radius ρ > 0 in x.

Proof. From Definition 5.1 and Theorem 3.1 (i).

Theorem 5.2.

(i) Let D be a nonempty bounded open Lipschitzian holdall in RN and let A ⊂ D,∂A = ∅. If ∇bA ∈ BV(D)N , then

∥∇χ∂A∥M1(D) ≤ 2 ∥D2bA∥M1(D), χ∂A ∈ BV(D),

and ∂A has finite perimeter with respect to D.

(ii) For any subset A of RN, ∂A = ∅,

∇bA ∈ BVloc(RN)N ⇒ χ∂A ∈ BVloc(RN),

and ∂A has locally finite perimeter.

5. Boundary of Bounded and Locally Bounded Curvature 355

Proof. Given ∇bA in BV(D)N , there exists a sequence uk in C∞(D)N such that

uk → ∇bA in L1(D)N and ∥Duk∥M1(D) → ∥D2bA∥M1(D)

as k goes to infinity, and since |∇bA(x)| ≤ 1, this sequence can be chosen in such away that

∀k ≥ 1, |uk(x)| ≤ 1.

This follows from the use of mollifiers (cf. E. Giusti [1, Thm. 1.17, p. 15]). For allV in D(D)N

−∫

Dχ∂A div V dx =

∫

D(|∇bA|2 − 1) div V dx =

∫

D|∇bA|2 div V dx.

For each uk

∫

D|uk|2div V dx = −2

∫

D[ ∗Duk]uk · V dx = −2

∫

Duk · [Duk]V dx,

where ∗Duk is the transpose of the Jacobian matrix Duk and∣∣∣∣∫

D|uk|2 div V dx

∣∣∣∣ ≤ 2∫

D|uk| |Duk| |V | dx

≤ 2 ∥Duk∥L1∥V ∥C(D) ≤ 2 ∥Duk∥M1∥V ∥C(D)

since for W 1,1(D)-functions ∥∇f∥L1(D)N = ∥∇f∥M1(D)N . Therefore, as k goes toinfinity,

∣∣∣∣∫

Dχ∂A div V dx

∣∣∣∣ =∣∣∣∣∫

D|∇bA|2 div V dx

∣∣∣∣ ≤ 2∥D2bA∥M1∥V ∥C(D).

Therefore, ∇χ∂A ∈M1(D)N .

5.1 Examples and Limit of Tubular Norms as h Goes to Zero

It is informative to compute the Laplacian of bA for a few examples. The first threeexamples are illustrated in Figure 7.1.

Example 5.1 (Half-plane in R2).(cf. Example 5.1 in Chapter 6). Consider the domain

A = (x1, x2) : x1 ≤ 0, ∂A = (x1, x2) : x1 = 0.

It is readily seen that

bA(x1, x2) = x1, ∇bA(x1, x2) = (1, 0), ∆bA(x) = 0,


and

b∂A(x1, x2) = |x1|, ∇b∂A(x1, x2) =(

x1

|x1|, 0)

,

⟨∆b∂A,ϕ⟩ = 2∫

∂Aϕ dx.

Figure 7.1. ∇bA for Examples 5.1, 5.2, and 5.3.

Example 5.2 (Ball of radius R > 0 in R2).(cf. Example 5.2 in Chapter 6). Consider the domain

A = x ∈ R2 : |x| ≤ R, ∂A = x ∈ R2 : |x| = R.

Clearly,

bA(x) = |x|−R, ∇bA(x) =x

|x| ,

⟨∆bA,ϕ⟩ =∫

R2

1|x|ϕ dx.

Also

b∂A(x) =∣∣|x|−R

∣∣, ∇b∂A(x) =

x

|x| , |x| > R,

− x|x| , |x| < R,

⟨∆b∂A,ϕ⟩ = 2∫

∂Aϕ ds−

∫

A

1|x|ϕ dx +

∫

!A

1|x|ϕ dx.

Again ∆b∂A contains twice the boundary measure on ∂A.

Example 5.3 (Unit square in R2).(cf. Example 5.3 in Chapter 6). Consider the domain

A = x = (x1, x2) : |x1| ≤ 1, |x2| ≤ 1 .

Since A is symmetrical with respect to both axes, it is sufficient to specify bA in thefirst quadrant. We use the notation Q1, Q2, Q3, and Q4 for the four quadrants in

5. Boundary of Bounded and Locally Bounded Curvature 357

the counterclockwise order and c1, c2, c3, and c4 for the four corners of the squarein the same order. We also divide the plane into three regions:

D1 = (x1, x2) : |x2| ≤ min1, |x1| ,

D2 = (x1, x2) : |x1| ≤ min1, |x2| ,

D3 = (x1, x2) : |x1| ≥ 1 and |x2| ≥ 1 .

Hence for 1 ≤ i ≤ 4

bA(x) =

⎧⎪⎨

⎪⎩

|x2|− 1, x ∈ D2 ∩Qi,

|x− ci|, x ∈ D3 ∩Qi,

|x1|− 1, x ∈ D1 ∩Qi,

∇bA(x) =

⎧⎪⎨

⎪⎩

(0, 1), x ∈ D2 ∩Qi,x−ci

|x−ci| , x ∈ D3 ∩Qi,

(1, 0), x ∈ D1 ∩Qi,

and for the whole plane

⟨∆bA,ϕ⟩ =4∑

i=1

∫

D3∩Qi

1|x− ci|

ϕ dx +√

2∫

D1∩D2

ϕ dx,

where D1 ∩ D2 is made up of the two diagonals of the square where ∇bA has asingularity (that is, the skeleton). Moreover, for 1 ≤ i ≤ 4

b∂A(x) =

⎧⎪⎨

⎪⎩

||x2|− 1| , x ∈ D2 ∩Qi,

|x− ci|, x ∈ D3 ∩Qi,

||x1|− 1| , x ∈ D1 ∩Qi,

and

⟨∆b∂A,ϕ⟩ =4∑

i=1

∫

D3∩Qi

1|x− ci|

ϕ dx−√

2∫

D1∩D2

ϕ dx + 2∫

∂Aϕ dx.

Notice that the structures of the Laplacian are similar to the ones observed in theprevious examples except for the presence of a singular term along the two diagonalsof the square.

All C2-domains with a compact boundary belong to all the categoriesof Definition 5.1 and of Definition 5.1 of Chapter 6. The h-dependent norms∥D2bA∥M1(Uh(∂A)), ∥D2d!A∥M1(Uh(!A)), and ∥D2dA∥M1(Uh(A)) are all decreasingas h goes to zero. The limit is particularly interesting since it singles out the be-havior of the singular part of the Hessian matrix in a shrinking neighborhood of theboundary ∂A.

Example 5.4.If A ⊂ RN is of class C2 with compact boundary, then

limh0∥D2bA∥M1(Uh(∂A)) = 0.

Example 5.5.Let A = xiI

i=1 be I distinct points in RN. Then ∂A = A, and

limh0∥D2bA∥M1(Uh(∂A)) =

2 I − 1, N = 1,

0, N ≥ 2.


Example 5.6.Let A be a closed line in RN of length L > 0. Then ∂A = A, and

limh0∥D2bA∥M1(Uh(∂A)) =

2 L, N = 2,

0, N ≥ 3.

Example 5.7.Let N = 2. For the finite square and the ball of finite radius

limh0∥∆dA∥M1(Uh(∂A)) = H1(∂A),

where H1 is the one-dimensional Hausdorff measure (cf. Examples 5.2 and 5.3 inChapter 6).

Also, looking at ∆bA in Uh(A) as h goes to zero provides information aboutSk (A).

Example 5.8.Let A be the unit square in R2. Then

limh0∥∆bA∥M1(Uh(A)) =

√2

2H1(Sk (A)),

where H1 is the one-dimensional Hausdorff measure and Sk(A) = Skint(A) is theskeleton of A made up of the two interior diagonals (cf. Example 5.3). It wouldseem that in general

limh0∥∆bA∥M1(Uh(A)) =

∫

Sk(A)| [∇bA] · n| dH1,

where [∇bA] is the jump in ∇bA and n is the unit normal to Sk(A) (if it exists!).

6 Approximation by Dilated Sets/TubularNeighborhoods

We have seen in Theorem 7.1 of Chapter 6 that dA can be approximated by thedistance function dAr of the dilated set Ar = x ∈ RN : dA(x) ≤ r, r > 0, of Ain the W 1,p-topology. The approximation can also be achieved with the orienteddistance function: bAr (x) → bA(x) uniformly in RN. But it requires that themeasure of the boundary ∂A be zero to get the approximation in the W 1,p-topology.

Lemma 6.1. Given A ⊂ RN, ∂A = ∅ if and only if A = ∅ and there exists h > 0such that Ah = RN.

Proof. If ∂A = ∅, then A = ∅ and !A = ∅, A = ∅ and !A = ∅, and

RN = A = ∩h>0Ah ⇒ ∃h > 0 such that Ah = RN .

Conversely, A ⊃ A = ∅ and, for some h > 0, A = ∩k>0Ak ⊂ Ah = RN. ThereforeA = ∅ and !A = ∅ imply that ∂A = ∅.

6. Approximation by Dilated Sets/Tubular Neighborhoods 359

Theorem 6.1. Given A ⊂ RN such that ∂A = ∅, there exists h > 0 such that forall r, 0 < r ≤ h, ∅ = A ⊂ A ⊂ Ar ⊂ Ah = RN.

(i) For all r, 0 < r ≤ h, !A ⊃ !Ar = ∅, ∂Ar = ∅, bUr(A) = bAr , and Ur(A)and Ar are invariant open and closed representatives in the equivalence class[Ar]b. Moreover,

d!Ur(A)(x) = d!Ar(x) = 0, if dA(x) ≥ r,

d!Ur(A)(x) = d!Ar(x) ≥ r − dA(x), if 0 < dA(x) < r, (6.1)

d!Ur(A)(x) = d!Ar(x) ≥ d!A(x) + r, if dA(x) = 0, (6.2)

∀x ∈ RN, 0 ≤ bA(x)− bAr (x) ≤ r, (6.3)

and, as r → 0,

bUr(A)(x) = bAr (x)→ bA(x) uniformly in RN,

χAr → χA in Lploc(R

N), ∀p, 1 ≤ p <∞.

(ii) The following conditions are equivalent:

(a) m(∂A) = 0.

(b) As r → 0, bUr(A) = bAr → bA in W 1,ploc (RN) for some p, 1 ≤ p <∞.

(c) As r → 0, d!Ur(A) = d!Ar→ d!A in W 1,p

loc (RN) for some p, 1 ≤ p <∞.

If (a), (b), or (c) is verified, then (b) and (c) are verified for all p, 1 ≤ p <∞.

In particular, (a) is verified for all bounded sets A = ∅ in RN such that m(∂A) = 0.

Proof. The existence of the h > 0 such that Ah = RN and A = ∅ is a consequenceof Lemma 6.1. From this for all r, 0 < r ≤ h, ∅ = A ⊂ A ⊂ Ar ⊂ Ah = RN.

(i) From the initial remarks, RN = !A ⊃ !A ⊃ !Ar ⊃ !Ah = ∅ and ∂Ar = ∅and ∂A = ∅, and the functions d!A, d!Ar

, bA, and bAr are well-defined. FromTheorem 3.1 of Chapter 6, Ar = Ur(A), and !Ar = !Ur(A), and ∂Ar = d−1

A r.Hence, dAr = dUr(A). Hence, dAr = dUr(A), d!Ar

= d!Ur(A), and bAr = bUr(A).For x ∈ RN such that dA(x) ≥ r, x ∈ !Ur(A) and d!Ur(A)(x) = 0.Inequality (6.1). For all y ∈ !Ur(A) and all p ∈ A,

r < dA(y) ≤ |y − p| ≤ |y − x| + |x− p|,r ≤ inf

y∈!Ur(A)|y − x| + inf

p∈A|x− p| = d!Ur(A)(x) + dA(x).

Inequality (6.2). Since A ∩ !Ur(A) = ∅, for x ∈ A, there exists pr ∈∂!Ur(A) = d−1

A r such that d!Ur(A)(x) = |pr − x| > 0. Therefore dA(pr) = r.Either d!A(x) = 0 or d!A(x) > 0. If d!A(x) = 0, then

d!Ur(A)(x) = |pr − x| ≥ dA(pr) = r = d!A(x) + r.


If d!A(x) > 0, then x /∈ !A and x ∈ int A = A\∂A, Bxdef= B(x, d!A(x)) ⊂ intA,

and Bx ⊂ A. Define

pdef= x + d!A(x)

pr − x

|pr − x| ∈ Bx ⇒ |p− x| = d!A(x) and p ∈ Bx ⊂ A.

By construction

d!Ur(A)(x) = |x− pr| = |x− p| + |p− pr| = d!A(x) + |p− pr|,|p− pr| ≥ inf

p′∈Bx

|p′ − pr| ≥ infp′∈A

|p′ − pr| = dA(pr) = r

and d!Ur(A)(x) ≥ d!A(x) + r.Inequality (6.3). From Theorem 2.1 (ii), A ⊂ Ar implies that bA(x) ≥ bAr (x).

As for the second part of inequality (6.3), we use identity (7.2) from Theorem 7.1in Chapter 6 for dUr(A) = dAr combined with (6.2) for x ∈ A and with (6.1) forx ∈ RN \A:

r ≤ dA(x) ⇒ x ∈ !Ur(A) ⊂ !A ⇒ d!Ur(A)(x) = d!A(x) = 0

⇒ bUr(A)(x) = dUr(A)(x) = dA(x)− r = bA(x)− r,

0 < dA(x) < r ⇒ dUr(A)(x) = 0 and d!A(x) = 0⇒ bUr(A)(x) = −d!Ur(A)(x) ≤ dA(x)− r = bA(x)− r,

dA(x) = 0 ⇒ x ∈ A and dUr(A)(x) = 0⇒ bUr(A)(x) = −d!Ur(A)(x) ≤ −d!A(x)− r = bA(x)− r.

(ii) If m(∂A) = 0, from part (i), d!Ur(A)(x)→ d!A(x) implies that ∇d!Ur(A)

∇d!A in L2(D)N -weak for all bounded open sets. Consider the estimate∫

D|∇d!Ur(A) −∇d!A|2 dx =

∫

D

(|∇d!Ur(A)|2 + |∇d!A|2 − 2∇d!Ur(A) ·∇d!A

)dx

=∫

D

(1− χ!Ur(A) + 1− χ!A

− 2∇d!Ur(A) ·∇d!A

)dx

from the identities of Theorem 3.3 (viii) in Chapter 6. As r → 0, !Ur(A) is afamily of increasing closed sets and

!Ur(A) ∪r>0!Ur(A) = ! ∩r>0 Ur(A) = !A ⇒ m(D ∩ Ur(A))→ m(D ∩ !A).

Going to the limit∫

D

(1− χ!Ur(A) + 1− χ!A

− 2∇d!Ur(A) ·∇d!A

)dx

→∫

D

(2− χ!A − χ!A

− 2 |∇d!A|2)

dx =∫

D

(χ!A− χ!A

)dx =

∫

Dχ∂A dx = 0,

since |∇d!A| = 1 − χ!A, ∂A = !A\!A, and m(∂A) = 0. Therefore d!Ur(A) → d!A

in W 1,2(D) and hence in W 1,ploc (RN) for all p, 1 ≤ p < ∞. So, from Theorem 7.1

in Chapter 6, bUr(A) = dUr(A) − d!Ur(A) → dA − d!A = bA in W 1,ploc (RN) for all p,

1 ≤ p <∞.

7. Federer’s Sets of Positive Reach 361

7 Federer’s Sets of Positive Reach7.1 Approximation by Dilated Sets/Tubular NeighborhoodsWe have seen in Theorem 6.1 that the oriented distance function bA can be approx-imated uniformly in RN by the oriented distance function bAr of its dilated set Ar,r > 0, and that the W 1,p-convergence can be achieved if and only if m(∂A) = 0.It turns out that the W 1,p-convergence can be obtained for sets with positive reachsuch that ∂A = ∅ without assuming a priori that m(∂A) = 0.

Theorem 7.1. Let A ⊂ RN such that ∂A = ∅ and reach (A) > 0.

(i) A = ∅ and there exists h such that 0 < h ≤ reach (A) and Ah = RN. For allr, 0 < r < h, the sets Ur(A) are of class C1,1, bUr(A) = bAr , and

bAr (x) = bA(x)− r in RN . (7.1)

As r goes to zero, bAr (x)→ bA(x) uniformly in RN,

bAr → bA in W 1,ploc (RN), (7.2)

dAr → dA, d!Ar→ d!A, d∂Ar → d∂A in W 1,p

loc (RN), (7.3)

m(∂A) = 0, and χAr → χA = χint A in Lploc(R

N) for all p, 1 ≤ p <∞.

(ii) Furthermore, ∂A is of locally bounded curvature and of locally finite perimeter:

∇bA ∈ BVloc(RN)N and χ∂A ∈ BVloc(RN). (7.4)

Moreover, for all r, 0 < r < h,

bA(x) =1r

[f∂Ar (x)− f∂A(x)

]+ r =

1r

[b2A(x)− b2

Ar(x)]

+ r, (7.5)

∀x ∈ RN, ∀v ∈ RN, dHbA(x; v) exists. (7.6)

Remark 7.1.Recall from Theorem 3.1 (iii) that the Hadamard semiderivative of bA always existsat points off the boundary. The last statement in part (ii) says that for a closed setwith nonempty boundary and positive reach, the Hadamard semiderivative existsin all points of its boundary and there is an explicit expression for it. In particular,this is true for all closed convex sets.

Proof. (i) The uniform convergence of bAr to bA in RN was proved in Theorem 6.1 (i),and it remains to prove the W 1,p-convergence without the a priori assumption thatm(∂A) = 0 that will come as a consequence rather than as a hypothesis. To do thatwe first prove the convergence d!Ur(A) → d!A in W 1,p

loc (RN).From Theorem 6.1 (i), it sufficient to show that inequalities (6.1) and (6.2)

become equalities for x ∈ Uh(A): for 0 < r < h

d!Ur(A)(x) = 0, if dA(x) ≥ r,

d!Ur(A)(x) ≥ r − dA(x), if 0 < dA(x) < r, (7.7)

d!Ur(A)(x) ≥ d!A(x) + r, if dA(x) = 0. (7.8)


For x ∈ RN such that dA(x) ≥ r, d!A(x) = d!Ur(A)(x) = 0 and, from identity (7.2)of Chapter 6 in Theorem 6.1, bAr (x) = dAr (x) = dA(x) − r = bA(x) − r. Forx ∈ RN such that 0 < dA(x) < r, d!A(x) = 0 = dAr (x). From Theorem 6.2 (iii) ofChapter 6 for all r, 0 < dA(x) < r < h ≤ reach (A),

x(r) def= pA(x) + rx− pA(x)

dA(x)⇒ dA(x(r)) = r and pA(x(r)) = pA(x)

and x(r) ∈ ∂Ar = ∂Ur(A). Since x /∈ !Ur(A), d!Ur(A)(x) = infp∈∂Ar |x− p| and

d!Ur(A)(x) = d!Ar(x) = d∂Ar (x) ≤ |x(r)− x| = r − dA(x).

Combining this with inequality (7.7), bAr (x) = bA(x) − r. Finally, for x ∈ RN

such that 0 = dA(x), x ∈ A, and since ∂A = ∅, there exists p ∈ ∂A such thatd!A(x) = |p − x|. There exists a sequence yn ⊂ Ur(A)\A such that yn → p andcr such that

|pA(yn)− pA(p)| ≤ cr |yn − p| ⇒ pA(yn)→ pA(p) = p.

Associate with each yn the points

qndef= pA(yn) + r∇dA(yn).

Since 0 < dA(qn) ≤ r < h ≤ reach (A), by Theorem 6.2 (iii) of Chapter 6

pA(qn) = pA(yn), dA(qn) = r, qn ∈ ∂Ar.

The sequence qn is bounded since

|qn − p| ≤ |qn − pA(qn)| + |pA(qn)− p| = |qn − pA(qn)| + |pA(yn)− pA(p)|≤ r + cr|yn − p|.

There exist q and a subsequence, still indexed by n, such that

qn → q ⇒ r = dA(qn)→ dA(q) ⇒ pA(yn) = pA(qn)→ pA(q)⇒ dA(q) = r and pA(q) = p, r = dA(q) = |q − pA(q)| = |q − p|.

Finally, since x ∈ A, A ∩ !Ar = ∅, and ∂!Ar = ∂Ar = d−1A r

d!Ar(x) = inf

dA(pr)=r|x− pr| ≤ |x− q| ≤ |x− p| + |p− q| = d!A(x) + r.

Hence, from inequality (7.8), d!Ar(x) = d!A(x) + r, and since dA(x) = 0 = dAr (x),

bAr (x) = bA(x)− r.Since ∇bAr = ∇bA, for all bounded open subsets D of RN as r → 0,

∥bAr − bA∥W 1,p(D) = ∥bAr − bA∥Lp(D) ≤ r m(D)1/p → 0.

7. Federer’s Sets of Positive Reach 363

From Theorem 7.1 of Chapter 6

dAr → dA in W 1,ploc (RN) and χAr → χA in Lp

loc(RN)

for all p, 1 ≤ p <∞. For all 0 < r < h ≤ reach (A), d!Ar= dAr − bAr → dA− bA =

d!A in W 1,ploc (RN) and d∂Ar = dAr + d!Ar

→ dA + d!A = d∂A in W 1,ploc (RN). Finally,

since bAr → bA in W 1,ploc (RN), 0 = χ∂Ar = 1 − |∇bAr | → 1 − |∇bA| = χ∂A in

Lploc(R

N) and χ∂A = 0 and m(∂A) = 0.(ii) For 0 < r < h < reach (A), the open set Ur(A) is defined via the r-level

set d−1A r = x ∈ RN : dA(x) = r of the function dA. From Theorem 6.2 (vi) in

Chapter 6, for each x ∈ d−1A r = ∂Ur(A) there exists a neighborhood Bρ(x), ρ =

minr, h−r/2 > 0, of x in Uh(A)\A such that dA ∈ C1,1(Bρ(x)). From part (i) forall 0 < r < h and y ∈ Bρ(x) ⊂ Uh(A)\A, d!A(y) = 0, bAr (y) = bA(y)−r = dA(y)−r.Therefore, bAr ∈ C1,1(Bρ(x)), and, a fortiori, for all x ∈ ∂Ar, there exists Bρ(x)such that ∇bAr = ∇dA ∈ BV(Bρ(x)). By Theorem 5.1, ∂Ar is of locally boundedcurvature. By Definition 5.1, ∇bAr ∈ BVloc(RN)N and, since bAr = bA − r in RN,we also have ∇bA = ∇bAr ∈ BVloc(RN)N . By Theorem 5.2 (ii), χ∂A ∈ BVloc(RN).Finally, since bAr (x) = bA(x)− r in RN from part (i)

|x|2 − b2Ar

(x) = |x|2 − b2A(x)− r2 + 2r bA(x),

f∂Ar (x)− f∂A(x) + r2 = r bA(x)

and bA is the difference of two convex continuous functions for which the Hadamardsemiderivative exists in all points x ∈ RN and for all directions v ∈ RN.

7.2 Boundaries with Positive ReachThe next theorem is a specialization of Theorem 6.2 of Chapter 6 to the orienteddistance function.

Theorem 7.2. Let A ⊂ RN, ∂A = ∅.

(i) Associate with a ∈ ∂A the sets

P (a) def=v ∈ RN : Π∂A(a + v) = a

, Q(a) def=

v ∈ RN : d∂A(a + v) = |v|

.

Then P (a) and Q(a) are convex and P (a) ⊂ Q(a) ⊂ (−Ta∂A)∗.

(ii) Given a ∈ ∂A and v ∈ RN, assume that

0 < r(a, v) def= sup t > 0 : Π∂A(a + tv) = a .

Then for all t, 0 ≤ t < r(a, v), Π∂A(a + tv) = a and d∂A(a + tv) = t |v|.Moreover, if r(a, v) < +∞, then a + r(a, v) v ∈ Sk (∂A).

(iii) If there exists h such that 0 < h ≤ reach (∂A), then for all x ∈ Uh(∂A)\∂Aand 0 < |t| < h, Π∂A(x(t)) = p∂A(x) and bA(x(t)) = t, where

x(t) def= p∂A(x) + t∇bA(x) = p∂A(x) + tx− p∂A(x)

bA(x). (7.9)


(iv) Given b ∈ ∂A, x ∈ RN \Sk (∂A), a = p∂A(x), such that reach (∂A, a) > 0,then

(x− a) · (a− b) ≥ − |a− b|2|x− a|2 reach (∂A, a)

. (7.10)

(v) Given 0 < r < q <∞ and

x ∈ RN \Sk (∂A) such that |bA(x)| ≤ r and reach (∂A, p∂A(x)) ≥ q,

y ∈ RN \Sk (∂A) such that |bA(y)| ≤ r and reach (∂A, p∂A(y)) ≥ q,

then

|p∂A(y)− p∂A(x)| ≤ q

q − r|y − x|. (7.11)

(vi) If 0 < reach (∂A) < +∞, then for all h, 0 < h < reach (∂A), and all x, y ∈(∂A)h = x ∈ RN : |bA(x)| ≤ h

|p∂A(y)− p∂A(x)| ≤ reach (∂A)reach (∂A)− h

|y − x|,

∣∣∇b2A(y)−∇b2

A(x)∣∣ ≤ 2

(1 +

reach (∂A)reach (∂A)− h

)|y − x|; (7.12)

for all s, 0 < s < h < reach (∂A), x, y ∈ ∂As,h =x ∈ RN : s ≤ |bA(x)| ≤ h

,

|∇bA(y)−∇bA(x)| ≤ 1s

(2 +

reach (∂A)reach (∂A)− h

)|y − x| .

If reach (∂A) = +∞,

∀x, y ∈ RN, |p∂A(y)− p∂A(x)| ≤ |y − x|, (7.13)∣∣∇b2

A(y)−∇b2A(x)

∣∣ ≤ 4 |y − x|;

for all 0 < s and all x, y ∈ x ∈ RN : s ≤ d∂A(x)

|∇bA(y)−∇bA(x)| ≤ 3s

|y − x| .

(vii) If reach (∂A) > 0, then, for all 0 < r < reach (∂A), Ur(∂A) is a set of classC1,1, Ur(∂A) = (∂A)r, ∂Ur(∂A) = x ∈ RN : |bA(x)| = r, and !(∂A)r =!Ur(∂A) = x ∈ RN : |bA(x)| ≥ r.

(viii) For all a ∈ ∂A,

Ta∂A =

h ∈ RN : lim inft0

|bA(a + th)|t

= 0

.

In particular, if a ∈ int ∂A, Ta∂A = RN.

8. Boundary Smoothness and Smoothness of bA 365

The next theorem is a specialization of Theorem 6.3 of Chapter 6 to theoriented distance function.

Theorem 7.3. Given A ⊂ RN such that ∂A = ∅, the following conditions areequivalent:

(i) ∂A has positive reach, that is, reach (∂A) > 0.

(ii) There exists h > 0 such that b2A ∈ C1,1(Uh(∂A)).

(iii) There exists h > 0 such that for all s, 0 < s < h, bA ∈ C1,1(Us,h(∂A)).

(iv) There exists h > 0 such that bA ∈ C1(Uh(∂A)\∂A).

(v) There exists h > 0 such that for all x ∈ Uh(∂A), Π∂A(x) is a singleton.

Recall from Theorem 3.1 (viii) and (3.15) the following properties. If ∂A = ∅,

Sk (∂A) = Sk (A) ∪ Sk (!A) ⊂ RN \∂A,

Ck b(A) ⊂ Ck (∂A) = Ck (A) ∪ Ck (!A) ⊂ ∂(∂A).

If ∂A has positive reach, then both A and !A have positive reach.

8 Boundary Smoothness and Smoothness of bA

In this section the smoothness of the boundary ∂A is related to the smoothness ofthe function bA in a neighborhood of ∂A. The next two theorems give a completecharacterization of sets A such that m(∂A) = 0 and bA ∈ Ck,ℓ in a neighborhoodof ∂A. We first present two equivalence theorems that will follow from the moredetailed Theorems 8.3 and 8.4. It is useful to recall that for a set that is locally ofclass Ck,ℓ in a point of ∂A, ∂A is locally a Ck,ℓ-submanifold of dimension N − 1 inthat point, but the converse is not true (cf. Definitions 3.1 and 3.2 in Chapter 2)4.

Theorem 8.1 (Local version). Let A ⊂ RN, ∂A = ∅, let k ≥ 1 be an integer, let0 ≤ ℓ ≤ 1 be a real number, and let x ∈ ∂A.

4It is important to note that Theorem 8.1 is not true when bA is replaced by d∂A since its gra-dient ∇d∂A is discontinuous across ∂A. Part (iii) of Theorem 8.1 in the direction (⇒) was assertedby J. Serrin [1] in 1969 for N = 3 and proved in 1977 by D. Gilbarg and N. S. Trudinger [1] fork ≥ 2 (provided that d∂A is replaced by bA in Lemma 1, p. 382 in [1], and Lemma 14.16, p. 355 in[2]) with a different proof by S. G. Krantz and H. R. Parks [1] in 1981. Another proof with thefunction bA was given in 1994 by M. C. Delfour and J.-P. Zolesio [17], who extended the resultfrom (iii) to (ii) in the direction (⇒) down to the C1,1 case and established the equivalence (⇔)in the whole range from C∞ to C1,1. The counterexample of Example 6.2 for domains of classC1,1−ϵ is the same as the one provided earlier in S. G. Krantz and H. R. Parks [1], where theyobserve only that the domain is C2−ε, leaving the reader under the misleading impression thatpart (ii) of Theorem 8.1 would not be true for domains ranging from class C1,1 to C2. Part (i) ofTheorem 8.1 in the direction (⇒) was proved in the C1 case by S. G. Krantz and H. R. Parks [1]in 1981. The equivalence (⇔) here for C1,λ, 0 ≤ λ < 1, was given in 2004 by M. C. Delfour andJ.-P. Zolesio [40].


(i) (k = 1, 0 ≤ ℓ < 1). A is a set of class C1,ℓ in a bounded open neighborhoodV (x) of x and Sk (∂A) ∩ V (x) = ∅ if and only if

∃ρ > 0 such that bA ∈ C1,ℓ(Bρ(x)) and m(∂A ∩Bρ(x)) = 0. (8.1)

(ii) (k = 1, ℓ = 1). A is locally a set of class C1,1 at x if and only if

∃ρ > 0 such that bA ∈ C1,1(Bρ(x)) and m(∂A ∩Bρ(x)) = 0. (8.2)

(iii) (k ≥ 2, 0 ≤ ℓ ≤ 1). A is locally a set of class Ck,ℓ at x if and only if

∃ρ > 0 such that bA ∈ Ck,ℓ(Bρ(x)) and m(∂A ∩Bρ(x)) = 0. (8.3)

Moreover, in all cases, ∇bA = np∂A in Bρ(x), where n is the unit exterior normalto A on ∂A, and ∂A is locally a Ck,ℓ-submanifold of dimension N − 1 at x.

Remark 8.1.The condition Sk (∂A) ∩ V (x) = ∅ in part (i) is really necessary. Example 6.2in Chapter 6 gives a family of two-dimensional sets of class C1,λ, 0 ≤ λ < 1, forwhich bA ∈ C1,λ and d2

∂A = b2A /∈ C1 in any neighborhood of the point (0, 0) of its

boundary ∂A.

From Theorem 8.1 we get the global version that has had and still has aninteresting history of inaccurate statements or incomplete proofs

Theorem 8.2 (Global version). Let A ⊂ RN, ∂A = ∅, let k ≥ 1 be an integer,and let 0 ≤ ℓ ≤ 1 be a real number.

(i) (k = 1, 0 ≤ ℓ < 1). A is a set of class C1,ℓ and ∂A∩ Sk (∂A) = ∅ if and onlyif

m(∂A) = 0 and ∀x ∈ ∂A, ∃ρ > 0 such that bA ∈ C1,ℓ(Bρ(x)). (8.4)

(ii) (k = 1, ℓ = 1). A is a set of class C1,1 if and only if

m(∂A) = 0 and ∀x ∈ ∂A, ∃ρ > 0 such that bA ∈ C1,1(Bρ(x)). (8.5)

(iii) (k ≥ 2, 0 ≤ ℓ ≤ 1). A is a set of class Ck,ℓ if and only if

m(∂A) = 0 and ∀x ∈ ∂A, ∃ρ > 0 such that bA ∈ Ck,ℓ(Bρ(x)). (8.6)

Moreover, in all cases, ∇bA = np∂A in Bρ(x), where n is the unit exterior normalto A on ∂A, and ∂A is a Ck,ℓ-submanifold of dimension N − 1.

Remark 8.2.Theorems 8.1 and 8.2 deal only with sets of class C1,ℓ whose boundary is a C1,ℓ-submanifold of dimension (N − 1). For arbitrary closed submanifolds M of RN ofcodimension greater than or equal to 1, ∇bM generally does not exist on M . Inthat case the smoothness of M is related to the existence and the smoothness of


∇d2M in a neighborhood of M that implies that M is locally of positive reach. The

reader is referred to Theorem 6.5 in section 6.2 of Chapter 6.

The above two equivalence theorems follow from the following two local the-orems in each direction.

Theorem 8.3. Let A ⊂ RN be such that ∂A = ∅. Given x ∈ ∂A, assume that

∃ a bounded open neighborhood V (x) of x such that bA ∈ Ck,ℓ(V (x)) (8.7)

for some integer k ≥ 1 and some real ℓ, 0 ≤ ℓ ≤ 1, and that m(∂A ∩ V (x)) = 0.

(i) Then A is locally a set of class Ck,ℓ in V (x) and bA = bint A = bA in V (x).

(ii) Π∂A(y) = p∂A(y) is a singleton for all y ∈ V (x) and

b2A ∈ C1,1

loc (V (x)) ∩ Ck,ℓloc (V (x)) and p∂A ∈ C0,1

loc (V (x))N ∩ Ck−1,ℓloc (V (x))N .

Remark 8.3.If condition (8.7) is verified, the condition m(∂A) = 0 is equivalent to ∂A = RN.The condition bA ∈ C0

b (RN) is necessary to rule out the case ∅ = ∂A = RN thatyields bA = 0 and A = !A = RN that trivially satisfies condition (8.7).

Proof. (i) Since ∇bA exists and is continuous everywhere in V (x), |∇bA| is contin-uous in V (x). But |∇bA(x)| = 1 in V (x)\∂A and |∇bA(x)| = 0 almost everywherein V (x) ∩ ∂A. By continuity, either |∇bA| = 1 or |∇bA| = 0 in V (x). If ∇bA = 0 inV (x), then bA is constant in V (x) and equal to bA(x) = 0. Therefore, V (x) ⊂ ∂Aand this yields the contradiction 0 < m(V (x)) ≤ m(∂A ∩ V (x)) = 0. Considerthe set Ω def= x ∈ RN : −bA(x) > 0. By Theorem 4.2 in Chapter 2, Ω is aset of class Ck,ℓ since bA ∈ Ck,ℓ(V (x)) and ∇bA = 0 on b−1

A 0 ∩ V (x). In par-ticular, ∂Ω = b−1

A 0 = ∂A, intA = Ω, and int !A = int !Ω in V (x). FinallybA = bΩ = bint A = bA.

(ii) By assumption, bA ∈ Ck,ℓloc (V (x)) and hence b2

A belongs to Ck,ℓloc (V (x)). By

Theorem 3.1 (iv), Π∂A(y) = p∂A(y) is a singleton in V (x). By Theorem 7.3,b2A ∈ C1,1

loc (V (x)) and, a fortiori, in Ck,ℓloc (V (x)). This yields p∂A ∈ C0,1

loc (V (x)) ∩Ck−1,ℓ

loc (V (x)).

In the other direction the smoothness of the domain implies the smoothnessof bA in a neighborhood of ∂A when the domain is at least of class C1,1. A coun-terexample will be given in dimension 2.

Theorem 8.4. Let A ⊂ RN, ∂A = ∅, let k ≥ 1 be an integer, let 0 ≤ ℓ ≤ 1 be areal number, and let x ∈ ∂A if it is not.

(i) (k = 1, 0 ≤ ℓ < 1). If A is locally a set of class C1,ℓ in a bounded openneighborhood V (x) of x and Sk (∂A)∩V (x) = ∅, then m(∂A∩V (x)) = 0 and

∃ρ > 0 such that bA ∈ C1,ℓ(Bρ(x)) and p∂A ∈ C0,1(Bρ(x)). (8.8)


(ii) (k = 1, ℓ = 1). If A is locally a set of class C1,1 in a bounded open neighbor-hood V (x) of x, then m(∂A ∩ V (x)) = 0 and

∃ρ > 0 such that bA ∈ C1,1(Bρ(x)) and p∂A ∈ C0,1(Bρ(x)). (8.9)

(iii) (k ≥ 2, 0 ≤ ℓ ≤ 1). If A is locally a set of class Ck,ℓ in a bounded openneighborhood V (x) of x, then m(∂A ∩ V (x)) = 0 and

∃ρ > 0 such that bA ∈ Ck,ℓ(Bρ(x)) and p∂A ∈ Ck−1,ℓ(Bρ(x)). (8.10)

Moreover, in all cases, ∇bA = np∂A in Bρ(x), where n is the unit exterior normalto A on ∂A.

Proof. (i) From Definition 3.2 of Chapter 2 of a locally a C1,ℓ-submanifold of di-mension N − 1, m(∂A ∩ V (x)) = 0 in a bounded open neighborhood V (x) and theset A can be locally described by the level sets of the C1,ℓ-function

f(y) def= gx(y) · eN , (8.11)

since by definition

intA ∩ V (x) = y ∈ V (x) : f(y) > 0 , ∂A ∩ V (x) = y ∈ V (x) : f(y) = 0 .

The boundary ∂A in V (x) is the zero level set of f and the gradient

∇f(y) = Dgx(y)∗eN = 0 in V (x)

is normal to that level set. Thus the outward unit normal n to A on V (x) ∩ ∂A isgiven by

n(y) = − ∇f(y)|∇f(y)| = − Dgx(y)∗eN

|Dgx(y)∗eN | = − Dhx(gx(y))−∗eN

|Dhx(gx(y))−∗eN | . (8.12)

Since V (x) ∩ Sk (∂A) = ∅, for all y ∈ V (x),

∃ unique p∂A(y) ∈ ∂A, d∂A(y) = infp∈∂A

|y − p| = |y − p∂A(y)|,

p∂A(y) = y − 12∇d2

∂A(y) = y − 12∇b2

A(y). (8.13)

Choose ρ > 0 such that B3ρ(x) ⊂ V (x). Then for all y ∈ Bρ(x)

|p∂A(y)− x| ≤ |p∂A(y)− y| + |y − x| ≤ 2 |y − x| ≤ 2r

⇒ p∂A(y) ∈ B2ρ(x) ∩ ∂A ⊂ V (x) ∩ ∂A = V (x) ∩ f−10.


Therefore, for each y ∈ Bρ(x), there exists a unique p∂A(y) ∈ V (x) ∩ f−10such that

d2∂A(y) = inf

p∈V (x), f(p)=0|p− y|2 = |y − p∂A(y)|2.

By using the Lagrange multiplier theorem for the Lagrangian

|z − y|2 + λ f(z)

there exists a λ ∈ R such that

2(p∂A(y)− y) + λ∇f(p∂A(y)) = 0 and f(p∂A(y)) = 0

⇒ 2d∂A(y) = |λ| |∇f(p∂A(y))|

⇒ p∂A(y) = y + bA(y)∇f(p∂A(y))|∇f(p∂A(y))| = y − bA(y) n(p∂A(y))

since ∇f(p∂A(y)) = 0 on ∂A ∩ V (x). Combining this with (8.13)

∀y ∈ Br(x), y − 12∇b2

A(y) = p∂A(y) = y − bA(y) n(p∂A(y))

⇒ ∀y ∈ Br(x)\∂A, ∇bA(y) = n(p∂A(y)) (8.14)

since V (x)∩Sk (∂A) = ∅ implies that∇bA(y) exists in V (x)\∂A by Theorem 3.1 (vii).Consider the normal n(z) = −∇f(z)/|∇f(z)| in V (x)\∂A. For z2 and z1 in

∂A ∩B2ρ(x) the difference

|n(z2)− n(z1)| =∣∣∣∣∇f(z2)|∇f(z2)|

− ∇f(z1)|∇f(z1)|

∣∣∣∣

=∣∣∣∣

1|∇f(z2)|

(∇f(z2)−∇f(z1)) +(

1|∇f(z2)|

− 1|∇f(z1)|

)∇f(z1)

∣∣∣∣

=1

|∇f(z2)|

∣∣∣∣∇f(z2)−∇f(z1) + (|∇f(z1)|− |∇f(z2)|)∇f(z1)|∇f(z1)|

∣∣∣∣

≤ 21

|∇f(z2)||∇f(z2)−∇f(z1)| ≤ 2

1inf

z∈∂A∩B2ρ(x) |∇f(z)| |∇f(z2)−∇f(z1)| .

But f ∈ C1,ℓ(B2ρ(x)) and since |∇f | = 0 is continuous on V (x),

|∇f(z2)−∇f(z1)| ≤ cx |z2 − z1|ℓ and αdef= inf

z∈∂A∩B2ρ(x)|∇f(z)| > 0

and the outward unit normal is ℓ-Holderian,

∀z2, z1 ∈ B2ρ(x) ∩ ∂A, |n(z2)− n(z1)| ≤2α

cx |z2 − z1|ℓ,

where cx is the Lipschitz constant of f in B2ρ(x). By construction, B3ρ(x) ∩Sk (∂A) ⊂ V (x) ∩ Sk (∂A) = ∅ and for all z ∈ Bρ(x), B2ρ(z) ∩ Sk (∂A) = ∅


implies that reach (∂A, z) ≥ 2r > 0. By Theorem 6.2 (v) in Chapter 6 applied to∂A with 0 < r = ρ < q = 2ρ,

∀z1, z2 ∈ Bρ(x), |p∂A(z1)− p∂A(z2)| ≤ 2 |z1 − z2| (8.15)

and hence p∂A ∈ C0,1(Bρ(x))N . So the composition np∂A belongs to C0,ℓ(Bρ(x))N

since we have already shown that n ∈ C0,ℓ(B2ρ(x) ∩ ∂A)N . Finally, ∇bA ∈L∞(Bρ(x))N , n(p∂A) ∈ C0,ℓ(Bρ(x))N , m(∂A) = 0, and from (8.14)

∇bA(y) = n(p∂A(y)) a.e. in Bρ(x)

imply that ∇bA ∈ C0,ℓ(Bρ(x))N , bA ∈ C1,ℓ(Bρ(x)), and ∇bA = n p∂A in Bρ(x).(ii) The first steps are the same as in part (i) up to (8.12) with C1,1 in place

of C1,ℓ. For each y ∈ V (x), Π∂A(y) is compact and nonempty and

∀p ∈ Π∂A(y), d2∂A(y) = inf

a∈∂A|a− y|2 = |p− y|2.

Choose r > 0 such that B3r(x) ⊂ V (x). Then for all y ∈ Br(x) and p ∈ Π∂A(y)

|p− x| ≤ |p− y| + |y − x| ≤ 2 |y − x| ≤ 2r

⇒ p ∈ B2r(x) ∩ ∂A ⊂ V (x) ∩ ∂A = V (x) ∩ f−10.

Therefore, for each y ∈ Br(x) and p ∈ Π∂A(y), p ∈ V (x) ∩ f−10 such that

d2∂A(y) = inf

p∈V (x), f(p)=0|p− y|2 = |y − p∂A(y)|2.

By using the Lagrange multiplier theorem for the Lagrangian

|a− y|2 + λ f(a)

there exist λ ∈ R and p ∈ Π∂A(y) such that

2(p− y) + λ∇f(p) = 0 and f(p) = 0 ⇒ 2d∂A(y) = |λ| |∇f(p)|

⇒ p = y + bA(y)∇f(p)|∇f(p)| = y − bA(y) n(p)

⇒ ∀y ∈ Br(x), ∀p ∈ Π∂A(y), p = y − bA(y) n(p), (8.16)

since ∇f(p) = 0 on ∂A ∩ V (x). By the same proof as in part (i) with ℓ = 1

∀z2, z1 ∈ B2r(x) ∩ ∂A, |n(z2)− n(z1)| ≤2α

cx |z2 − z1|,

where cx is the Lipschitz constant associated with f in B2r(x). Therefore, n ∈C0,1(B2r(x) ∩ ∂A). From (8.16) for y ∈ Br(x) and for all p1, p2 ∈ Π∂A(y),

|p2 − p1| ≤ |bA(y)| |n(p2)− n(p1)| ≤ |bA(y)|2 cx

α|p2 − p1|.


By choosing h such that 0 < h < minr,α/(2 cx), h 2 cx/α < 1, for y ∈ Bh(x) ⊂Br(x) ∩ z ∈ RN : |bA(z)| ≤ h and all p1, p2 ∈ Π∂A(y), p2 = p1, Π∂A(y) is asingleton, and

p∂A(y) = y − bA(y) n(p∂A(y)).

By construction, Bh(x)∩Sk (∂A) = ∅ and for all z ∈ Bh/3(x), B2h/3(z)∩Sk (∂A) =∅ implies that reach (∂A, z) ≥ 2h/3 > 0. By Theorem 6.2 (v) in Chapter 6 appliedto ∂A with 0 < r = h/3 < q = 2h/3

∀z1, z2 ∈ Bh/3(x), |p∂A(z1)− p∂A(z2)| ≤ 2 |z1 − z2| (8.17)

and hence p∂A ∈ C0,1(Bh/3(x))N . Thus n p∂A belongs to C0,1(Bh/3(x))N sincewe have already shown that n ∈ C0,1(B2r(x) ∩ ∂A)N . The remainder and theconclusion of the proof is the same as in part (i) with ρ = h/3.

(iii) When A is of class Ck,ℓ in V (x) for an integer k, k ≥ 2, and a real numberℓ, 0 ≤ ℓ ≤ 1, it is C1,1 and the previous constructions and conclusions of (ii) areverified. There exists ρ > 0 such that for each y ∈ Bρ(x) there exists a uniquep∂A(y) ∈ B2ρ(x) such that

p∂A(y) = y − bA(y)∇f(p∂A(y))|∇f(p∂A(y))| .

Since ∇f is continuous and ∇f = 0 on ∂A ∩ V (x), there exists r, 0 < 2r < ρ,such that

infz∈B2r(x)

|∇f(z)| > 0

and defining N(z) = ∇f(z)/|∇f(z)|,

∇N(z) =1

|∇f(z)|

[D2f(z)− ∇f(z)

|∇f(z)|∗(

D2f(z)∇f(z)|∇f(z)|

)]

belongs to Ck−1,ℓ(B2r(x)), where ∗v denotes the transpose of a vector v ∈ RN, sothat (u ∗v)ij = uivj is the tensor product. Consider the map

(y, z) .→ G(y, z) def= z − y − bA(y)∇f(z)|∇f(z)| : Br(x)×B2r(x)→ RN .

From part (ii), there exists a unique p(y) = p∂A(y) ∈ B2r(x) ∩ ∂A such thatp∂A ∈ C0,1(B2r(x))N , bA ∈ C1,1(Br(x)), and

G(y, p(y)) = 0 ⇒ DGy(y, p(y)) + DGz(y, p(y))Dp(y) = 0,

where

DGz(y, z) = I − bA(y)|∇f(z)|

[D2f(z)− ∇f(z)

|∇f(z)|∗(

D2f(z)∇f(z)|∇f(z)|

)],

DGy(y, z)) = −I −N(z) ∗∇bA(y).


DGz(y, z) is at least C1,1 in (y, z) since bA ∈ C1,1 and N is at least C1,1 for k ≥ 2.Since f is at least C2 in V (x), f and its first and second derivatives are bounded insome smaller neighborhood Br′(x) of x. By reducing the upper bound on |bA(y)|,there exists a new smaller neighborhood Br′′(x) of x such that DzG(y, z) is invertiblefor all (y, z) ∈ Br′′(x)×B2r′′(x). So the conditions of the implicit function theoremare met. There exist a neighborhood W (x) ⊂ Br′′(x) of x and a unique C1-mapping

p : W (x)→ RN such that ∀y ∈W (x), G(y, p(y)) = 0

that coincides with p∂A (cf., for instance, L. Schwartz [3, Thm. 26, p. 283]).Therefore

Dp∂A(y) = (DGz(y, p∂A(y)))−1 [I + N(p∂A(y)) ∗∇bA(y)]

and Dp∂A is C0 and p∂A is C1. Thus ∇bA = N p∂A ∈ C1 and bA is now C2. SinceN ∈ Ck−1,ℓ and D2f ∈ Ck−2,ℓ, we can repeat this argument until we get Dp∂A inCk−2,ℓ, and finally p∂A ∈ Ck−1,ℓ, ∇bA = N p∂A ∈ Ck−1,ℓ, and bA ∈ Ck,ℓ.

From Theorem 8.1 (ii) additional information is available on the flow of ∇bΩ.

Theorem 8.5. Let Ω ⊂ RN be of class C1,1. Denote its boundary by Γ def= ∂Ω.

(i) For each x ∈ Γ there exists a bounded open neighborhood W (x) of x such that

bΩ ∈ C1,1(W (x)), ∇bΩ = n pΓ = ∇bΩ pΓ, |∇bΩ| = 1 in W (x), (8.18)

where n is the unit exterior normal to Ω on Γ.

(ii) Associate with x ∈ Γ and y in W (x) the flow Tt(y) of ∇bΩ in W (x), that is,

dx

dt(t) = ∇bΩ(x(t)), x(0) = y, Tt(y) def= x(t). (8.19)

Then for t in a neighborhood of 0 we have the following properties:

pΓ Tt = pΓ, ∇bΩ Tt = ∇bΩ, Tt(y) = y + t∇bΩ(y). (8.20)

(iii) Almost everywhere in W (x)

DTt = I + t D2bΩ,d

dtDTt = D2bΩ (8.21)

⇒ D2bΩ Tt DTt = D2bΩ, D2bΩ Tt = D2bΩ[I + tD2bΩ

]−1, (8.22)

(DTt)−1∇bΩ = ∇bΩ = ∗(DTt)−1∇bΩ. (8.23)

Proof. For simplicity, we use the notation b = bΩ and p = pΓ.(i) The result follows from Theorem 8.4 (i).(ii) By assumption from part (i) the function b belongs to C1,1(W (x)) and

hence p ∈ C0,1(W (x);RN). Thus almost everywhere in W (x),

Dp = I −∇b ∗∇b− bD2b ⇒ Dp∇b = 0.

9. Sobolev or Wm,p Domains 373

Nowd

dt(p Tt) = Dp Tt

dTt

dt= Dp Tt∇b Tt = (Dp∇b) Tt = 0

⇒ p Tt = p in W (x).

But since p and Tt belong to C0,1(W (x);RN), the identity necessarily holds every-where in W (x). Moreover,

∇b Tt = n p Tt = n p = ∇b in W (x).

Finally

dTt

dt= ∇b Tt = ∇b ⇒ Tt(y) = y + t∇b(y) in W (x).

In particular, almost everywhere in W (x),

DTt = I + t D2b ⇒ d

dtDTt = D2b,

DTt∇b = ∇b ⇒ (DTt)−1∇b = ∇b = ∗(DTt)−1∇b.

9 Sobolev or W m,p DomainsThe N -dimensional Lebesgue measure of the boundary of a Lipschitzian subset ofRN is zero. However, this is generally not true for sets of locally bounded curvature,as can be seen from the following example.

Example 9.1.Let B be the open unit ball centered in 0 of R2 and define

A = x ∈ B : x with rational coordinates.

Then ∂A = B, bA = dB , and for all h > 0

∇bA ∈ BV(Uh(∂A))2,

⟨∆bA,ϕ⟩ =∫

∂Bϕ dx +

∫

!B

1|x|ϕ dx.

The next natural question that comes to mind is whether the boundary or theskeleton has locally finite (N − 1)-Hausdorff measure. Then come questions aboutthe mean curvature of the boundary. For the function bA, ∆bA = tr D2bA is propor-tional to the mean curvature of the boundary, which is only a measure in RN for setsof bounded local curvature. This calls for the introduction of some classificationof sets that would complement the classical Holderian terminology introduced inChapter 2, would fill the gap in between, and would possibly say something aboutsets whose boundary is not even continuous. The following definition seems to havebeen first introduced in M. C. Delfour and J.-P. Zolesio [32].


Definition 9.1 (Sobolev domains).Given m > 1 and p ≥ 1, a subset A of RN is said to be an (m, p)-Sobolev domainor simply a Wm,p-domain if ∂A = ∅ and there exists h > 0 such that

bA ∈Wm,ploc (Uh(∂A)).

This classification gives an analytical description of the smoothness of bound-aries in terms of the derivatives of bA. The definition is obviously vacuous for m = 1since bA ∈ W 1,∞

loc (RN) for any set A such that ∂A = ∅. For 1 < m < 2 we shallsee that it encompasses sets that are less smooth than sets of locally bounded cur-vature. For m = 2 the W 2,p-domains are of class C1,1−N/p for p > N . For m > 2they are intertwined with sets of class Ck,ℓ.

Theorem 9.1. Given any subset A of RN, ∂A = ∅,

∇bA ∈ BVloc(RN)N

⇒ ∀p, 1 ≤ p <∞, ∀η, 0 ≤ η <1p, bA ∈W 1+η,p

loc (RN).

Proof. Since |∇bA(x)| ≤ 1 almost everywhere, ∇bA ∈ BVloc(RN)N ∩ L∞loc(R

N)N

and the theorem follows directly from Theorem 6.9 in Chapter 5.

It is quite interesting that a domain of locally bounded curvature is a W 2−ε,1-domain for any arbitrary small ε > 0. So it is almost a W 2,1-domain, and domainsof class W 2−ε,1 seem to be a larger class than domains of locally bounded curvature.In addition, their boundary does not generally have zero measure.

Now consider the “threshold case” m = 2.

Theorem 9.2. Given an integer N ≥ 1, let A be a subset of RN such that ∂A = ∅.

(i) If m(∂A) = 0 and there exist p > N and h > 0 such that

bA ∈W 2,ploc (Uh(∂A)), (9.1)

then

reach (∂A) ≥ h > 0, bA ∈ C1,1−N/ploc (Uh(∂A)),

and A is a Holderian set of class C1,1−N/p.

(ii) In dimension N = 2 the condition bA ∈W 2,ploc (Uh(∂A)) is equivalent to ∆bA ∈

Lploc(Uh(∂A)).

Proof. (i) Consider the function |∇bA|2. Since |∇bA| ≤ 1, then

bA ∈W 2,ploc (Uh(∂A)) ∩W 1,∞

loc (Uh(∂A)) ⇒ b2A ∈W 2,p

loc (Uh(∂A)) ∩W 1,∞loc (Uh(∂A)).

In particular, for all x ∈ ∂A,

bA, b2A ∈W 2,p(B(x, h)) ∩W 1,∞(B(x, h)).

10. Characterization of Convex and Semiconvex Sets 375

For p > N , from R. A. Adams [1, Thm. 5.4, Part III, and Rem. 5.5 (3), p. 98],

bA, b2A ∈ C1,λ(B(x, h)), 0 < λ ≤ 1−N/p

⇒ bA, b2A ∈ C1,λ

loc (Uh(A)), 0 < λ ≤ 1−N/p.

From Theorem 8.2 (i), if m(∂A) = 0, the set A is a set of class C1,1−N/p, ∂A ∩Sk (∂A) = ∅, and reach (∂A) ≥ h.

(ii) From part (i), |∇bA(x)|2 = 1 in Uh(∂A), and D2bA(x)∇bA(x) = 0 almosteverywhere. Hence in dimension N = 2,

∂212bA(x) = ∂2

21bA(x) = −∂1bA(x) ∂2bA(x) ∆bA(x),

∂211bA(x) = (∂2bA(x))2 ∆bA(x),

∂222bA(x) = (∂1bA(x))2 ∆bA(x).

As can be seen, a certain amount of work would be necessary to characterizeall the Sobolev domains and answer the many associated open questions.

10 Characterization of Convex and Semiconvex Sets10.1 Convex Sets and Convexity of bA

We have seen in Chapter 6 that the convexity of dA is equivalent to the convexityof A. This characterization remains true with bA in place of dA. However, if in bothcases the convexity of dA or bA is sufficient to get the convexity of A, it is not thecase for A as can be seen from the following example.

Example 10.1.Let B be the open unit ball in RN and A be the set B minus all the points in Bwith rational coordinates. By definition,

A = B, !A = RN, ∂A = B, ∂A = ∂B = ∂A,

dA = dB = dA, d!A = d!A= dRN = 0

⇒ bA = dB = bB = bA.

By Theorem 8.1 of Chapter 6 the functions dB and, a fortiori, bA are convex, butA is not convex and ∂A = ∂A.

Theorem 10.1.

(i) Let A be a convex subset of RN. Then

bA = bA and [A]b = [A]b. (10.1)

Hence, the equivalence class [A]b can be identified with the unique closed setA in the class.


(ii) Let A be a convex subset of RN such that intA = ∅. Then

bint A = bA = bA and [intA]b = [A]b = [A]b. (10.2)

Hence, the equivalence class [A]b can be identified with either the unique closedset A or the unique open set intA = !!A = ∅ in the class. If, in addition,A = RN, then ∂A = ∅ and int !A = ∅.

(iii) Let A be a subset of RN such that ∂A = ∅.5 Then

bA is convex ⇐⇒ A is convex. (10.3)

Proof. (i) If A = ∅, then A = ∅, !A = RN, and !A = RN. Hence bA = bA.If A = RN, we use the identity bA = −b!A. If ∂A = ∅, then dA = dA and, byTheorem 3.5 (ii) in Chapter 5, !A = !A, and bA = bA.

(ii) From Theorem 3.5 (ii) in Chapter 5, intA = A and !intA = !!!A = !A.Hence bA = bint A. From part (i), bA = bA. If, in addition, A = RN, always fromTheorem 3.5 (ii) in Chapter 5, ∂A = ∅ and int !A = ∅.

(iii) (⇒) Denote by xλ the convex combination λx + (1− λ)y of two points xand y in A for some λ ∈ [0, 1]. By convexity of bA,

bA(xλ) ≤ λbA(x) + (1− λ)bA(y) = − [λd!A(x) + (1− λ)d!A(y)] ≤ 0

⇒ dA(xλ) = b+A(xλ) = maxbA(xλ), 0 = 0 ⇒ xλ ∈ A

and A is convex.(⇐) We first prove that for A closed the function bA is convex. From this we

conclude that if A is convex, then bA is convex and from part (i) bA = bA is convex.So for A closed, consider three cases. The first one deserves a lemma.

Lemma 10.1. Let A be a subset of RN such that ∂A = ∅ and let A be convex.Then

bA = −d!A is convex in A.

Proof. Again we can assume that A is closed. Associate with x and y in A the radiirx = d!A(x), ry = d!A(y), and rλ = λrx + (1 − λ)ry and the closed balls Bx ofcenter x and radius rx, By of center y and radius ry, and Bλ of center xλ and radiusrλ. By the definition of d!A, Bx ⊂ A and By ⊂ A since A is closed. Associate with

5In 1985 D. H. Armitage and U. Kura [1] showed that a proper closed domain Ω is convex ifand only if −bA is superharmonic and that, for N = 2, the result still holds if −bA is superharmoniconly on A. They also showed that, for A compact, d∂A is subharmonic on !A if and only if A isconvex. This work was pursued in 1987 by M. J. Parker [1] and in 1988 by M. J. Parker [2].In 1994 it was shown in M. C. Delfour and J.-P. Zolesio [17] that the property that A is convexif and only if dA is convex remains true with bA in place of dA.


each z ∈ Bλ the points zx = x and zy = y if rλ = 0, and if rλ > 0 the points

zxdef= x +

rx

rλ(z − xλ) ⇒ zx ∈ Bx,

zydef= y +

ry

rλ(z − xλ) ⇒ zy ∈ By

⇒ λzx + (1− λ)zy = xλ +λrx + (1− λ)ry

rλ(z − xλ) = z.

Obviously z = xλ if rλ = 0. Therefore,

Bλ ⊂ λBx + (1− λ)By ⊂ A,

since A is closed and convex. In particular

d!A(xλ) ≥ d!Bλ(xλ) ≥ rλ = λd!A(x) + (1− λ)d!A(y),

and this proves the lemma.

For the second case consider x and y in !A. By definition

bA(x) = dA(x) and bA(y) = dA(y).

By Theorem 8.1 of Chapter 6, dA is convex when A is convex and

bA(xλ) ≤ dA(xλ) ≤ λdA(x) + (1− λ)dA(y) = λbA(x) + (1− λ)bA(y).

The third and last case is the mixed one for x ∈ !A and y ∈ A. Define

xλdef= λx + (1− λ)y, pλ

def= p∂A(xλ).

Since A is convex, denote by H the tangent hyperplane to A through pλ, by H+

the closed half-space containing A, and by H− the other closed subspace associatedwith H. By the definition of H,

d∂A(xλ) = dH(xλ), H ⊂ H±,

and by convexity of A, A ⊂ H+ and H− ⊂ !A. The projection onto H is a linearoperator and

pλ = pH(xλ) = λpH(x) + (1− λ)pH(y),pλ − xλ = λ(pH(x)− x) + (1− λ)(pH(y)− y).

If y ∈ H+ and x ∈ H+,

dH(xλ) = λdH(x) + (1− λ)dH(y), (10.4)


and if y ∈ H+ and x ∈ H−,

dH(xλ) =

λdH(x)− (1− λ)dH(y), if xλ ∈ H−,

(1− λ)dH(y)− λdH(x), if xλ ∈ H+.(10.5)

By convexity any y ∈ A belongs to H+ and since H− ⊂ !A, we readily have

dH(y) = dH−(y) ≥ d!A(y) = −bA(y).

First consider the case xλ ∈ !A. Then dA(xλ) > 0, xλ ∈ H−, and necessarilyx ∈ H−. From (10.5)

bA(xλ) = dA(xλ) = dH(xλ) = λdH(x)− (1− λ)dH(y).

But since x ∈ H− and A ⊂ H+

dH(x) = dH+(x) ≤ dA(x)

and

bA(xλ) = dH(xλ) ≤ λdA(x)− (1− λ)d!A(y) = λbA(x) + (1− λ)bA(y).

Next consider the case xλ ∈ A for which xλ ∈ H+ and

−bA(xλ) = d!A(xλ) = dH(xλ).

If x ∈ H−, then since A ⊂ H+,

dH(x) = dH+(x) ≤ dA(x),

and from (10.5)

−bA(xλ) = dH(xλ)= (1− λ)dH(y)− λdH(x)≥ (1− λ)d!A(y)− λdA(x) = −[(1− λ)bA(y) + λbA(x)].

If, on the other hand, x ∈ H+, then x, y ∈ H+ and xλ ∈ A ⊂ H+. So from (10.4)

−bA(xλ) = dH(xλ)= (1− λ)dH(y) + λdH(x)≥ (1− λ)d!A(y)− λdA(x) = −[(1− λ)bA(y) + λbA(x)].

This covers all cases and concludes the proof.


10.2 Families of Convex Sets Cb(D), Cb(E; D), and Cb,loc(E; D)

We have seen in Theorem 10.1 (i) that for a convex subset A of D, bA = bA and thatA is the unique closed representative in the equivalence class. To work with openconvex subsets is more delicate. Always from Theorem 10.1 (ii), intA becomes theunique open representative in the equivalence class when the convex set A has anonempty interior. In order to work with families of open rather than closed convexsubsets, it will be necessary to add a constraint on the interior of the subsets toprevent the occurrence of subsets with an empty interior.

Theorem 10.2. Let D be a nonempty open (resp., bounded open) subset of RN.

(i) The family

Cb(D) def=bA : A ⊂ D, ∂A = ∅, A convex

(10.6)

is closed in Cloc(D) (resp., compact in C(D)). It coincides with the familybA : A ⊂ D, ∂A = ∅, A closed and convex

.

(ii) Let E be a nonempty open subset of D. Then the families

Cb(E;D) def= bΩ : E ⊂ Ω ⊂ D, ∂Ω = ∅, Ω open and convex , (10.7)

Cb,loc(E;D) def=

bΩ :

∃x ∈ RN, ∃A ∈ O(N) such that x+AE⊂Ωand Ω is open and convex ⊂ D

(10.8)

are closed in Cloc(D) (resp., compact in C(D)).

(iii) The conclusions of parts (i) and (ii) remain true with W 1,ploc (D)- (resp.,

W 1,p(D)-) strong in place of Cloc(D) (resp., C(D)).

Proof. It is sufficient to prove the result for D bounded. Furthermore, by compact-ness of Cb(D) in C(D), it is sufficient to prove that Cb(D), Cb(E;D), and Cb,loc(E;D)are closed in C(D).

(i) Let bAn be a Cauchy sequence in Cb(D). It converges to some bA ∈ Cb(D).The function bA is convex as the pointwise limit of a family of convex functions.By Theorem 10.1 (iii), A is convex, and by Theorem 10.1 (i) bA = bA is convex.Therefore the Cauchy sequence converges to bA in Cb(D) and Cb(D) is closed inC(D).

(ii) From part (i), given a Cauchy sequence bΩn in Cb(E;D), there exists anopen convex subset Ω of D such that bΩn → bΩ. Moreover, since d!Ωn

→ d!Ω,

E ⊂ Ωn ⇒ !E ⊃ !Ωn ⇒ d!E ≤ d!Ωn⇒ d!E ≤ d!Ω ⇒ !E ⊃ !Ω ⇒ E ⊂ Ω.

The proof for Cb,loc(E;D) is similar to the proof for Ccd,loc(E;D) in Theorem 2.6 of

Chapter 6.(iii) Let bAn ⊂ Cb(D) be a Cauchy sequence in W 1,p(D). From Theo-

rem 4.2 (ii), it is also Cauchy in C(D). Hence from part (i) there exists a convex set


A, ∂A = ∅, such that bAn → bA in C(D) and, a fortiori, in W 1,p(D). This showsthat Cb(D) is closed in W 1,p(D). To show that it is compact consider a sequencebAn ⊂ Cb(D). From part (i) there exist a subsequence, still indexed by n, and aconvex subset A, ∂A = ∅, of D such that bAn → bA in C(D). Since D is bounded,it also converges in Lp(D). Since |∇bAn | is pointwise bounded by one, there existsanother subsequence of bAn, still indexed by n, such that ∇bAn converges to ∇bA

in W 1.p(D)N -weak. But An and A are convex. Thus m(∂A) = 0 = m(∂An) and|∇bAn | = 1 = |∇bA| almost everywhere in RN. As a result for p = 2

∫

D|∇bAn −∇bA|2 dx =

∫

D|∇bAn |2 + |∇bA|2 dx− 2∇bAn ·∇bA dx

=∫

D2− 2∇bAn ·∇bA dx→ 0.

Hence we get the compactness in W 1,2(D)-strong and by Theorem 4.1 (i) in W 1,p(D)-strong, 1 ≤ p <∞.

10.3 BV Character of bA and Semiconvex SetsThe next theorem is a consequence of the fact that the gradient of a convex functionis locally of bounded variations.

Theorem 10.3. For all convex sets A such that ∂A = ∅,

(i) ∇bA ∈ BVloc(RN)N , ∇χ∂A ∈ BVloc(RN), ∇dA ∈ BVloc(RN)N , ∇χA ∈BVloc(RN), and for all x ∈ RN and v ∈ RN, dHbA(x, v) exists.

(ii) The Hessian matrix D2bA of second-order derivatives is a matrix of signedRadon measures which are nonnegative on the diagonal.

(iii) bA has a second-order derivative almost everywhere, and for almost all x andy in RN,∣∣∣∣bA(y)− bA(x)−∇bA(x) · (y − x)− 1

2(y − x) · D2bA(x) (y − x)

∣∣∣∣ = o(|y − x|2)

as y → x.

Proof. Same argument as in the proof of Theorem 8.2 of Chapter 6.

When A is convex and of class C2, then for any x0 ∈ ∂A, there exists a strictlyconvex neighborhood N(x0) of x0 such that

bA ∈ C2(N(x0)) and ∀x ∈ N(x0), ∀ξ ∈ RN, D2bA(x)ξ · ξ ≥ 0, (10.9)

or, since D2bA(x)∇bA(x) = 0 in N(x0), then for all x ∈ N(x0),

∀ξ ∈ RN such that ξ ·∇bA(x) = 0, D2bA(x)ξ · ξ ≥ 0. (10.10)

This is related to the notion of strong elliptic midsurface in the theory of shells:there exists c > 0

∀x ∈ ∂A, ∀ξ ∈ RN such that ξ ·∇bA(x) = 0, D2bA(x) ξ · ξ ≥ c|ξ|2.

All this motivates the introduction of the following notions.

11. Compactness and Sets of Bounded Curvature 381

Definition 10.1.Let A be a closed subset of RN such that ∂A = ∅.

(i) The set A is locally convex (resp., locally strictly convex ) if for each x0 ∈ ∂Athere exists a convex neighborhood N(x0) of x0 such that

bA is convex (resp., strictly convex) in N(x0).

(ii) The set A is semiconvex if

∃α ≥ 0, bA(x) + α|x|2 is convex in RN .

(iii) The set A is locally semiconvex if for each x0 ∈ ∂A there exists a convexneighborhood N(x0) of x0 and

∃α ≥ 0, bA(x) + α|x|2 is convex in N(x0).

Remark 10.1.When A is a C2 domain with a compact boundary, D2bA is bounded in a boundedneighborhood of ∂A and A is necessarily locally semiconvex. When A is a locallysemiconvex set, for each x0 ∈ ∂A there exists a convex neighborhood N(x0) of x0such that ∇bA ∈ BV(N(x0))N . If, in addition, ∂A is compact, then there existsh > 0 such that ∇bA ∈ BV(Uh(∂A))N .

Remark 10.2.Given a fixed constant β > 0, consider all the subsets of D that are semiconvex withconstant 0 ≤ α ≤ β. Then this set is closed for the uniform and the W 1,p-topologies,1 ≤ p <∞.

11 Compactness and Sets of Bounded CurvatureFor the family of sets with bounded curvature, the key result is the compactness ofthe embedding

BCb(D) =bA ∈ Cb(D) : ∇bA ∈ BV(D)N

→W 1,p(D) (11.1)

for bounded open Lipschitzian subsets of RN and p, 1 ≤ p <∞. It is the analogueof the compactness Theorem 6.3 of Chapter 5 for Caccioppoli sets

BX(D) = χ ∈ X(D) : χ ∈ BV(D)→ Lp(D), (11.2)

which is a consequence of the compactness of the embedding

BV(D)→ L1(D) (11.3)

for bounded open Lipschitzian subsets of RN (cf. C. B. Morrey, Jr. [1, Def. 3.4.1,p. 72, Thm. 3.4.4, p. 75] and L. C. Evans and R. F. Gariepy [1, Thm. 4, p. 176]).

As in the case of characteristic functions in Chapter 5, we give a first versioninvolving global conditions on a fixed bounded open Lipschitzian holdall D. Inthe second version the sets are contained in a bounded open holdall D with localconditions in the tubular neighborhood of their boundary.


11.1 Global Conditions on D

Theorem 11.1. Let D be a nonempty bounded open Lipschitzian subset of RN. Theembedding (11.1) is compact. Thus for any sequence An, ∂An = ∅, of subsets ofD such that

∃c > 0,∀n ≥ 1, ∥D2bAn∥M1(D) ≤ c, (11.4)

there exist a subsequence Ank and a set A, ∂A = ∅, such that ∇bA ∈ BV(D)N

andbAnk

→ bA in W 1,p(D)-strong

for all p, 1 ≤ p <∞. Moreover, for all ϕ ∈ D0(D),

limn→∞

⟨∂ijbAnk,ϕ⟩ = ⟨∂ijbA,ϕ⟩, 1 ≤ i, j ≤ N, ∥D2bA∥M1(D) ≤ c. (11.5)

Proof. Given c > 0 consider the set Scdef=bA ∈ Cb(D) : ∥D2bA∥M1(D) ≤ c

. By

compactness of the embedding (11.3), given any sequence bAn there exist a subse-quence, still denoted by bAn, and f ∈ BV(D)N such that ∇bAn → f in L1(D)N .But by Theorem 2.2 (ii), Cb(D) is compact in C(D) for bounded D and there existanother subsequence bAnk

and bA ∈ Cb(D) such that bAnk→ bA in C(D) and,

a fortiori, in L1(D). Therefore, bAnkconverges in W 1,1(D) and also in L1(D).

By uniqueness of the limit, f = ∇bA and bAnkconverges in W 1,1(D) to bA. For

Φ ∈ D1(D)N×N as k goes to infinity∫

D∇bAnk

·−→div Φ dx→

∫

D∇bA ·

−→div Φ dx

⇒∣∣∣∣∫

D∇bA ·

−→div Φ dx

∣∣∣∣ = limk→∞

∣∣∣∣∫

D∇bAnk

·−→div Φ dx

∣∣∣∣ ≤ c∥Φ∥C(D),

∥D2bA∥M1(D) ≤ c, and ∇bA ∈ BV(D)N . This proves the sequential compactness ofthe embedding (11.1) for p = 1 and properties (11.5). The conclusions remain truefor p ≥ 1 by the equivalence of the W 1,p-topologies on Cb(D) in Theorem 4.1 (i).

11.2 Local Conditions in Tubular NeighborhoodsThe global condition (11.4) is now weakened to a local one in a neighborhood of eachset of the sequence. Simultaneously, the Lipschitzian condition on D is removedsince only the uniform boundedness of the sets of the sequence is required.

Theorem 11.2. Let ∅ = D ⊂ RN be bounded open and An, ∂An = ∅, be asequence of subsets of D. Assume that there exist h > 0 and c > 0 such that

∀n, ∥D2bAn∥M1(Uh(∂An)) ≤ c. (11.6)

Then, there exist a subsequence Ank and A ⊂ D, ∅ = ∂A, such that ∇bA ∈BVloc(RN)N and for all p, 1 ≤ p <∞,

bAnk→ bA in W 1,p(Uh(D))-strong. (11.7)

11. Compactness and Sets of Bounded Curvature 383

Moreover, for all ϕ ∈ D0(Uh(∂A)),

limk→∞

⟨∂ijbAnk,ϕ⟩ = ⟨∂ijbA,ϕ⟩, 1 ≤ i, j ≤ N, ∥D2bA∥M1(Uh(∂A)) ≤ c, (11.8)

and χ∂A belongs to BVloc(RN).

Proof. (i) By assumption, An ⊂ D implies that Uh(An) ⊂ Uh(D). Since Uh(D)is bounded, there exist a subsequence, still indexed by n, and a subset A of D,∂A = ∅, such that

bAn → bA in C(Uh(D))-strong

and another subsequence, still indexed by n, such that

bAn → bA in H1(Uh(D))-weak.

For all ε > 0, 0 < 3ε < h, there exists N > 0 such that for all n ≥ N and allx ∈ Uh(D),

d∂An(x) ≤ d∂A(x) + ε, d∂A(x) ≤ d∂An(x) + ε. (11.9)

Therefore,

∂An ⊂ Uh−2 ε(∂An) ⊂ Uh−ε(∂A) ⊂ Uh(∂An), (11.10)!Uh−ε(∂A) ⊂ !Uh−2 ε(∂An) ⊂ !∂An. (11.11)

From (11.6) and (11.10)

∀n ≥ N, ∥D2bAn∥M1(Uh−ε(∂A)) ≤ c.

In order to use the compactness of the embedding (11.3) as in the proof of The-orem 11.1, we would need Uh−ε(∂A) to be Lipschitzian. To get around this weconstruct a bounded Lipschitzian set between Uh−2ε(∂A) and Uh−ε(∂A). Indeedby definition,

Uh−ε(∂A) = ∪x∈∂AB(x, h− ε) and Uh−2ε(∂A) ⊂ Uh−ε(∂A),

and by compactness there exists a finite sequence of points xini=1 in ∂A such that

Uh−2ε(∂A) ⊂ UBdef= ∪n

i=1B(xi, h− ε) ⊂ Uh(D).

Since UB is Lipschitzian as the union of a finite number of balls, it now follows bycompactness of the embedding (11.3) for UB that there exist a subsequence, stilldenoted by bAn, and f ∈ BV(UB)N such that ∇bAn → f in L1(UB)N . SinceUh(D) is bounded, Cb(Uh(D)) is compact in C(Uh(D)) and there exist anothersubsequence, still denoted by bAn, and A ⊂ D, ∂A = ∅, such that bAn → bA inC(Uh(D)) and, a fortiori, in L1(Uh(D)). Therefore, bAn converges in W 1,1(UB) andalso in L1(UB). By uniqueness of the limit, f = ∇bA on UB and bAn converges tobA in W 1,1(UB). By Definition 5.1 and Theorem 3.1 (i), ∇bA and ∇bAn all belong


to BVloc(RN)N since they are of bounded variation in tubular neighborhoods oftheir respective boundaries. Moreover, by Theorem 5.2 (ii), χ∂A ∈ BVloc(RN).The above conclusions also hold for the subset Uh−2ε(∂A) of UB .

(ii) Convergence in W 1,p(Uh(D)). Consider the integral

∫

Uh(D)|∇bAn −∇bA|2 dx

=∫

Uh−2ε(∂A)|∇bAn −∇bA|2 dx +

∫

Uh(D)\Uh−2ε(∂A)|∇bAn −∇bA|2 dx.

From part (i), the first integral on the right-hand side converges to zero as n goesto infinity. The second integral is on a subset of !Uh−2ε(∂A). From (11.11) for alln ≥ N ,

|∇bAn(x)| = 1 a.e. in !∂An ⊃ !Uh−3ε(∂An) ⊃ !Uh−2ε(∂A),|∇bA(x)| = 1 a.e. in !∂A ⊃ !Uh−2ε(∂A).


Uh(D)\Uh−2ε(∂A)|∇bAn −∇bA|2 dx =

∫

Uh(D)\Uh−2ε(∂A)2 (1−∇bAn ·∇bA) dx,

which converges to zero by weak convergence of ∇bAn to ∇bA in the spaceL2(Uh(D))N in part (i) and the fact that |∇bA| = 1 almost everywhere in Uh(D)\Uh−2ε(∂A). Therefore, since bAn → bA in C(Uh(D))

bAn → bA in H1(Uh(D))-strong,

and by Theorem 4.1 (i) the convergence is true in W 1,p(Uh(D)) for all p ≥ 1.(iii) Properties (11.8). Consider the initial subsequence bAn which converges

to bA in H1(Uh(D))-weak constructed at the beginning of part (i). This sequenceis independent of ε and the subsequent constructions of other subsequences. Byconvergence of bAn to bA in H1(Uh(D))-weak for each Φ ∈ D1(Uh(∂A))N×N ,

limn→∞

∫

Uh(∂A)∇bAn ·

−→div Φ dx =

∫

Uh(∂A)∇bA ·

−→div Φ dx.

Each such Φ has compact support in Uh(∂A), and there exists ε = ε(Φ) > 0,0 < 3ε < h, such that

supp Φ ⊂ Uh−2ε(∂A).


∀n ≥ N(ε), Uh−2 ε(∂An) ⊂ Uh−ε(∂A) ⊂ Uh(∂An).

12. Finite Density Perimeter and Compactness 385


Uh(∂A)∇bAn ·

−→div Φ dx =

∫

Uh−2ε(∂A)∇bAn ·

−→div Φ dx =

∫

Uh(∂An)∇bAn ·

−→div Φ dx

⇒∣∣∣∣∫

Uh(∂A)∇bAn ·

−→div Φ dx

∣∣∣∣ ≤ ∥D2bAn∥M1(Uh(∂An)) ∥Φ∥C(Uh(∂An))

≤ c ∥Φ∥C(Uh−2ε(∂A)) = c ∥Φ∥C(Uh(∂A)).

By convergence of ∇bAn to ∇bA in the space L2(Uh(D))-weak, then for all Φ ∈D1(Uh(∂A))N×N

∣∣∣∣∫

Uh(∂A)∇bA ·

−→div Φ dx

∣∣∣∣ ≤ c ∥Φ∥C(Uh(∂A)) ⇒ ∥D2bA∥M1(Uh(∂A)) ≤ c.

Finally, the convergence remains true for all subsequences constructed in parts (i)and (ii). This completes the proof.

12 Finite Density Perimeter and CompactnessThe density perimeter introduced by D. Bucur and J.-P. Zolesio [14] is a re-laxation of the (N − 1)-dimensional upper Minkowski content which leads to thecompactness Theorem 12.1.

Definition 12.1.Let h > 0 be a fixed real number and let ∅ = A ⊂ RN, ∂A = ∅. Consider thequotient

Ph(∂A) def= sup0<k<h

mN (Uk(∂A))2k

, (12.1)

where Uk(∂A) = x ∈ RN : d∂A(x) < k. We say that A has a finite h-densityperimeter if Ph(∂A) is finite.

Clearly for all k, 0 < k < h,∫

∂AdmN ≤

∫

Uk(∂A)dmN ≤ c 2k ⇒ mN (∂A) = 0.

The compactness result of D. Bucur and J.-P. Zolesio [14] has been revisitedand established in the W 1,p-topology by M. C. Delfour and J.-P. Zolesio [38],from which convergence in all other topologies of Theorem 4.1 follows.

Theorem 12.1. Let D = ∅ be a bounded open subset of RN and An, ∂An = ∅,be a sequence of subsets of D. Assume that

∃h > 0 and c > 0 such that ∀n, Ph(∂An) ≤ c. (12.2)

Then there exist a subsequence Ank and A ⊂ D, ∂A = ∅, such that

Ph(∂A) ≤ lim infn→∞

Ph(∂An) ≤ c, (12.3)

∀p, 1 ≤ p <∞, bAnk→ bA in W 1,p(Uh(D))-strong. (12.4)


Proof. The proof essentially rests on Theorem 4.3 since Ph(∂A) ≤ c impliesmN (∂A) = 0. Since D is bounded, the family of oriented distance functions Cb(D)is compact in C(D) and W 1,p(D)-weak for all p, 1 ≤ p <∞ (cf. Theorems 2.2 (ii)and 4.2 (iii)). So there exist bA ∈ Cb(D) and a subsequence, still indexed by n, suchthat bAn → bA in the above topologies. Moreover, for all k, 0 < k < h, and all ε,0 < ε < h− k,

∃N(ε) > 0 such that ∀n ≥ N(ε), Uk−ε(∂An) ⊂ Uk(∂A) ⊂ Uk+ε(∂An)

(cf. proof of part (i) of Theorem 11.2). As a result for all n ≥ N(ε),

mN (Uk−ε(∂An))2(k − ε)

k − εk≤ mN (Uk(∂A))

2k≤ mN (Uk+ε(∂An))

2(k + ε)k + ε

k≤ c

k + ε

k

⇒ mN (Uk(∂A))2k

≤ mN (Uk+ε(∂An))2(k + ε)

k + ε

k≤ Ph(∂An)

k + ε

k

⇒ mN (Uk(∂A))2k

≤ lim infn→∞

Ph(∂An)k + ε

k. (12.5)

Going to the limit as ε goes to zero in the second and fourth terms

∀k, 0 < k < h,mN (Uk(∂A))

2k≤ lim inf

n→∞Ph(∂An) ≤ c

⇒ Ph(∂A) ≤ lim infn→∞

Ph(∂An) ≤ c⇒ mN (∂A) = 0.

The theorem now follows from the fact that mN (∂A) = 0 and Theorem 4.3.

Corollary 1. Let D = ∅ be a bounded open subset of RN and A, ∂A = ∅, be asubset of D such that

∃h > 0 and c > 0 such that Ph(∂A) ≤ c. (12.6)

Then the mapping bA′ → Ph(∂A′) : Cb(D)→ R∪+∞ is lower semicontinuous inA for the W 1,p(D)-topology.

Proof. Since we have a metric topology, it is sufficient to prove the property forW 1,p(D)-converging sequences bAn to bA. From that point on the argument isthe same as the one used to get (12.5) in the proof of Theorem 12.1 after theextraction of the subsequence.

Remark 12.1.It is important to notice that even if An is a W 1,p-convergent sequence of boundedopen subsets of RN with a uniformly bounded perimeter, the limit set A need notbe an open set or have a nonempty interior intA such that bA = bint A. It wouldbe tempting to say that bAn → bA implies d!An

→ d!A and use the open setintA = !!A for which d!A = d!int A to conclude that bA = bint A. This is incorrectas can be seen in the following example shown in Figure 7.2. Consider a family Anof open rectangles in R2 of width equal to 1 and height 1/n, n ≥ 1, an integer.Their density perimeter is bounded by 4, and the bAn ’s converge to bA for A equalto the line of length 1 which has an empty interior.

13. Compactness and Uniform Fat Segment Property 387

1n

11

AAn

Figure 7.2. W 1,p-convergence of a sequence of open subsets An : n ≥ 1of R2 with uniformly bounded density perimeter to a set with empty interior.

13 Compactness and Uniform Fat Segment Property13.1 Main TheoremIn Theorem 6.11 of section 6.4 of Chapter 5, we have seen a compactness theorem forthe family of subsets of a bounded holdall D satisfying the uniform cone property.This compactness theorem is no longer true when the uniform cone property isreplaced by a uniform segment property. This is readily seen by considering thefollowing example.

Example 13.1.Given an integer n ≥ 1, consider the following sequence of open domains in R2:

Ωndef=

(x, y) ∈ R2 : |x| < 1 and |x|1/n < y < 2

.

They satisfy the uniform segment property of Definition 6.1 (ii) of Chapter 2 bychoosing λ = r = 1/4. The sequence Ωn converges to the closed set

Adef=(x, y) ∈ R2 : |x| ≤ 1 and 1 ≤ y ≤ 2

∪ L,

Ldef=(0, y) ∈ R2 : 0 ≤ y ≤ 1

in the uniform topologies associated with dΩn and bΩn or in the Lp-topologies asso-ciated with χΩn and χ!Ωn

. However, the segment property is not satisfied along theline L and the corresponding family of subsets of the holdall D = B(0, 4) satisfyingthe uniform segment property with r = λ = 1/4 is not closed and, a fortiori, notcompact.

This example shows that a uniform segment property is too meager to makethe corresponding family compact. Looking back at the proof of Theorem 6.11 ofsection 6.4 in Chapter 5, the fact that the cone is an open set around the segmentis used critically. That was the motivation to introduce the more general uniformfat segment property (Definition 6.1 (iii) in section 6 of Chapter 2) that includesas special cases both the uniform cone and the uniform cusp properties (cf. Defini-tions 6.3 and 6.4 in section 6.4 of Chapter 2). However, the proof of Theorem 6.11in Chapter 5 was based on the fact that the perimeter of all the sets in the family


were uniformly bounded. There is no hope of getting that property for generalHolderian domains, and a completely different proof is necessary.

Given a bounded open subset D of RN consider the family

L(D, r,O,λ) def=

Ω ⊂ D :

Ω satisfies the uniform fatsegment property for (r, O,λ)

. (13.1)

We first give a general proof of the compactness of L(D, r,O,λ) in the C(D)- andW 1,p(D)-topologies associated with the oriented distance function bΩ. As a conse-quence, we get the compactness for the C(D)- and W 1,p(D)-topologies associatedwith dΩ and d!Ω and the Lp(D)-topologies associated with χΩ and χ!Ω. We recoverthe compactness Theorem 6.11 of Chapter 5 under the uniform cone property forχΩ in Lp(D) as a special case. In the last part of this section, we give several otherways to specify the compact families by using the equivalent conditions of Chapter 2of the uniform fat segment property in terms of conditions on the local epigraphsvia the dominating function and equicontinuity of the local graph functions.

Theorem 13.1. Let D be a nonempty bounded open subset of RN and let 1 ≤p < ∞. Assume that L(D, r,O,λ) is not empty for r > 0, λ > 0, and an opensubset O of RN such that (0, eN ) ⊂ O and 0 /∈ O. Then the family

B(D, r, O,λ) def= bΩ : ∀Ω ∈ L(D, r,O,λ)

is compact in C(D) and W 1,p(D). As a consequence the families

Bd(D, r, O,λ) def= dΩ : ∀Ω ∈ L(D, r,O,λ),

Bcd(D, r, O,λ) def= d!Ω : ∀Ω ∈ L(D, r,O,λ),

B∂d (D, r, O,λ) def= d∂Ω : ∀Ω ∈ L(D, r,O,λ)

are compact in C(D) and W 1,p(D), and the families

X(D, r, O,λ) def= χΩ : ∀Ω ∈ L(D, r,O,λ),

Xc(D, r, O,λ) def= χ!Ω : ∀Ω ∈ L(D, r,O,λ)

are compact in Lp(D).

From Lemma 6.1 of Chapter 2, Theorem 13.1 can be specialized to structuredopen sets O.

Corollary 1. The conclusions of Theorem 13.1 hold when O is specialized to openregions of the form

O(h, ρ,λ) =ξ ∈ RN : ξ′ ∈ BH(0, ρ) and h(|ξ′|) < ξN < λ

(13.2)

for dominating functions h ∈ H.

Theorem 13.1 can be further specialized to Lipschitzian and Holderian sets.


Corollary 2. The conclusions of Theorem 13.1 hold under the uniform cone prop-erty of Definition 6.3 (ii) in section 6.4 of Chapter 2 with

O = O(h, ρ,λ) with h(θ) = θ/ tanω and ρ = λ tanω, 0 < ω < π/2, (13.3)

and the uniform cusp property of Definition 6.4 (ii) of Chapter 2 with

O = O(hℓ, ρ,λ) with hℓ(θ) = λ (θ/ρ)ℓ , 0 < ℓ < 1.

The proof of Theorem 13.1 will require Theorem 4.3 and the following lemma.

Lemma 13.1. Given a sequence bΩn ⊂ Cb(D) such that bΩn → bΩ in C(D) forsome bΩ ∈ Cb(D), we have the following properties:

∀x ∈ Ω, ∀R > 0, ∃N(x, R) > 0, ∀n ≥ N(x, R), B(x, R) ∩ Ωn = ∅,

and for all x ∈ !Ω,

∀R > 0, ∃N(x, R) > 0, ∀n ≥ N(x, R), B(x, R) ∩ !Ωn = ∅. (13.4)

Moreover,

∀x ∈ ∂Ω, ∀R > 0, ∃N(x, R) > 0, ∀n ≥ N(x, R),B(x, R) ∩ Ωn = ∅ and B(x, R) ∩ !Ωn) = ∅,

and B(x, R) ∩ ∂Ωn = ∅.

Proof. We proceed by contradiction. Assume that

∃x ∈ Ω, ∃R > 0, ∀N > 0, ∃n ≥ N, B(x, R) ∩ Ωn = ∅.

So there exists a subsequence Ωnk, nk →∞, such that

bΩnk(x) = dΩnk

(x) ≥ R = 0 = dΩ(x) ≥ bΩ(x),

which contradicts the fact that bΩnk→ bΩ. Of course, the same assertion is true

for the complements and for all x ∈ !Ω,

∀R > 0, ∃N(x, R) > 0, ∀n ≥ N(x, R), B(x, R) ∩ !Ωn = ∅.

When x ∈ ∂Ω, x ∈ Ω ∩ !Ω and we combine the two results. For the last result usethe fact that the open ball cannot be partitioned into two nonempty disjoint opensubsets.

Proof of Theorem 13.1. (i) Compactness in C(D). Since for all Ω in L(D, r,O,λ),Ω is locally a C0-epigraph, ∂Ω = ∅, bΩ ∈ Cb(D), and mN (∂Ω) = 0. Consideran arbitrary sequence Ωn in L(D, r,O,λ). For D compact, Cb(D) is compactin C(D) and there exist Ω ⊂ D, ∂Ω = ∅, and a subsequence Ωnk such thatbΩnk

→ bΩ in C(D). It remains to prove that Ω ∈ L(D, r,O,λ), that is,

∀x ∈ ∂Ω, ∃A ∈ O(N) such that ∀y ∈ Ω ∩B(x, r), y + λAO ⊂ int Ω.


From Lemma 13.1, for each x ∈ ∂Ω, for all k ≥ 1, there exists nk ≥ k such that

B(x, r/2k

)∩ ∂Ωnk = ∅.

Denote by xk an element of that intersection:

∀k ≥ 1, xk ∈ B(x, r/2k

)∩ ∂Ωnk .

By construction xk → x. Next consider y ∈ B(x, r) ∩ Ω. From the first part ofthe lemma, there exists a subsequence of Ωnk, still denoted by Ωnk, such thatfor all k ≥ 1, B(y, r/2k) ∩ Ωnk = ∅. For each k ≥ 1 denote by yk a point of thatintersection. By construction

yk ∈ Ωnk → y ∈ Ω ∩B(x, r).

There exists K > 0 large enough such that for all k ≥ K, yk ∈ B(xk, r). To seethis, note that y ∈ B(x, r) and that

∃ρ > 0, B(y, ρ) ⊂ B(x, r) and |y − x| + ρ/2 < r.

Now

|yk − xk| ≤ |yk − y| + |y − x| + |x− xk|

≤ r

2k+ r − ρ

2+

r

2k≤ r +

[ r

2k−1 −ρ

2

]< r.

Since r/ρ > 1 the result is true for

r

2k−1 −ρ

2< 0 ⇒ k > 2 + log

(r

ρ

).

So we have constructed a subsequence Ωnk such that for k ≥ K

xk ∈ ∂Ωnk → x ∈ ∂Ω and yk ∈ Ωnk ∩B(xk, r)→ y ∈ Ω ∩B(x, r).

For all k, there exists Ak ∈ O(N), Ak∗Ak = ∗Ak Ak = I, such that yk + λAkO ⊂

int Ωnk . Pick another subsequence of Ωnk, still denoted by Ωnk, such that

∃A ∈ O(N), A ∗A = ∗A A = I, Ak → A.

Now consider z ∈ y + λAO. Since y + λAO is open there exists ρ > 0 such thatB(z, ρ) ⊂ y + λAO, and there exists K ′ ≥ K such that

∀k ≥ K ′, B(z, ρ/2) ⊂ yk + λAkO ⊂ int Ωnk = !!Ωnk

⇒ !B(z, ρ/2) ⊃ !Ωnk ⇒ 0 < ρ/2 = d!B(z,ρ/2)(z) ≤ d!Ωnk(z)→ d!Ω(z)

⇒ 0 < ρ/2 ≤ d!Ω(z) ⇒ z ∈ !!Ω = int Ω ⇒ y + λAO ⊂ int Ω.

Hence Ω ⊂ D satisfies the uniform fat segment property.


(ii) W 1,p(D)-compactness. From Theorem 4.1 (i), it is sufficient to provethe result for p = 2. Consider the subsequence Ωnk ⊂ L(D,λ,O, r) and letΩ ∈ L(D,λ,O, r) be the set previously constructed such as bΩnk

→ bΩ in C(D).Hence bΩnk

→ bΩ in L2(D). Since D is compact

∀Ω ⊂ D,

∫

D|bΩnk

|2 dx ≤∫

Ddiam (D)2 dx ≤ diam (D)2mN (D),

∫

D|∇bΩnk

|2 dx ≤∫

Ddx = mN (D),

and there exists a subsequence, still denoted by bΩnk, which converges weakly to

bΩ. Since all the sets in L(D, r,O,λ) verify a uniform segment property, they arelocally C0-epigraphs, mN (∂Ωnk) = 0 = mN (∂Ω) and, by Theorem 4.3, bΩnk

→ bΩ

in W 1,p(D)-strong, 1 ≤ p <∞.(iii) The other compactness follows by continuity of the maps (4.6) and (4.7)

in Theorem 4.1 and the fact that mN (∂Ω) = 0 implies χint Ω = χΩ and χint !Ω = χ!Ωalmost everywhere for Ω ∈ L(D,λ,O, r).

13.2 Equivalent Conditions on the Local Graph FunctionsWe have shown in Theorem 6.2 of Chapter 2 that the uniform fat segment property isequivalent to the equi-C0 epigraph property. This means that we can equivalentlycharacterize the compact family L(D, r,O,λ) in terms of conditions on the localgraph functions such as the ones introduced in Theorem 5.1 of Chapter 2. Recallthe following equivalent conditions:

(i) Ω verifies the fat segment property.

(ii) Ω is an equi-C0-epigraph; that is, Ω is a C0-epigraph and the local graphfunctions are uniformly bounded and equicontinuous.

(iii) Ω is a C0-epigraph and there exist ρ > 0 and h ∈ H such that BH(0, ρ) ⊂ Vand for all x ∈ ∂Ω

∀ζ ′, ξ′ ∈ BH(0, ρ) such that |ξ′ − ζ ′| < ρ, |ax(ξ′)− ax(ζ ′)| ≤ h(|ξ′ − ζ ′|). (13.5)

(iv) Ω is a C0-epigraph and there exist ρ > 0 and h ∈ H such that BH(0, ρ) ⊂ Vand for all x ∈ ∂Ω

∀ξ′ ∈ BH(0, ρ), ax(ξ′) ≤ h(|ξ′|). (13.6)

Recall also from Definition 5.1 (ii) in Chapter 2 that Ω is said to be a C0

epigraph if it is locally a C0-epigraph and the neighborhoods U(x) and Vx can bechosen in such a way that Vx and A−1

x (U(x)− x) are independent of x: there existbounded open neighborhoods V of 0 in H and U of 0 in RN such that PH(U) ⊂ Vand

∀x ∈ ∂Ω, Vx = V and ∃Ax ∈ O(N) such that U(x) = x + AxU.


By applying each of the last three characterizations to all members of a fam-ily of subsets of a bounded holdall D, we get the analogue of the compactnessTheorem 13.1 under the uniform fat segment property.

Theorem 13.2. Let D be a bounded open nonempty subset of RN and consider thefamily L(D, ρ, U) of all subsets Ω of D that verify the C0-epigraph property: thereexist ρ > 0 and a bounded open neighborhood U of 0 such that

U ⊂ζ ∈ RN : PH(ζ) ∈ BH(0, ρ)

, V

def= BH(0, ρ), (13.7)

and for each Ω ∈ L(D, ρ, U) and each x ∈ ∂Ω, there exists AΩx ∈ O(N) such that

UΩ(x) def= x + AΩx U (13.8)

and a C0-graph function aΩx : V → R such that aΩ

x (0) = 0 and

UΩ(x) ∩ ∂Ω = UΩ(x) ∩x + Ax(ζ ′ + ζNeN ) : ζ ′ ∈ V, ζN = aΩ

x (ζ ′)

,

UΩ(x) ∩ int Ω = UΩ(x) ∩x + Ax(ζ ′ + ζNeN ) : ζ ′ ∈ V, ζN > aΩ

x (ζ ′)

.(13.9)

Then, for all h ∈ H, the subfamilies

L0(D, ρ, U) def=

⎧⎪⎨

⎪⎩Ω ∈ L(D, ρ, U) :

aΩx : Ω ∈ L(D, ρ, U), x ∈ ∂Ω

is uniformly boundedand equicontinuous

⎫⎪⎬

⎪⎭, (13.10)

L1(D, ρ, U, h) def=

Ω ∈ L(D, ρ, U) : ∀x ∈ ∂Ω, ∀ζ ′ ∈ V, aΩx (ζ ′) ≤ h(|ζ ′|)

, (13.11)

L2(D, ρ, U, h) def=

⎧⎪⎨

⎪⎩Ω ∈ L(D, ρ, U) :

∀x ∈ ∂Ω, ∀ζ ′, ξ′ ∈ V

such that |ξ′ − ζ ′| < ρ

|ax(ξ′)− ax(ζ ′)| ≤ h(|ξ′ − ζ ′|)

⎫⎪⎬

⎪⎭(13.12)

are compact (possibly empty) in C(D) and W 1,p(D), 1 ≤ p < ∞, and the conclu-sions of Theorem 13.1 hold.

Remark 13.1.The geometric fat segment property of Corollary 1 was introduced in the form (13.2)in the book of M. C. Delfour and J.-P. Zolesio [37] under the name uniformcusp property in 2001, where the condition was specialized to (13.3) for Lipschitzianand Holderian domains as stated in Corollary 2.

Condition (13.10) is a streamlined version of the sufficient condition givenin an unpublished report by D. Tiba [2], announced without proof in the noteby W. Liu, P. Neittaanmaki, and D. Tiba [1] in 2000, and later documentedin D. Tiba [3] in 2003. He gets a compactness result by introducing the followingconditions on the local C0-epigraphs: the boundaries of the domains belong (aftera rotation and a translation) to a family F of uniformly equicontinuous functionsdefined in a fixed neighborhood V of 0 in RN−1. This means that there exists amodulus of continuity µ(ε) > 0 such that

∀ε > 0, ∀f ∈ F , ∀x, y ∈ V, |x− y| < µ(ε) ⇒ |f(y)− f(x)| < ε.

Then the compactness follows by Ascoli–Arzela Theorem 2.4 of Chapter 2.

14. Compactness under Uniform Fat Segment Property and Bound on a Perimeter 393

The connection between that condition and our geometric condition wasfirst established in M. C. Delfour, N. Doyon, and J.-P. Zolesio [1, 2] in2005. Condition (13.11) that relaxes the equicontinuity condition (13.10) to thesimpler dominating function condition and condition (13.12) were both introducedin M. C. Delfour and J.-P. Zolesio [43] in 2007 along with the general formof the uniform fat segment property that involves only the unparametrizedopen set O.

14 Compactness under the Uniform Fat SegmentProperty and a Bound on a Perimeter

Domains that are locally Lipschitzian epigraphs or equivalently satisfy the localuniform cone property enjoy the additional property that for any bounded open setD the (N − 1)-Hausdorff measure of D ∩ ∂Ω is finite. In general this is no longertrue for domains verifying a uniform cusp property for some function h(θ) = |θ/ρ|α,0 < α < 1 (cf., for instance, Example 6.2 given6 in section 6.5 of Chapter 2 of abounded domain Ω in RN for which HN−1(∂Ω) = +∞ and the Hausdorff dimensionof ∂Ω is exactly N − α). Yet, there are many domains with cusps whose boundaryhas finite HN−1 measure.

14.1 De Giorgi Perimeter of Caccioppoli Sets

One of the classical notions of perimeter is the one introduced in Definition 6.2in section 6 of Chapter 5 in the context of the problem of minimal surfaces forCaccioppoli sets. Theorem 13.1 extends to the following subfamily of L(D, r,O,λ)for a bounded open subset D of RN and a constant c′:

L(D, r,O,λ, c′) def=

⎧⎪⎨

⎪⎩Ω ⊂ D :

Ω satisfies the uniform fatsegment property for (r, O,λ)and PD(Ω) ≤ c′

⎫⎪⎬

⎪⎭. (14.1)

Theorem 14.1. Let D be a nonempty bounded open subset of RN and let 1 ≤p < ∞. Assume that L(D, r,O,λ, c′) is not empty for r > 0, λ > 0, c′ > 0, and abounded open subset O of RN such that (0, eN ) ⊂ O and 0 /∈ O. Then the family

B(D, r, O,λ, c′) def= bΩ : ∀Ω ∈ L(D, r,O,λ, c′)

is compact in C(D) and W 1,p(D). As a consequence the families

Bd(D, r, O,λ, c′) def= dΩ : ∀Ω ∈ L(D, r,O,λ, c′),

Bcd(D, r, O,λ, c′) def= d!Ω : ∀Ω ∈ L(D, r,O,λ, c′),

B∂d (D, r, O,λ, c′) def= d∂Ω : ∀Ω ∈ L(D, r,O,λ, c′)

6This was originally in M. C. Delfour, N. Doyon, and J.-P. Zolesio [1].


are compact in C(D) and W 1,p(D), and the families

X(D, r, O,λ, c′) def= χΩ : ∀Ω ∈ L(D, r,O,λ, c′) ,

Xc(D, r, O,λ, c′) def= χ!Ω : ∀Ω ∈ L(D, r,O,λ, c′)

are compact in Lp(D).

14.2 Finite Density Perimeter

It is also possible to use the density perimeter introduced in Definition 12.1 in placeof the De Giorgi perimeter.

The proof of the next result combines Theorem 14.1, which says that thefamily L(D, r,O,λ, c′) is compact, with Theorem 12.1, which says that the familyof sets verifying (12.2) is compact in W 1,p(D). The intersection of the two familiesof oriented distance functions is compact in W 1,p(D).

Theorem 14.2. For fixed h > 0 and an open subset Ω of RN, Theorem 14.1remains true when PD(Ω) is replaced by the density perimeter Ph(∂Ω).

15 The Families of Cracked SetsIn this section we introduce new families of thin sets, that is, sets A such thatmN (∂A) = 0, by introducing conditions on the semiderivative of the distance func-tion at points of the boundary. They have been used in M. C. Delfour andJ.-P. Zolesio [38] in the context of the image segmentation problem of D. Mum-ford and J. Shah [2]. Cracked sets are more general than sets which are locallythe epigraph of a continuous function in the sense that they include domains withcracks and sets that can be made up of components of different codimensions. TheHausdorff (N −1) measure of their boundary is not necessarily finite. Yet, compactfamilies (in the W 1,p-topology) of such sets can be constructed.

First recall lower and upper semiderivatives7 of Dini type for the differentialquotient of a function f : V (x) ⊂ RN → R defined in a neighborhood V (x) of apoint x ∈ RN in the direction d ∈ RN:

lim inft0

f(x + td)− f(x)t

def= limδ0

inf0<t<δ

f(x + td)− f(x)t

,

lim supt0

f(x + td)− f(x)t

def= limδ0

sup0<t<δ

f(x + td)− f(x)t

,

lim inft0w→d

f(x + tw)− f(x)t

def= limδ0

inf0<t<δ

|w−d|RN<δ(t,w)=(0,v)

f(x + tw)− f(x)t

,

7Note that the (t, d), 0 < t < δ, is allowed in the inf and the sup of the last two definitions andthe constraint (t, w) = (0, v) can be removed.

15. The Families of Cracked Sets 395

lim supt0w→d

f(x + tw)− f(x)t

def= limδ0

sup0<t<δ

|w−d|RN<δ(t,w)=(0,v)

f(x + tw)− f(x)t

.

Definition 15.1.Let Ω ⊂ RN be such that Γ def= ∂Ω = ∅.

(i) Ω is said to be weakly cracked if

∀x ∈ Γ, ∃d ∈ RN, |d| = 1, such that lim supt0w→d

dΓ(x + tw)t

> 0. (15.1)

(ii) Ω is said to be cracked if

∀x ∈ Γ, ∃d ∈ RN, |d| = 1, such that lim inft0

dΓ(x + td)t

> 0. (15.2)

(iii) Ω is said to be strongly cracked8 if

∀x ∈ Γ, ∃d ∈ RN, |d| = 1, such that lim inft→0

dΓ(x + td)|t| > 0. (15.3)

(iv) A set ∅ = A ⊂ RN is said to be thin if mN (A) = 0.

Strongly cracked implies cracked, and cracked implies weakly cracked since

lim inft0w→d

dΓ(x + tw)t

≤ lim inft0

dΓ(x + td)t

≤ lim supt0

dΓ(x + td)t

≤ lim supt0w→d

dΓ(x + tw)t

.

The special terminology of Definition 15.1 is introduced to provide an intuitivedescription of the sets. It is motivated by the fact that the boundary of such a sethas zero N -dimensional Lebesgue measure (cf. Lemma 15.1) and can be made upof internal cracks, external hairs, cusps, or points, as shown in Figure 7.3.

The weakly cracked property is verified in any point of the boundary wherethe gradient of dΓ does not exist; in boundary points where the gradient exists itis not identically 0. This is a very large family of sets that includes domains whichare locally the epigraph of a continuous function. There are obvious variations ofthe above definitions and the forthcoming compactness Theorem 15.1 by replacingdΓ by dΩ, d!Ω, or bΩ.

8Here the definition is

lim inft→0

f(x + td) − f(x)|t|

def= limδ0

inf0<|t|<δ

f(x + td) − f(x)t

.


Figure 7.3. Example of a two-dimensional strongly cracked set.

Lemma 15.1. Let Ω ⊂ RN, let Γ = ∅, let x ∈ Γ, and let d, |d| = 1, be a directionin RN. If the semiderivative ddΓ(x; d) does not exist, then

lim supt0w→d

dΓ(x + tw)t

> 0.

Proof. Since the function dΓ is Lipschitzian, the limit of the quotient

ddΓ(x; d) def= limt0

dΓ(x + td)− dΓ(x)t

exists if and only if the limit of the quotient

dHdΓ(x; d) def= limt0w→d

dΓ(x + tw)− dΓ(x)t

exists (cf. Theorem 2.4 (i) in Chapter 9). Moreover, by the Lipschitz continuity,∣∣∣∣dΓ(x + tw)− dΓ(x)

t

∣∣∣∣ ≤∣∣∣∣tw

t

∣∣∣∣ = |w|→ |d|

and hence the liminf and the limsup of the quotient exist and are finite.For x ∈ Γ, dΓ(x) = 0. Therefore ddΓ(x; d) does not exist if and only if

lim supt0w→d

dΓ(x + tw)t

> lim inft0w→d

dΓ(x + tw)t

≥ 0

since the last term is nonnegative. This completes the proof.

Theorem 15.1. Let ∅ = Ω ⊂ RN.

(i) Given a weakly cracked set Ω,

mN (Γ) = 0 and Γ ⊂ int Ω ∪ int !Ω.

Moreover in any point x ∈ Γ either ∇bΩ(x) exists and is different from zeroor ∇bΩ(x) does not exists (set of cracks).


(ii) Given a cracked set Ω, for each x ∈ Γ

∃ a direction d ∈ RN, |d| = 1, such that 1 ≥ ℓ(x) def= lim inft0

dΓ(x + td)t

> 0

and for all ε, 0 < ε < ℓ(x), there exists δ > 0 such that

∀t, 0 < t < δ, dΓ(x + td) ≥ (ℓ(x)− ε)t

⇒ x + C(δ cosω,ω, d) ⊂ RN \Γ, sinω def= ℓ(x)− ε, 0 < ω ≤ π/2

(C(λ,ω, d) is the open cone in 0 of direction d, height λ, and aperture ω).

(iii) Given a strongly cracked set Ω, for each x ∈ Γ there exists a direction d ∈ RN,|d| = 1, such that

1 ≥ ℓ(x) def= lim inft→0

dΓ(x + td)|t| > 0


∀t, 0 < |t| < δ, dΓ(x + td) ≥ (ℓ(x)− ε)|t|⇒ x ± C(δ cosω,ω, d) ⊂ RN \Γ, sinω = ℓ(x)− ε, 0 < ω ≤ π/2.

This means that in each point of the boundary of a cracked set, it is possibleto place an open cone that does not intersect the boundary. If, in addition, the setis strongly cracked, this cone can be replaced by a double cone.

Proof. (i) We already know that ∇dΓ exists almost everywhere in RN and that,whenever it exists,

|∇dΓ(x)| =

0, if x ∈ Γ,

1, if x /∈ Γ

(cf. Theorem 3.3 (vi) in Chapter 6). Therefore, if ∇dΓ(x) exists in a point x ∈ Γ,∇dΓ(x) = 0 and for all d, |d| = 1,

lim supt0w→d

dΓ(x + tw)t

= lim supt0w→d

dΓ(x + tw)− dΓ(x)t

= ∇dΓ(x) · d = 0,

which contradicts the weakly cracked property. Hence the points of Γ are pointswhere ∇dΓ(x) does not exist, which is itself a set of zero measure. Moreover, if∇bΩ(x) exists in a point x ∈ ∂Ω, then bΩ(x) = 0 and for all d, |d| = 1,

limt0w→d

bΩ(x + tw)t

= limt0w→d

bΩ(x + tw)− bΩ(x)t

= ∇bΩ(x) · d.


If ∇bΩ(x) = 0, then for all d

limt0w→d

bΩ(x + tw)t

= limt0w→d

bΩ(x + tw)− bΩ(x)t

= ∇bΩ(x) · d = 0

⇒ limt0w→d

dΓ(x + tw)t

= limt0w→d

∣∣∣∣bΩ(x + tw)

t

∣∣∣∣ = 0

and ∇d∂Ω(x) = 0, which contradicts our assumption. Therefore if ∇bΩ(x) exists ina point x ∈ ∂Ω, ∇bΩ(x) = 0. For any x ∈ Γ introduce the notation

ℓ(x) def= lim supt0w→d

dΓ(x + tw)t

.

By assumption ℓ(x) > 0 and for all δ > 0

ℓ(x) ≤ sup0<t<δ

|w−d|<δ

dΓ(x + tw)t

.

Hence there exist sequences tn, tn → 0, and wn, wn → d, such that

0 < ℓ(x)/2 ≤ dΓ(x + tnwn)t

⇒ RN \Γ ∋ xndef= x + tnwn → x

and necessarily Γ ⊂ int Ω ∪ int !Ω ⊂ int Ω ∪ int !Ω.(ii) Given a cracked set Ω, for each x ∈ Γ there exists a direction d ∈ RN,

|d| = 1, such that

ℓ(x) def= lim inft0

dΓ(x + td)t

> 0


∀t, 0 < t < δ, dΓ(x + td) ≥ (ℓ(x)− ε)t.

Recall that since dΓ is Lipschitzian of constant 1, we necessarily have 1 ≥ ℓ(x) for|d| = 1. Therefore

x + C(δ cosω,ω, d) ⊂ RN \Γ, sinω = ℓ(x)− ε, 0 < ω ≤ π/2.

(iii) This is similar to the proof of part (ii).

Theorem 15.2. Let D be a bounded open subset of RN and α, 0 < α ≤ 1, andh > 0 be real numbers.9 Consider the families

F(D, h,α) def=

⎧⎨

⎩Ω ⊂ D :Γ = ∅ and ∀x ∈ Γ, ∃d, |d| = 1,

such that inf0<t<h

dΓ(x + td)t

≥ α

⎫⎬

⎭ ,

Ch,αb (D) def= bΩ : Ω ∈ F(D, h,α) ,

(15.4)

9In view of the fact that the distance function dΓ is Lipschitzian with constant 1, we necessarilyhave 0 < α ≤ 1.


Fs(D, h,α) def=

⎧⎨

⎩Ω ⊂ D :Γ = ∅ and ∀x ∈ Γ, ∃d, |d| = 1,

such that inf0<|t|<h

dΓ(x + td)|t| ≥ α

⎫⎬

⎭ ,

(Ch,αb )s(D) def= bΩ : Ω ∈ Fs(D, h,α) .

(15.5)

Then Ch,αb (D) and (Ch,α

b )s(D) are compact in W 1,p(D), 1 ≤ p <∞.

Proof. (i) The family Ch,αb (D) is contained in Cb(D), which is compact in the uni-

form topology of C(D) and in the weak topology of W 1,p(D), 1 ≤ p < ∞ (cf.Theorem 2.2 (ii) and Theorem 4.2 (iii) in Chapter 6). Therefore, given a se-quence bΩn in Ch,α

b (D), there exist a subsequence, still denoted by bΩn, andbΩ ∈ Cb(D) such that bΩn → bΩ in W 1,p(D)-weak and C(D). In addition, bydefinition of the elements of Ch,α

b (D), condition (15.2) is verified and each Ωn hasthin boundary. We want to show that bΩ ∈ Ch,α

b (D). Once this is proved, fromTheorem 15.1 (i), Ω has a thin boundary. Hence the weak W 1,p(D) convergenceimplies the strong W 1,p(D) convergence by Theorem 4.3. In view of the continuityof the map bΩ → dΓ = |bΩ| : C(D)→ C(D), dΓn → dΓ in C(D). From Lemma 13.1,given x ∈ Γ, there exists a subsequence of bΩn, still denoted by bΩn, and foreach n ≥ 1 points xn ∈ Γn ∩B(x, 1/n). Hence xn → x. By assumption

∀n ≥ 1, ∃dn ∈ RN, |dn| = 1, such that inf0<t<h

dΓn(xn + tdn)t

≥ α.

Since the dn’s have norm 1, there exist a subsequence, still denoted by dn, andd, |d| = 1, such that dn → d. Fix t, 0 < t < δ. Given ε > 0, there exists N suchthat for all n ≥ N

|xn − x| < εt, |dn − d| < δn < ε, ∥dΓn − dΓ∥C(D) < εt.

Fix n = N and consider the following estimates:

dΓ(x + td)t

≥ dΓn(xn + tdn)t

− |dΓn(x + tdn)− dΓn(xn + td)|t

− |dΓ(x + td)− dΓ(xn + td)|t

− |dΓ(xn + td)− dΓn(xn + td)|t

≥α− ε− |d− dn|− |x− xn|t

−∥dΓ − dΓn∥CD)

t≥ α− 4ε

⇒ ∀ε > 0, inf0<t<h

dΓ(x + td)t

≥ α− 4ε ⇒ inf0<t<h

dΓ(x + td)t

≥ α.

Therefore bΩ ∈ Ch,αb (D) and this completes the proof of the compactness.

(ii) The proof for (Ch,αb )s(D) is identical with obvious changes.


16 A Variation of the Image Segmentation Problemof Mumford and Shah

16.1 Problem Formulation

In this section we specialize to the segmentation of (N -dimensional) images wherethe segmentation can be composed of objects of codimension greater than or equalto one. To represent the set of Figure 7.3 as the boundary of an open set one canuse the unbounded plane R2 minus all the lines and the points in Figure 7.3. If it isimportant that the open set be bounded, a fixed open frame D is introduced. Theopen set Ω in Figure 7.4 is then defined as the interior of the bounded open frameD minus all the lines and points used to draw the picture.

open frame D

Figure 7.4. The two-dimensional strongly cracked set of Figure 7.3 in anopen frame D.

Definition 16.1.Let D be a bounded open subset of RN with Lipschitzian boundary.

(i) An image in the frame D is specified by a function f ∈ L2(D).

(ii) We say that Ωii∈I is an open partition of D if Ωii∈I is a family of disjointconnected open subsets of D such that

mN (∪i∈IΩi) = mN (D) and mN (∂ ∪i∈I Ωi) = 0.

Denote by P(D) the family of all such open partitions of D.

Given an open partition Ωii∈I of D, associate with each i ∈ I a functionϕi ∈ H1(Ωi). In its intuitive form the problem formulated by D. Mumford andJ. Shah [2] aims at finding an open partition P = Ωii∈I in P(D) solution of thefollowing minimization problem:

infP∈P(D)

∑

i∈I

infϕi∈H1(Ωi)

∫

Ωi

ε |∇ϕi|2 + |ϕi − f |2 dx (16.1)

16. A Variation of the Image Segmentation Problem of Mumford and Shah 401

for some fixed constant ε > 0. The term in ε times the norm of the gradientcan be seen as a Tikhonov regularization on each Ωi. Observe that without thecondition mN (∪i∈IΩi) = mN (D) the empty set would be a solution of the problemor a phenomenon of the type discussed in Remark 12.1 and the phenomenon ofFigure 7.2 could occur.

The question of existence requires a more specific family of open partitions or apenalization term which preserves the “length” of the interfaces in some appropriatesense:

infP∈P(D)

∑

i∈I

infϕi∈H1(Ωi)

∫

Ωi

ε |∇ϕi|2 + |ϕi − f |2 dx + c HN−1(∂ ∪i∈I Ωi) (16.2)

for some c > 0. The choice of the relaxation of the (N − 1)-Hausdorff measureHN−1 is critical as discussed in section 9 of Chapter 1. Another way of lookingat the problem would be to minimize the number I of open subsets of the openpartition, but this seems more difficult to formalize.

16.2 Cracked Sets without the PerimeterIn this section we specialize the compact family of Theorem 15.2 to get the existenceof a solution to the following minimization problem:

infΩ∈F(D,h,α)

Ω open ⊂D, mN (Ω)=mN (D)

infϕ∈H1(Ω)

∫

Ωε |∇ϕ|2 + |ϕ− f |2 dx. (16.3)

This allows for open sets Ω with HN−1(Γ) = ∞. The pair (h,α) are the controlparameters of the segmentation. Recall the characterization of Theorem 15.1 (ii),which says that in each point of the boundary there exists a small open cone ofuniform height and aperture which does not intersect the boundary.

Theorem 16.1. Given a bounded open frame D ⊂ RN with a Lipschitzian boundaryand real numbers h, α, and ε > 0, there exist an open subset Ω∗ of D in F(D, h,α)such that mN (Ω∗) = mN (D) and a function y ∈ H1(Ω∗) that are solutions ofproblem (16.3).

16.2.1 Technical Lemmas

The proof of the existence theorems will require the following technical results.

Lemma 16.1. Given a subset A of RN with nonempty boundary ∂A,

∃ an open subset Ω ⊂ RN such that bA = bΩ

if and only if dA = dint A or equivalently A = intA.

Proof. By definition for Ω open

bA = bΩ ⇐⇒ dA = dΩ and d!Ω = d!A,


which is also equivalent to

A = Ω and !Ω = !A ⇐⇒ A = Ω and Ω = intA.

Hence the necessary and sufficient condition finally reduces to intA = A.

Lemma 16.2. Let D ⊂ RN be bounded open with Lipschitzian boundary. Then

Ω ⊂ D, Γ = ∅, mN (Γ) = 0, and mN (Ω) = mN (D) ⇒ int Ω = Ω.

Proof. By contradiction. If int Ω # Ω, there exists x ∈ Ω such that dint Ω(x) = ρ > 0and hence B(x, ρ) ⊂ !Ω. Therefore Ω ∩ B(x, ρ) ⊂ Γ and mN (Ω ∩ B(x, ρ)) ≤mN (Γ) = 0. By assumption mN (D) = mN (Ω) = mN (Ω) implies mN (D∩B(x, ρ)) =mN (Ω∩B(x, ρ)) = 0. Since Ω⊂D, there exists x∈D such that mN (D∩B(x, ρ)) = 0.But this is a contradiction since D is an open set with Lipschitzian boundary.

16.2.2 Another Compactness Theorem

The compactness of the following special family of cracked sets contained in a frameis a corollary to the compactness Theorem 15.2.

Theorem 16.2. Let D be a bounded open subset of RN with Lipschitzian boundaryand h > 0 and α > 0 be real numbers. Consider the family

Fc(D, h,α) def=

⎧⎪⎪⎪⎨

⎪⎪⎪⎩Ω ⊂ D :

Ω open ⊂ D and mN (Ω) = mN (D)and ∀x ∈ Γ, ∃d, |d| = 1,

such that inf0<t<h

dΓ(x + td)t

≥ α

⎫⎪⎪⎪⎬

⎪⎪⎪⎭,

Cc,h,αb (D) def= bΩ : Ω ∈ Fc(D, h,α) .

(16.4)

Then Cc,h,αb (D) is compact in W 1,p(D), 1 ≤ p <∞.

Proof. By standard arguments and Lemma 16.2. The conclusion follows from The-orem 15.2 by adding the constraint mN (Ωn) = mN (D) which will be verified forthe limit set Ω for which a subsequence of bΩn converges to bΩ in W 1,1(D) andhence χΩn converges to χΩ in L1(D).

16.2.3 Proof of Theorem 16.1

Proof of Theorem 16.1. (i) For each open Ω ∈ Fc(D, h,α), the problem

infϕ∈H1(Ω)

F (Ω,ϕ), F (Ω,ϕ) def=∫

Ωε|∇ϕ|2 + |ϕ− f |2 dx

has a unique solution y in H1(Ω) since the objective function F (Ω,ϕ) is continuousand coercive on H1(Ω). Define

mdef= inf

Ω∈Fc(D,h,α)inf

ϕ∈H1(Ω)

∫

Ωε |∇ϕ|2 + |ϕ− f |2 dx.


(ii) The minimum is finite since the objective function is positive and

∀ open Ω ∈ Fc(D, h,α), infϕ∈H1(Ω)

∫

Ωε |∇ϕ|2 + |ϕ− f |2 dx ≤

∫

Ω|f |2 dx

by choosing ϕ = 0. Let Ωn be a minimizing sequence of open subsets of D inFc(D, h,α) and for each n let yn ∈ H1(Ωn) be the minimizing element of F (Ωn,ϕ)over H1(Ωn). Therefore

∫

Ωn

ε |∇yn|2 + |yn − f |2 dx→ m

⇒ ∃c > 0 such that ∀n,

∫

Ωn

ε |∇yn|2 + |yn − f |2 dx ≤ c.

By coercivity the sequence yn is uniformly bounded in H1(Ωn); that is, thereexists a constant c > 0 such that

∀n, ∥yn∥L2(Ωn) ≤ c and ∥∇yn∥L2(Ωn) ≤ c.

By Theorem 16.2, there exist a subsequence of Ωn and an open set Ω ∈ Fc(D, h,α)such that bΩn → bΩ in H1(D) and C(D). In particular d!Ωn

→ d!Ω in C(D). Bythe compactivorous property of Theorem 2.4 (iii) in Chapter 6

∀K compact ⊂ int Ω, ∃N such that ∀n ≥ N, K ⊂ Ωn.

Moreover

χΩn → χΩ ∈ L2(D) ⇒ χΩn χΩ ∈ L∞(D)-weak∗ (16.5)

⇒ fχΩn fχΩ ∈ L2(D)-weak. (16.6)

Define the distributions

⟨yn,ϕ⟩ def=∫

Ωn

yn ϕ dx, ∀ϕ ∈ D(D),

⟨∇yn, Φ⟩ def=∫

Ωn

∇yn · Φ dx, ∀Φ ∈ D(D)N ,

⟨∇yn, Φ⟩ = −∫

Dyn div Φ dx, ∀Φ ∈ D(D)N .

It is readily seen that we can identify yn and ∇yn with the extensions of yn and∇yn by zero from Ωn to D. As a result there exist subsequences, still denoted byyn and ∇yn, and y ∈ L2(D) and Y ∈ L2(D)N such that yn y in L2(D)-weakand ∇yn Y in L2(D)N -weak.

By the compactivorous property, for all Φ ∈ D( Ω), there exists N such that

∀n > N, supp Φ ⊂ Ωn ⇒ Φ ∈ D(Ωn).


Therefore for all n > N

⟨∇yn − ∇yn, Φ⟩ = −∫

Ωn

yn div Φ dx−∫

Ωn

∇yn · Φ dx = 0

since yn ∈ H1(Ωn). But D(Ωn) ⊂ D(D) and by letting n go to infinity

0 = −∫

Ωn

yn div Φ dx−∫

Ωn

∇yn · Φ dx→ −∫

Dy div Φ dx−

∫

DY · Φ dx

⇒ ∀Φ ∈ D(Ω),∫

Dy div Φ dx +

∫

DY · Φ dx = 0.

Define the new distribution

⟨y,ϕ⟩ def=∫

Ωy ϕ dx, ∀ϕ ∈ D(Ω).

It is easy to check that y ∈ L2(Ω) and hence

∀Φ ∈ D(Ω), 0 =∫

Dy div Φ dx +

∫

DY · Φ dx =

∫

Ωy div Φ dx +

∫

ΩY · Φ dx

⇒ ∀Φ ∈ D(Ω), ⟨∇y, Φ⟩ = −∫

Ωy div Φ dx =

∫

ΩY · Φ dx

⇒ ∇y = Y |Ω ∈ L2(Ω) ⇒ y ∈ H1(Ω).

(iv) Coming back to our objective function

infϕ∈H1(Ωn)

∫

Ωn

ε |∇ϕ|2 + |ϕ− f |2 dx =∫

Ωn

ε |∇yn|2 + |yn − f |2 dx

=∫

Dε |∇yn|2 + |yn − fχΩn |2 dx.

By convexity and continuity of the objective function with respect to the pair (yn−fχΩn , ∇yn) in L2(D)×L2(D)N and the fact that, from (16.6), (yn−fχΩn , ∇yn) (y − fχΩ, Y ) in L2(D)× L2(D)N -weak,

∫

Dε |Y |2 + |y − fχΩ|2 dx ≤ lim inf

n→∞

∫

Dε |∇yn|2 + |yn − fχΩn |2 dx

= lim infn→∞

∫

Ωn

ε |∇yn|2 + |yn − f |2 dx = m

⇒∫

Ωε |∇y|2 + |y − f |2 dx =

∫

Dε |∇y|2 + |y − fχΩ|2 dx ≤ m.

By definition of the minimum we have the equality and there exist an open setΩ ∈ Fc(D, h,α) and y ∈ H1(Ω) solution of the segmentation problem.


16.3 Existence of a Cracked Set with Minimum DensityPerimeter

Theorem 16.1 gives an existence result for the family F(D, h,α) of open sets Ωsuch that mN (Ω) = mN (D) without constraint on the “perimeter” of Ω. Denote byF∗(D, h,α) the set of solutions to problem (16.3). In general, the perimeter can beinfinite as can be seen from the following example.

Example 16.1.The function f : D → R is defined as follows:

f(x) =

1, if x ∈ Ω1,

0, if x ∈ Ω2,(16.7)

where D = (x, y) : −2 < x < 3, −1 < y < 3, Ω = Ω1 ∪ Ω2, Ω2 = D\Ω1, and theopen set Ω1 is constructed below (see Figure 7.5). The set Ω with y = f is a solutionof problem (16.3) with infinite perimeter. The set Ω1 is a two-dimensional examplesimilar to Examples 6.1 and 6.2 of section 6.5 in Chapter 2 of an open domainsatisfying the uniform cusp condition for the function h(θ) = θα, 0 < α < 1. It caneasily be generalized to an N -dimensional example. Consider the open domain Ω1in R2

Ω1def= (x, y) : −1 < x ≤ 0 and 0 < y < 2∩ (x, y) : 0 < x < 1 and f(x) < y < 2∩ (x, y) : 1 ≤ x < 2 and 0 < y < 2 ,

where f : [0, 1]→ R is defined as follows:

f(x) def=∞∑

k=0

fk(x), fk :[1− 1

2k, 1− 1

2k+1

]→ R .

Associate with 0 < α < 1 and k ≥ 0 the even integer ηk = 2 [(2k+1)α

1−α ], where [β]is the smallest integer greater than or equal to β. Assume that for each k ≥ 0

fkdef=

ηk/2∑

j=1

gk,j gk,j : [xk,j−1, xk,j−1 + δk]→ R,

xk,jdef= 1− 1

2k+ (j − 1)2δk, 1 ≤ j ≤ ηk/2, δk

def=1

ηk 2k+1

and that the function gk,j is given by the expression

gk,j(x) def=

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

0, 0 ≤ x < xk,j−1,

(x− xk,j−1)α, xk,j−1 ≤ x ≤ xk,j−1 + δk,

(2xk,j−1 − x)α, xk,j−1 + δk ≤ x ≤ xk,j−1 + 2δk,

0, x > xk,j−1 + 2δk.


Ω1

Ω2

f = 1

f = 0

frame D

Figure 7.5. The two open components Ω1 and Ω2 of the open domain Ωfor N = 2.

Note that

gk,j(xk,j−1 + δk) = (δk)α

is independent of j and is the maximum of the function gk,j .The uniform cusp property is verified for ρ = 1/8, λ = h(ρ), and h(θ) = θα.

The boundary of Ω1 is made up of straight lines of total length 9 plus the length ofthe curve

Cdef= (x, f(x)) : 0 ≤ x < 1 , C = ∪∞

k=0Ck, Ck = ∪ηk/2j=1 Ck,j ,

Ck,jdef= (x, f(x)) : xk,j−1 ≤ x < xk,j−1 + δk .

The length of the curve Ck,j is bounded below by

HN−1(Ck,j) ≥ 2√

(δk)2 + (δk)2α ≥ 2 (δk)α ⇒ HN−1(Ck) ≥ηk/2∑

j=1

HN−1(Ck,j),

HN−1(Ck) ≥ ηk

22 (δk)α = ηk (δk)α = ηk

(1

ηk 2k+1

)α= ηk

1−α(

12k+1

)α,

HN−1(Ck) ≥ ηk1−α

(1

2k+1

)α=(2[(2k+1)

α1−α ]

)1−α(

12k+1

)α

≥ 21−α(2k+1)α(

12k+1

)α= 21−α

⇒ HN−1(C) =∞∑

k=0

HN−1(Ck) ≥ +∞ 21−α = +∞.

When at least one solution has a bounded perimeter, it is possible to showthat there is one that minimizes the h-density perimeter.


Theorem 16.3. Assume that the assumptions of Theorem 16.1 are verified. Thereexists an Ω∗ in F∗(D, h,α) which minimizes the h-density perimeter.

Proof. If for all Ω in F∗(D, h,α) the h-density perimeter is +∞, the theorem istrue. If for some Ω ∈ F∗(D, h,α), Ph(Γ) ≤ c, then there exists a sequence Ωn inF∗(D, h,α) such that

Ph(Γn)→ infΩ∈F∗(D,h,α)

Ph(Γ).

By the compactness Theorems 12.1 and 16.2, there exist a subsequence and Ω∗,Γ∗ = ∅, such that bΩn → bΩ∗ in W 1,p(D), Ω∗ ∈ F (D, h,α), mN (Ω) = mN (D), and

Ph(Γ∗) ≤ lim infn→∞

Ph(Γn) ≤ c.

Finally by going back to the proof of Theorem 16.1 and using the fact that all theΩn’s are already minimizers in F ∗(D, h,α), it can be shown that Ω∗ is indeed oneof the minimizers in the set F ∗(D, h,α).

16.4 Uniform Bound or Penalization Term in the ObjectiveFunction on the Density Perimeter

To complete the results on the segmentation problem, we turn to the existence of asegmentation for a family of sets with a uniform bound or with a penalization termin the objective function on the h-density perimeter.

Theorem 16.4. Given a bounded open frame ∅ = D ⊂ RN with a Lipschitzianboundary and real numbers h > 0 and c > 0,10 there exists an open subset Ω∗ of D,Γ∗ = ∅, with finite h-density perimeter such that mN (Ω∗) = mN (D) (Ph(Γ∗) ≤ cfor (16.8)), and y ∈ H1(Ω∗) solutions of the respective problems

infΩ open ⊂D, Ph(Γ)≤c

mN (Ω)=mN (D)

infϕ∈H1(Ω)

∫

Ωε |∇ϕ|2 + |ϕ− f |2 dx, (16.8)

infΩ open ⊂D

mN (Ω)=mN (D)

infϕ∈H1(Ω)

∫

Ωε |∇ϕ|2 + |ϕ− f |2 dx + c Ph(Γ). (16.9)

Proof. The proof for the objective function (16.8) is exactly the same as the oneof Theorem 16.1. It uses Lemma 16.2 to show that the minimizing set has a thinboundary and the compactness Theorem 12.1. The proof for the objective func-tion (16.9) uses the fact that there is a minimizing sequence for which the h-densityperimeter is uniformly bounded and the lower semicontinuity of the density perime-ter in the W 1,p-topology is given by Corollary 1.

Problem (16.9) was originally considered in D. Bucur and J.-P. Zolesio [8].The above two identification problems can be further specialized to the family ofcracked sets F(D, h,α).

10Note that the constant c must be large enough to take into account the contribution of theboundary of D.


Corollary 1. Given a bounded open frame D ⊂ RN with a Lipschitzian boundaryand real numbers h > 0, α > 0, and c > 0, there exists an open subset Ω∗ of D inF(D, h,α) such that mN (Ω∗) = mN (D) (Ph(Γ∗) ≤ c for (16.10)), and y ∈ H1(Ω∗)solutions of the problem

infΩ open ⊂D, Ω∈F(D,h,α)Ph(Γ)≤c, mN (Ω)=mN (D)

infϕ∈H1(Ω)

∫

Ωε |∇ϕ|2 + |ϕ− f |2 dx, (16.10)

infΩ open ⊂D, Ω∈F(D,h,α)

mN (Ω)=mN (D)

infϕ∈H1(Ω)

∫

Ωε |∇ϕ|2 + |ϕ− f |2 dx + c Ph(Γ). (16.11)

Proof. Since the minimizing sequence bΩn constructed in the proof of Theo-rem 16.1 strongly converges to bΩ∗ in W 1,p(D) for all p, 1 ≤ p < ∞, from prop-erty (12.3) in Theorem 12.1, we have

Ph(Γ∗) ≤ lim infn→∞

Ph(Γn) ≤ c

and the optimal Ω∗ constructed in the proof of the theorem satisfies the additionalconstraint on the density perimeter.

Chapter 8

Shape Continuity andOptimization

1 Introduction and Generic ExamplesThe underlying philosophy behind Chapters 5, 6, and 7 was to introduce metrictopologies on sets associated with families of functions parametrized by sets ratherthan parametrize sets by functions. In the case of the characteristic function we haveseen several examples where it appears explicitly in the modeling of the problem.The distance functions provide other topologies and constructions. For instancethe oriented distance function gives a direct analytic access to geometric entitiessuch as the normal and the fundamental forms on the boundary of a geometricdomain without ad hoc local bases or Christoffel symbols. Many of the advantagesof working in a truly intrinsic framework for shape derivatives will be illustrated inChapters 9 and 10.

Now that we have meaningful topologies on sets, we can consider the continuityof a geometric objective functional such as the volume, the perimeter, the meancurvature, etc. In this chapter we concentrate on continuity issues related to shapeoptimization problems under state equation constraints. A special family of stateconstrained problems are the ones for which the objective function is defined as aninfimum over a family of functions over a fixed domain or set such as eigenvalueand compliance problems. What is nice about them is that no adjoint systemis necessary to characterize the minimizing function. Other problems have thestructure of optimal control theory, that is, a state equation depending on the controlthat describes the evolution of the state and an objective functional that depends onthe state and the control. In that case, the characterization of the optimal controlinvolves an adjoint state equation coupled with the state equation.

In the context of shape and geometric optimization the control will be theunderlying geometry. The state equation will be a static or dynamical partial dif-ferential equation on the domain, and the objective functional will depend on thedomain and the state that itself depends on the domain. As in control theory, weneed continuity of the objective functional and the state with respect to the geom-etry and compactness of the family of domains over which the optimization takes

409

A special family of state constrained problems are the ones for which the objective function is defined as an infimum over a family of functions over a fixed domain or set such as eigenvalue and compliance problems. What is nice about them is that no adjoint system is necessary to characterize the minimizing function.

410 Chapter 8. Shape Continuity and Optimization

place. Since it is not possible to give a complete general theory of shape optimiza-tion problems within this book, we choose to concentrate on a few simple genericexamples that are illustrative of the underlying technicalities involved.

Several examples of optimization problems involving the Lp-topology on mea-surable sets via the characteristic function were given in Chapter 5. In this chapter,we first characterize the continuity of the transmission problem and the upper semi-continuity of the first eigenvalue of the generalized Laplacian with respect to thedomain. We then study the continuity of the solution of the homogeneous Dirichletand Neumann boundary value problems with respect to their underlying domain ofdefinition since they require different constructions and topologies that are genericof the two types of boundary conditions even for more complex nonlinear partialdifferential equations. In the above problems the strong continuity of solutions ofthe elliptic equation or the eigenvector equation with respect to the underlying do-main is the key element in the proof of the existence of optimal domains. To getthat continuity, some extra conditions have to be imposed on the family of opendomains.

Those issues have received a lot of attention for the Laplace equation withhomogeneous Dirichlet boundary conditions. With a sequence of domains in afixed holdall D associate a sequence of extensions by zero of the solutions in thefixed space H1

0 (D). The Poincare inequality is uniform for that sequence, as thefirst eigenvalue of the Laplace equation in each domain is dominated by the oneassociated with the larger holdall D. By a classical compactness argument, thesequence of extensions converges to some limiting element y in H1

0 (D).To complete the proof of the continuity, two more fundamental questions re-

main. Is y the solution of the Laplace equation for the limit domain? Does y satisfythe Dirichlet boundary condition on the boundary of the limit domain?

The first issue can be resolved by assuming that the Sobolev spaces associatedwith the moving domains converge in the Kuratowski sense, i.e., with the propertythat any element in the Sobolev space associated with the limit domain can beapproached by a sequence of elements in the moving Sobolev spaces. That propertyis obtained in examples where the compactivorous property1 follows from the choiceof the definition of convergence of the domains. If the domains are open subsetsof D, then the complementary Hausdorff topology has that property. The family ofsubsets of D satisfying a uniform fat segment property has that property.

For the second issue it is necessary to impose some constraints at least on thelimit domain. The stability condition introduced by J. Rauch and M. Taylor [1]and used by E. N. Dancer [1] and D. Daners [1] precisely assumes that thelimit domain is sufficiently smooth, just enough to have the solution in H1

0 (Ω). Ofcourse, assuming such a regularity on the limit domain makes things easier but isa little bit artificial. The more fundamental issue is to identify the families of do-mains for which this stability property is preserved in the limit. The uniform coneproperty was used in the context of shape optimization by D. Chenais [1, 2] in1973. In 1994 D. Bucur and J.-P. Zolesio [3, 5, 6, 8, 9] showed that this is trueunder general capacitary conditions and have constructed new compact subfamilies

1A sequence of open sets converging to a limit open set in some topology has the compactivorousproperty if any compact subset of the limit set is contained in all sets of the sequence after a certainrank: the sequence eats up the compact set after a certain rank (cf. Theorem 2.4 (iii) in Chapter 6).

1. Introduction and Generic Examples 411

of domains with respect to the Hausdorff complementary topology. FurthermoreD. Bucur [1] proved that the condition given in 1993 by V. Sverak [1, 2] indimension 2 involving a bound on the number of connected components of the com-plement of the domain can be recovered from the more general capacity conditionsthat are sufficient in the case of the Laplacian with homogeneous Dirichlet boundaryconditions. In a more recent paper D. Bucur [5] proved that they are almost nec-essary. Intuitively, those capacity conditions are such that, locally, the complementof the domains in the family under consideration has enough capacity to preserveand retain the homogeneous Dirichlet boundary condition in the limit.

1.1 First Generic ExampleThe first example is the optimization of the first eigenvalue of an associated ellipticdifferential operator:

supΩ∈A(D)

λA(Ω)

infΩ∈A(D)

λA(Ω)

∣∣∣∣∣∣∣λA(Ω) def= inf

0 =ϕ∈H10 (Ω)

∫Ω A∇ϕ ·∇ϕ dx∫

Ω |ϕ|2 dx, (1.1)

where A(D) is a family of admissible open subsets of D.

1.2 Second Generic ExampleAs a second generic example, consider the transmission problem over the fixedbounded open nonempty domain D associated with the Laplace equation and ameasurable subset Ω of D: find y = y(χΩ) ∈ H1

0 (D) such that

∀ϕ ∈ H10 (D),

∫

D(k1χΩ + k2(1− χΩ))∇y ·∇ϕ dx = ⟨f,ϕ⟩ (1.2)

for some strictly positive constants k1 and k2, α = maxk1, k2 > 0, f ∈ H−1(D),H−1(D) the topological dual of H1

0 (D), and ⟨·, ·⟩ denotes the duality pairing betweenH−1(D) and H1

0 (D). This problem was studied in section 4 of Chapter 5 in thecontext of the compliance problem with two materials.

1.3 Third Generic ExampleAs a third example, consider the minimization of the objective function

J(Ω) def=12

∫

Ω|uΩ − g|2 dx (1.3)

over the family of open subsets Ω of a bounded open holdall D of RN, where g inL2(D), uΩ ∈ H1

0 (Ω) is the solution of the variational problem

∀ϕ ∈ H10 (Ω),

∫

ΩA∇uΩ ·∇ϕ dx = ⟨f |Ω,ϕ⟩H−1(Ω)×H1

0 (Ω), (1.4)

A ∈ L∞(D;L(RN,RN)) is a matrix function on D such that ∗A = A and αI ≤A ≤ βI for some coercivity and continuity constants 0 < α ≤ β, and f |Ω denotesthe restriction of f in H−1(D) to H−1(Ω).


1.4 Fourth Generic Example

As a fourth generic example, consider the previous example but with the homo-geneous Dirichlet boundary value problem replaced by the homogeneous Neumannboundary value problem of the mathematician for the Laplacian

−∆y + y = f in Ω,∂y

∂n= 0 on ∂Ω. (1.5)

The classical (physical) homogeneous Neumann boundary value problem

−∆y = f in Ω,∂y

∂n= 0 on ∂Ω (1.6)

has a unique solution up to a constant if∫

D f dx = 0.

2 Upper Semicontinuity and Maximization of theFirst Eigenvalue

Given a bounded open nonempty domain D ⊂ RN, consider the minimization ofthe Raleigh quotient for an open nonempty subset Ω of D:

λA(Ω) def= inf0 =ϕ∈H1

0 (Ω;D)

∫D A∇ϕ ·∇ϕ dx∫

D |ϕ|2 dx= infϕ∈H1

0 (Ω;D)∥ϕ∥L2=1

∫

DA∇ϕ ·∇ϕ dx, (2.1)

where A ∈ L∞(D;L(RN,RN)) is a matrix function on D such that

∗A(x) = A(x) and αI ≤ A(x) ≤ βI (2.2)

for some coercivity and continuity constants 0 < α ≤ β. We set λA(∅) = +∞. Theminimizers are characterized as follows.

Theorem 2.1. (i) Given a bounded open nonempty domain Ω in RN, there existuΩ ∈ H1

0 (Ω; D), uΩ = 0, and λA(Ω) > 0 such that

∀ϕ ∈ H10 (Ω; D),

∫

ΩA∇uΩ ·∇ϕ− λA(Ω) uΩ ϕ dx = 0 (2.3)

and

∀ϕ ∈ H10 (Ω; D), λA(Ω) =

∫D A∇uΩ ·∇uΩ dx∫

D |uΩ|2 dx≤∫

D A∇ϕ ·∇ϕ dx∫D |ϕ|2 dx

. (2.4)

Conversely, if there exist u ∈ H10 (Ω; D), u = 0, and λ such that

∀ϕ ∈ H10 (Ω; D),

∫

ΩA∇u ·∇ϕ− λuϕ dx = 0 (2.5)

2. Upper Semicontinuity and Maximization of the First Eigenvalue 413

and

∀ϕ ∈ H10 (Ω; D), λ ≤


D |ϕ|2 dx, (2.6)

then

λ =∫

D A∇u ·∇u dx∫D |u|2 dx

= infϕ∈H1

0 (Ω;D)


D |ϕ|2 dx,

u is a nonzero solution of the minimization problem (2.1), and λ is equal tothe first eigenvalue λA(Ω) associated with Ω.

(ii) Given a bounded open nonempty domain D, there exists λAD > 0 (that depends

only on the diameter of D and A) such that for all open subsets Ω of D,

λA(Ω) ≥ λA(D) ≥ λAD > 0,

where λD(Ω) is the first eigenvalue of the symmetrical linear operator ϕ ,→−div (A∇ϕ) on D.

Proof. (i) For any bounded open Ω, the infimum is bounded below by 0 and hencefinite. Let ϕn be a minimizing sequence such that ∥ϕn∥L2(Ω) = 1. By coercivity,α ∥∇ϕn∥2L2(Ω) is bounded. Hence ϕn is a bounded sequence in H1

0 (Ω) and thereexist ϕ ∈ H1

0 (Ω) and a subsequence, still indexed by n, such that ϕn ϕ inH1(Ω)-weak. By the Rellich–Kondrachov compactness theorem, the subsequencestrongly converges in L2(Ω). Then

1 =∫

Ωϕ2

n dx→∫

Ωϕ2 dx and

∫

ΩA∇ϕ ·∇ϕ dx ≤ lim inf

n→∞

∫

ΩA∇ϕn ·∇ϕn dx

⇒∫Ω A∇ϕ ·∇ϕ dx∫

Ω ϕ2 dx

≤ lim infn→∞

∫Ω A∇ϕn ·∇ϕn dx∫

Ω ϕ2n dx

= λA(Ω).

By definition of λA(Ω), 0 = ϕ ∈ H10 (Ω) is a minimizing element and inequality

(2.4) is verified. Since ϕ = 0, the Rayleigh quotient is differentiable in ϕ and itsdirectional derivative in the direction ψ is given by

2∫Ω A∇ϕ ·∇ψ dx

∥ϕ∥2L2(Ω)− 2

∫

ΩA∇ϕ ·∇ϕ dx

∫Ω ϕψ dx

∥ϕ∥4L2(Ω).

A nonzero solution ϕ of the minimization problem on Ω is necessarily a stationarypoint, and for all ψ ∈ H1

0 (Ω),∫

ΩA∇ϕ ·∇ψ dx =

∫Ω A∇ϕ ·∇ϕ dx

∥ϕ∥2L2(Ω)

∫

Ωϕψ dx = λA(Ω)

∫

Ωϕψ dx

and we get (2.3). Conversely, any λ and nonzero u ∈ H10 (Ω; D) that verify (2.5)

yield λ ≥ λA(Ω). But, from inequality (2.6), λ ≤ λA(Ω) and λ = λA(Ω). Hence, uis a minimizer of the Rayleigh quotient over H1

0 (Ω; D).


(ii) Let diam (D) be the diameter of D and x ∈ RN be a point such thatD ⊂ Bx = B(x,diam (D)). Therefore, from Theorem 2.3 in Chapter 2,

H10 (Ω; Bx) ⊂ H1

0 (D;Bx) ⊂ H10 (Bx), H1

0 (Ω; D) ⊂ H10 (D)

with the associated isometries. Hence, by definition of the minimum of a functionalover an increasing sequence of sets, λA(Ω) ≥ λA(D) ≥ λA(Bx). If λA(Bx) = 0, werepeat the above construction with H1

0 (Bx) in place of H10 (Ω; Bx) and end up with

an element ϕ ∈ H10 (Bx) such that

∫

Bx

ϕ2 dx = 1 and α

∫

Bx

|∇ϕ|2 dx ≤∫

Bx

A∇ϕ · ϕ dx = 0,

which is impossible in H10 (Bx). Finally λA(Bx) is independent of the choice of x

since the eigenvalue is invariant under a translation of the domain: it depends onlyon the diameter of D. The case λA(∅) = +∞ is compatible with the results.

Remark 2.1.For A equal to the identity operator I, denote by λ(Ω), λ(D), and λD the quantitiesλA(Ω), λA(D), and λA

D of Theorem 2.1. For any open Ω ⊂ D and any ϕ ∈ H10 (Ω; D)

∥ϕ∥L2(D) ≤1√λD∥∇ϕ∥L2(D) (2.7)

⇒ ∀Ω open ⊂ D, ∥ϕ∥H1(Ω) ≤√

1 +1λD∥∇ϕ∥L2(Ω) (2.8)

and the spaces H10 (Ω) and H1

0 (D) will be endowed with the respective equivalentnorms ∥∇ϕ∥L2(Ω) and ∥∇ϕ∥L2(D).

Before considering the existence of minimizers/maximizers, we investigate thequestion of the continuity of the first eigenvalue λA(Ω) with respect to Ω for theuniform complementary Hausdorff topology on

Ccd(D) def=

d!Ω : ∀Ω open subset in D and Ω = RN

(cf. (2.11) of Definition 2.2 in Chapter 6).

Theorem 2.2. Let D be a bounded open domain in RN and A be a matrix functionsatisfying assumption (2.2). The mapping

d!Ω ,→ λA(Ω) : Ccd(D) ⊂ C0(D)→ R∪+∞ (2.9)

is upper semicontinuous.

Proof. Since λA(Ω) is finite in Ω = ∅ and +∞ in Ω = ∅, it is upper semicontinuousin ∅. For Ω = ∅, consider a sequence Ωn of open subsets of D such that d!Ωn

→d!Ω in C(D) and the associated sequence λA(Ωn). We want to show that

lim supλA(Ωn) ≥ λA(Ω).

2. Upper Semicontinuity and Maximization of the First Eigenvalue 415

Consider the new sequence Ωn∩Ω. Since Ω ⊂ D, for all n, !(Ωn∩Ω) = !Ωn∪!Ω ⊃!D = ∅ and

d!(Ωn∩Ω) = d!Ωn∪!Ω = mind!Ωn, d!Ω→ d!Ω in C(D).

Let un ∈ H10 (Ωn ∩ Ω), ∥un∥L2(D) = 1, be an eigenvector for the eigenvalue λn =

λA(Ωn ∩ Ω). From Theorem 2.1 un is solution of the variational equation (2.3):

∀ϕn ∈ H10 (Ωn;D),

∫

DA∇un ·∇ϕn dx = λn

∫

Dun ϕn dx. (2.10)

Since Ω = ∅, there exists a compact ball Kρ = B(x, ρ) ⊂ Ω for some x ∈ Ω andρ > 0. By the compactivorous property of Theorem 2.4 in section 2.3 of Chapter 6,there exists an integer N(Kρ) such that for all n ≥ N(Kρ), Kρ ⊂ Ωn ∩ Ω and byTheorem 2.1 (ii), 0 < λn ≤ λA(B(x, ρ)). Therefore, the sequence of eigenvaluesλn is bounded. By coercivity

α∥∇un2L2(D) ≤ λn ∥∇un2

L2(D) = λn

and un is bounded in H10 (Ω; D) since Ωn ∩Ω ⊂ Ω. Letting λ∗ = lim supλn, there

exist subsequences, still indexed by n, and u ∈ H10 (Ω; D) such that

λn → λ∗, ∥un∥L2(D) = 1, un u in H10 (Ω; D)-weak ⇒ ∥u∥L2(D) = 1.

Since for all n, λA(Ωn) ≤ λA(Ωn ∩ Ω), it is readily seen that

lim supλA(Ωn) ≤ lim supλA(Ωn ∩ Ω) = λ∗

and that it is sufficient to show that λ∗ = λA(Ω) to get the upper semicontinuity.We cannot directly go to the limit in (2.10), since the test functions ϕn dependon Ωn. To get around this difficulty, recall the density of D(Ω) in H1

0 (Ω; D) and usethe compactivorous property a second time. For any ϕ ∈ D(Ω), K = suppϕ ⊂ Ωis compact and there exists N(K) such that for all n ≥ N(K), suppϕ ⊂ Ωn ∩ Ω.Therefore, for all n ≥ N(K)

un ∈ H10 (Ωn ∩ Ω; D) ⊂ H1

0 (Ω; D),∫

DA∇un ·∇ϕ dx = λn

∫

Dun ϕ dx.

By letting n→∞

∀ϕ ∈ D(Ω),∫

DA∇u ·∇ϕ dx = λ∗

∫

Duϕ dx,

and, by density of D(Ω) in H10 (Ω; D), the variational equation is verified in H1

0 (Ω; D).Hence, u ∈ H1

0 (Ω; D), ∥u∥L2(Ω) = 1, satisfies the variational equation

∀ϕ ∈ H10 (Ω; D),

∫

DA∇u ·∇ϕ dx = λ∗

∫

Duϕ dx.


Therefore, since λA(Ω) is the smallest eigenvalue with respect to H10 (Ω; D),

λA(Ω) ≤ λ∗. To show that λ∗ = λA(Ω), it remains to prove that λ∗ ≤ λA(Ω). Foreach n

∀ϕn ∈ H10 (Ωn ∩ Ω; D), λA(Ωn ∩ Ω) ≤

∫D A∇ϕn ·∇ϕn dx

∥ϕn∥2L2(D).

By the compactivorous property for each ϕ ∈ D(Ω), ϕ = 0, there exists N suchthat suppϕ ⊂ Ω ∩ Ωn for all n ≥ N . Therefore for all n ≥ N

λA(Ωn ∩ Ω) ≤∫

D A∇ϕ ·∇ϕ dx

∥ϕ∥2L2(D)

and by taking the lim sup

∀ϕ ∈ D(Ω), λ∗ ≤∫

D A∇ϕ ·∇ϕ dx

∥ϕ∥2L2(D.

By density this is true for all ϕ ∈ H10 (Ω; D), ϕ = 0, and λ∗ ≤ λA(Ω). As a result

λ∗ = λA(Ω). Finally,

λA(Ωn) ≤ λA(Ωn ∩ Ω) ⇒ lim supλA(Ωn) ≤ lim supλA(Ωn ∩ Ω) = λ∗ = λA(Ω),

and we have the upper semicontinuity of the map d!Ω ,→ λA(Ω).

In view of the previous theorem, we now deal with the maximization of thefirst eigenvalue λA(Ω) associated with the operator A with respect to a family ofopen subsets Ω of D.

Theorem 2.3. Let D be a bounded open domain in RN and A be a matrix functionsatisfying assumption (2.2).

(i) Then

supΩ∈Cc

d(D)

λA(Ω) = λA(∅) = +∞. (2.11)

The conclusion remains true when Ccd(D) is replaced by

Ccd(D) = d!Ω : Ω convex and open ⊂ D (2.12)

(cf. Theorem 8.6 in Chapter 6).

(ii) Given an open set E, ∅ = E ⊂ RN, consider the compact family

Ccd,loc(E;D) =

d!Ω :


of Theorem 2.6 in Chapter 6. Then, there exists an open Ω∗ such that d!Ω∗ ∈Cc

d,loc(E;D) and

λA(Ω∗) = supΩ∈Cc

d,loc(E;D)λA(Ω) ≤ β λI(E). (2.13)

3. Continuity of the Transmission Problem 417

The conclusion remains true when Ccd,loc(E;D) is replaced by Cc

d(E;D) orthe subfamilies of convex subsets Cc

d(E;D) and Ccd,loc(E;D) of Theorem 8.6 in

Chapter 6.

(iii) There exists Ω∗ ∈ L(D, r,O,λ) such that

λA(Ω∗) = supΩ∈L(D,r,O,λ)

λ(Ω) ≤ βλI(O). (2.14)

Proof. (i) From Theorem 2.2 and the compactness of Ccd(D) in C(D).

(ii) All the families are compact by Theorems 2.6 and 8.6 in Chapter 6. Sothere is a maximizer d!Ω∗ of λA(Ω). The upper bound is obvious.

(iii) By compactness of L(D, r,O,λ) in W 1,p(D) and hence in Ccd(D) for the

C(D)-topology.

3 Continuity of the Transmission ProblemAs a second generic example, consider the transmission problem over the fixedbounded open nonempty domain D associated with the Laplace equation and ameasurable subset Ω of D: find y = y(χΩ) ∈ H1

0 (D) such that

∀ϕ ∈ H10 (D),

∫

D(k1χΩ + k2(1− χΩ))∇y ·∇ϕ dx = ⟨f,ϕ⟩ (3.1)

for some strictly positive constants k1 and k2, α = maxk1, k2 > 0, f ∈ H−1(D),H−1(D) the topological dual of H1

0 (D), and ⟨·, ·⟩ denotes the duality pairing betweenH−1(D) and H1

0 (D). This problem was studied in section 4 of Chapter 5 in thecontext of the compliance problem with two materials.

Theorem 3.1. Let D be a bounded open domain in RN and k1 and k2 be somestrictly positive constants such that α = maxk1, k2 > 0 and f ∈ H−1(D). Denoteby y(χ) the solution of the transmission problem (3.1) associated with χ ∈ X(D).The mapping

χ ,→ y(χ) : X(D)→ H10 (D)

is continuous when X(D) is endowed with the strong Lp(D)-topology, 1 ≤ p < ∞.The same result is true with co X(D) in place of X(D).

Remark 3.1.This theorem applies to the compact families of sets of Theorems 13.1, 14.1, and14.2 in section 13 of Chapter 7 under the uniform fat segment property and forthe conditions on the local graph functions of Theorem 13.2 (cf. Theorem 5.1 ofChapter 2) since the convergence of bΩn → bΩ in W 1,p(D) implies the convergenceof χΩn → χΩ in Lp(D). It also applies to the compact family of convex sets C(D) =χΩ : Ω convex subset of D introduced in (3.24) in section 3.4 of Chapter 5 (cf.Theorem 3.5 (iii) and Corollary 1 to Theorem 6.3 in section 6.1 of Chapter 5).


Proof of Theorem 3.1. The proof follows the arguments of J.-P. Zolesio [2], [24,sect. 4.2.1, pp. 446–448]. It is sufficient to prove that for any convergent sequenceχn → χ in L2(D)-strong, the corresponding solutions yn = y(χn) ∈ H1

0 (D) convergeto y = y(χ) in H1

0 (D)-strong. First, the sequence yn is bounded in H10 (D):

α∥∇yn∥2L2(D) ≤∫

D(k1χn + k2(1− χn))∇yn ·∇yn dx

= ⟨f, yn⟩ ≤ ∥f∥H−1(D) ∥yn∥H10 (D)

⇒ α∥∇yn∥2L2(D) ≤ ∥f∥H−1(D) ∥yn∥H10 (D) ≤ ∥f∥H−1(D)

√1 +

1λD∥∇yn∥L2(Ω)

since we have the equivalence of norms from (2.7) in Remark 2.1. So there is asubsequence, still denoted by yn, that weakly converges to some y ∈ H1

0 (D). Toshow that y is solution of (3.1) corresponding to χ, we go to the limit in

∀ϕ ∈ H10 (D),

∫

D(k1χn + k2(1− χn))∇ϕ ·∇yn dx = ⟨f,ϕ⟩.

Since (k1χn+k21−χn))∇ϕ→ (k1χ+k2(1−χ))∇ϕ in L2(D)-strong and ∇yn → ∇yin L2(D)-weak, we get that y ∈ H1

0 (D) is the unique solution of

∀ϕ ∈ H10 (D),

∫

D(k1χ+ k2(1− χ))∇y ·∇ϕ dx = ⟨f,ϕ⟩.

Finally, we prove the strong convergence. Indeed

α ∥∇(yn − y)∥2L2(D) ≤∫

D(k1χn + k2(1− χn))∇(yn − y) ·∇(yn − y) dx

≤∫

D(k1χn + k2(1− χn))∇yn ·∇yn dx

− 2∫

D(k1χn + k2(1− χn))∇yn ·∇y dx

+∫

D(k1χn + k2(1− χn))∇y ·∇y dx

=⟨f, yn⟩ − 2 ⟨f, y⟩+∫

D(k1χn + k2(1− χn))∇y ·∇y dx

→ −⟨f, y⟩+∫

D(k1χ+ k2(1− χ))∇y ·∇y dx = 0.

Since every strongly convergent subsequence converges to the same limit, the wholesequence strongly converges to y in H1

0 (D).

4 Continuity of the Homogeneous DirichletBoundary Value Problem

4.1 Classical, Relaxed, and Overrelaxed ProblemsWe have seen in section 3.5 of Chapter 5 two relaxations of the homogeneous Dirich-let boundary value problem for the Laplacian to measurable subsets Ω of a bounded

4. Continuity of the Homogeneous Dirichlet Boundary Value Problem 419

open holdall D by introducing the following closed subspaces of H10 (D):

H1⋄ (Ω; D) def=

ϕ ∈ H1

0 (D) : (1− χΩ)ϕ = 0 a.e. in D

, (4.1)H1

• (Ω; D) def=ϕ ∈ H1

0 (D) : (1− χΩ)∇ϕ = 0 a.e. in D

(4.2)

(cf. (3.27) and (3.28) in Theorem 3.6 of Chapter 5). It is readily seen thatH1

⋄ (Ω; D) ⊂ H1• (Ω; D). For Ω open, H1

0 (Ω; D) ⊂ H1⋄ (Ω; D) ⊂ H1

• (Ω; D) and thethree closed subspaces of H1

0 (D) are generally not equal as can be seen from Ex-amples 3.3, 3.4, and 3.5 in section 3.5 of Chapter 5.

The choice of a relaxation is very much problem dependent and should beguided by the proper modeling of the underlying physical or technological phenom-enon at hand. In general, there is a price to pay for the above relaxations. BothH1

⋄ (Ω; D) and H1• (Ω; D) depend on the equivalence class of measurable sets [Ω] and

not specifically on Ω. In general, there is not even an open representative of Ω inthe class. From section 3.3 in Chapter 5, [Ω] = [I], where I is the measure theoreticinterior of Ω,

I =

x ∈ RN : limr0

m(B(x, r) ∩ Ω)m(B(x, r))

= 1

,

int I = Ω1, where

Ω1 =x ∈ RN : ∃ρ > 0 such that m

(Ω ∩B(x, ρ)

)= m

(B(x, ρ)

)

is open, and int Ω ⊂ int I and ∂I ⊂ ∂Ω (cf. Definition 3.3 and Theorem 3.3 inChapter 5). In view of the cleaning operation in the definition of I, all cracks ofzero measure will be deleted or not seen by the functions of the spaces H1

⋄ (Ω; D) andH1

• (Ω; D). When mN (∂Ω) = 0, [int I] = [Ω] and int I = Ω1 is an open representativein the class [Ω].

Let D be a bounded open subset of RN and let f ∈ H−1(D), where H−1(D)is the topological dual of H1

0 (D). For all open subsets Ω of D

ϕ ,→ ⟨f,ϕ⟩H−1(D)×H10 (D) : D(Ω)→ R (4.3)

is well-defined and continuous with respect to the H1(Ω)-topology and extends toan element f |Ω ∈ H−1(Ω). The map

ϕ ,→ ⟨f,ϕ⟩H−1(D)×H10 (D) : H1

⋄ (Ω; D)→ R

is also well-defined and continuous with respect to the closed subspace H1⋄ (Ω; D) of

H10 (D) and defines an element f |⋄Ω of the dual H1

⋄ (Ω; D)′ of H1⋄ (Ω; D). Similarly,

the map

ϕ ,→ ⟨f,ϕ⟩H−1(D)×H10 (D) : H1

• (Ω; D)→ R

is also well-defined and continuous with respect to the closed subspace H1• (Ω; D) of

H10 (D) and defines an element f |•Ω of the dual H1

• (Ω; D)′ of H1• (Ω; D).

Let A ∈ L∞(D;L(RN,RN)) be a symmetrical matrix function on D such that∗A = A and αI ≤ A ≤ βI (4.4)


for some coercivity and continuity constants 0 < α ≤ β. Consider the classicalhomogeneous Dirichlet boundary value problem for the generalized Laplacian

∃y(Ω) ∈ H10 (Ω; D) such that,

∀ϕ ∈ H10 (Ω; D),

∫

DA∇y(Ω) ·∇ϕ dx = ⟨f |Ω,ϕ⟩H−1(Ω)×H1

0 (Ω),(4.5)

the relaxed homogeneous Dirichlet boundary value problem

∃y(χΩ) ∈ H1⋄ (Ω; D) such that,

∀ϕ ∈ H1⋄ (Ω; D),

∫

DA∇y(χΩ) ·∇ϕ dx = ⟨f |⋄Ω,ϕ⟩H1

⋄(Ω;D)′×H1⋄(Ω;D),

(4.6)

and the overrelaxed homogeneous Dirichlet boundary value problem

∃y(χΩ) ∈ H1• (Ω; D) such that,

∀ϕ ∈ H1• (Ω; D),

∫

DA∇y(χΩ) ·∇ϕ dx = ⟨f |•Ω,ϕ⟩H1

•(Ω;D)′×H1•(Ω;D).

(4.7)

By convention, set y(∅) = 0 and y(χ∅) = 0 as elements of H10 (D). In addition we

shall keep the terminology homogeneous Dirichlet boundary value problem for whatwe just called the classical homogeneous Dirichlet boundary value problem for thegeneralized Laplacian.

Since, in general, H10 (Ω; D) is a distinct closed subspace of H1

⋄ (Ω; D), a solu-tion of (4.6) is not necessarily a solution of (4.5). In the other direction, a solutiony = y(Ω) ∈ H1

0 (Ω; D) of (4.5) belongs to H1⋄ (Ω; D) and is a solution of

∃y ∈ H1⋄ (Ω; D) such that,

∀ϕ ∈ H10 (Ω; D),

∫

DA∇y ·∇ϕ dx = ⟨f |⋄Ω,ϕ⟩H1

⋄(Ω;D)′×H1⋄(Ω;D),

(4.8)

but y is not necessarily unique in H1⋄ (Ω; D) since this variational equation is verified

only on the closed subspace H10 (Ω; D) of H1

⋄ (Ω; D). However, the space H1⋄ (Ω; D)

is very close to the space H10 (Ω; D) and the two spaces will coincide on smooth

sets and more generally on crack-free sets that will be introduced in section 7(cf. Definition 7.1 (ii) and Theorem 7.3 (ii)). The following terminology has beenintroduced by J. Rauch and M. Taylor [1].

Definition 4.1.An open subset Ω of D is said to be stable with respect to D if H1

0 (Ω; D) =H1

⋄ (Ω; D).

The same considerations apply to the overrelaxed problem (4.7) in H1• (Ω; D)

with the additional observation that even for sets with a very smooth boundary, thesolution does not coincide with the one of the (classical) Dirichlet problem whenthe !Ω has more than one connected component as seen from Examples 3.3, 3.4,and 3.5 in section 3.5 of Chapter 5. The problem in H1

• (Ω; D) is associated withdifferent physical phenomena.


4.2 Classical Dirichlet Boundary Value ProblemWe have the following general results.

Theorem 4.1. Let D be a bounded open nonempty subset in RN. Associate witheach open subset Ω of D the solution y(Ω) ∈ H1

0 (Ω; D) of the classical Dirichletproblem (4.5) on Ω.

(i) Let Ω and the sequence Ωn be open subsets of D such that d!Ωn→ d!Ω in

C(D). Then

y(Ωn ∩ Ω)→ y(Ω) in H10 (D)-strong (4.9)

and there exist a subsequence Ωnk and y0 ∈ H1⋄ (Ω; D) such that

y(Ωnk) y0 in H10 (D)-weak (4.10)

and y0 is a (not necessarily unique) solution of (4.8).

(ii) The function

d!Ω ,→ y(Ω) : Ccd(D)→ H1

0 (D)-strong (4.11)

is continuous for each Ω such that H10 (Ω; D) = H1

⋄ (Ω; D).

Remark 4.1.We have the additional information that y0 ∈ H1

⋄ (Ω; D) ⊂ H10 (int Ω; D) in part (i)

of the theorem (cf. Theorem 7.3 (i)).

Remark 4.2.The projection operator

PΩ : H10 (D)→ H1

0 (Ω; D) (4.12)

is defined for u ∈ H10 (D) as the solution of the variational equation

∃PΩu ∈ H10 (Ω; D), ∀ϕ ∈ H1

0 (Ω; D),∫

D∇(PΩu) ·∇ϕ dx =

∫

D∇u ·∇ϕ dx.

Applying the theorem to a sequence of open subsets Ωn of D converging to Ω inthe complementary Hausdorff topology with

⟨f,ϕ⟩ def=∫

D∇u ·∇ϕ dx,

there exist u0 ∈ H1⋄ (Ω; D) and a subsequence Ωnk such that

d!Ωn→ d!Ω in C(D), PΩnk

u u0 in H10 (D)-weak.

The projection operator PΩ is continuous at Ω if H10 (Ω; D) = H1

⋄ (Ω; D), but this isonly a sufficient condition.


Proof. (i) The proof of the continuity of (4.9) follows the same steps as the proofof the upper semicontinuity of the first eigenvalue of the Laplacian in Theorem 2.2.As for the convergence (4.10), it is readily seen that the sequence yn = y(Ωn) ofelements of H1

0 (Ωn;D) is bounded in the bigger space H10 (D): by coercivity

α ∥∇yn∥2L2(D) ≤ ∥f∥H−1(D) ∥yn∥H10 (D) ≤

√1 +

1λD∥f∥H−1(D) ∥∇yn∥L2(D)

from Remark 2.1, where λD > 0 is the first eigenvalue of the Laplacian on D. Thereexist a subsequence, still indexed by n, and some y0 in H1

0 (D) such that yn y0 inH1

0 (D)-weak and hence yn → y0 in L2(D)-strong. From Corollary 1 to Theorem 4.4in Chapter 6

∀x ∈ RN, lim infn→∞

χΩn(x) ≥ χΩ(x)

⇒ ∀ a.a. x ∈ RN, 0 = lim infn→∞

(1− χΩn(x))|yn(x)|2 ≥ (1− χΩ(x))|y0(x)|2

since (1 − χΩn(x)) ≥ 0, |yn|2 ≥ 0, and |yn|2 → 0 a.e. in D. By definition, y0 ∈H1

⋄ (Ω; D). Finally, by the compactivorous property of Theorem 2.4 in section 2.3 ofChapter 6, for each ϕ ∈ D(Ω) there exists N such that for all n > N , suppϕ ⊂ Ωn.Therefore, by substituting ϕ in (4.5) on Ωn, for all n > N

∫

DA∇yn ·∇ϕ dx = ⟨f |Ωn ,ϕ⟩H−1(Ωn)×H1

0 (Ωn) = ⟨f |Ω,ϕ⟩H−1(Ω)×H10 (Ω) (4.13)

and by going to the limit

∀ϕ ∈ D(Ω),∫

DA∇y0 ·∇ϕ dx = ⟨f |Ω,ϕ⟩H−1(Ω)×H1

0 (Ω) (4.14)

and, by density of D(Ω) in H10 (Ω; D), y0 ∈ H1

⋄ (Ω; D) is a solution of (4.8).(ii) If, in addition, H1

⋄ (Ω; D) = H10 (Ω; D), then y0 ∈ H1

0 (Ω; D) is the uniquesolution of (4.5). Hence y0 = y(Ω), solution of the classic problem (4.5), is inde-pendent of the choice of the converging subsequence. Therefore the whole sequenceyn converges to y0 = y(Ω) in H1

0 (D)-weak.The strong continuity in H1

0 (D) now follows from the convergence of un → u0in L2(D)-strong:

α∥∇(un − u0)∥2L2(D)

≤∫

DA∇(un − u0) ·∇(un − u0) dx

=∫

DA∇un ·∇un dx +

∫

DA∇u0 ·∇u0 dx− 2

∫

DA∇un ·∇u0 dx

= ⟨f, un⟩+ ⟨f, u0⟩ − 2∫

DA∇un ·∇u0 dx

→ 2 ⟨f, u0⟩ − 2∫

DA∇u0 ·∇u0 dx = 0

and the whole sequence yn converges to y0 = y(Ω) in H10 (D)-strong. Therefore,

the function (4.11) is continuous in Ω.


4.3 Overrelaxed Dirichlet Boundary Value ProblemIn this section we concentrate on the overrelaxation (4.7) of the homogeneousDirichlet boundary value problem. To show the continuity with respect to thecharacteristic function of domains, we first approximate its solution by transmis-sion problems.

4.3.1 Approximation by Transmission Problems

The overrelaxed Dirichlet problem (4.7) corresponding to χ = χΩ ∈ X(D) is firstapproximated by a transmission problem over D indexed by ε > 0 going to zero:

find yε = yε(χ) ∈ H10 (D) such that ∀ϕ ∈ H1

0 (D)∫

D

[χ+

1ε

(1− χ)]∇yε ·∇ϕ dx =

∫

Dχfϕ dx.

(4.15)

They are special cases of the general transmission problem (3.1) over the fixedbounded open nonempty domain D associated with the Laplace equation and ameasurable subset Ω of D for k1 = 1 and k2 = 1/ε and χ = χΩ.

Theorem 4.2. Let D be a bounded open nonempty subset in RN and Ω be a measur-able subset of D. Let y(χΩ) ∈ H1

• (Ω; D) be the solution of the overrelaxed Dirichletproblem (4.7). For ε, 0 < ε < 1, let yε be the solution of problem (4.15) in H1

0 (D).Then

y(χΩ) = limε→0

yε in H10 (D)-strong.

Proof. For ε, 0 < ε < 1, substitute ϕ = yε in (4.15) to get the bounds

∥∇yε∥L2(D) ≤ c(D) ∥f∥L2(D), ∥(1− χΩ)∇yε∥L2(D) ≤√ε c(D) ∥f∥L2(D).

There exist a sequence εn 0 and y0 ∈ H10 (D) such that yn = yεn y0 in H1

0 (D)-weak and (1 − χΩ)∇yn → 0 in L2(D)-strong. Therefore, (1 − χΩ)∇y0 = 0 andy0 ∈ H1

• (Ω; D). For ϕ ∈ H1• (Ω; D), (4.15) reduces to

∀ϕ ∈ H1• (Ω; D),

∫

D∇yε ·∇ϕ dx =

∫

DχΩfϕ dx

and as n→∞

∀ϕ ∈ H1• (Ω; D),

∫

D∇y0 ·∇ϕ dx =

∫

DχΩfϕ dx.

Hence y0 ∈ H1• (Ω; D) is the solution of (4.7). Since the limit is the same for

all convergent sequences, yε → y0 in H10 (D)-weak. By subtracting the last two

equations

∀ϕ ∈ H1• (Ω; D),

∫

D[∇yε −∇y0] ·∇ϕ dx = 0 ⇒

∫

D[∇yε −∇y0] ·∇y0 dx = 0.


Finally, we estimate the following term:∫

D|∇(yε − y0)|2dx =

∫

D[∇yε −∇y0] ·∇yε dx−

∫

D[∇yε −∇y0] ·∇y0 dx

=∫

D[∇yε −∇y0] ·∇yε dx

=∫

D∇yε ·∇yε dx−

∫

D∇y0 ·∇yε dx

=∫

DχΩ fyε dx−

∫

D∇y0 ·∇yε dx

→∫

DχΩ fy0 dx−

∫

D∇y0 ·∇y0 dx = 0.

As a result yε → y0 in H10 (D)-strong.

4.3.2 Continuity with Respect to X(D) in the Lp(D)-Topology


• (Ω; D) be the solution of the overrelaxed Dirichletproblem (4.7). Then the function

χΩ ,→ y(χΩ) : X(D)→ H10 (D)-strong (4.16)

is continuous for all Lp(D)-topologies, 1 ≤ p <∞, on X(D).

Proof. Given χ = χΩ ∈ X(D), let χΩn in X(D) be a converging sequence toχ = χΩ. In view of Theorem 4.2, let εn > 0 be such that the approximatingsolution yn ∈ H1

0 (D) of (4.15),

∀ϕ ∈ H10 (D),

∫

D

[χΩn +

1εn

(1− χΩn)]∇yn ·∇ϕ dx =

∫

DχΩnfϕ dx,

is such that

∥yn − y(χΩn)∥H1(D) < 1/n

and recall the bounds

∥∇yn∥L2(D) ≤ c(D) ∥f∥L2(D), (4.17)∥(1− χΩn)∇yn∥L2(D) ≤

√εn c(D) ∥f∥L2(D) ≤ c(D) ∥f∥L2(D). (4.18)

Therefore the sequence y(χΩn is also bounded in H10 (D) and there exist subse-

quences, still indexed by n, and y0 ∈ H10 (D) such that

yn y0 and y(χΩn) y0 in H10 (D)-weak.

Moreover, (1− χΩn)∇y(χΩn) = 0. Hence (1− χΩ)∇y0 = 0 and y0 ∈ H1• (Ω; D).

For ϕ ∈ H1• (Ω; D)

∀ϕ ∈ H10 (D),

∫

D

[χΩn +

1εn

(1− χΩn)]∇yn · (χΩ∇ϕ) dx =

∫

DχΩnfϕ dx.


The right-hand side converges to∫

D χΩfϕ dx. The left-hand side can be rewritten as∫

DχΩ χΩn ∇yn ·∇ϕ+ χΩ (1− χΩn)

[1εn

(1− χΩn)]∇yn ·∇ϕ dx

→∫

DχΩ∇y0 ·∇ϕ dx =

∫

D∇y0 ·∇ϕ dx

since χΩ χΩn → χΩ and χΩ (1 − χΩn) → 0 in L2(D)-strong and the bound (4.18).Finally, y0 ∈ H1

• (Ω; D) is solution of

∀ϕ ∈ H1• (Ω; D),

∫

D∇y0 ·∇ϕ dx =

∫

DχΩfϕ dx.

Since the limit point y0 = y(χΩ) is independent of the choice of the sequence,y(χΩn) y(χΩ) in H1

0 (D)-weak. Finally,∫

D|∇ (y(χΩn)− y(χΩ))|2 dx

=∫

D|∇y(χΩn)|2 dx +

∫

D|∇y(χΩ)|2 dx− 2

∫

D∇y(χΩn) ·∇y(χΩ) dx

=∫

DχΩn f y(χΩn) dx +

∫

DχΩ f y(χΩ) dx− 2

∫

D∇y(χΩn) ·∇y(χΩ) dx

→ 2∫

DχΩ f y(χΩ) dx− 2

∫

D∇y(χΩ) ·∇y(χΩ) dx = 0

and y(χΩn)→ y(χΩ) in H10 (D)-strong.

Remark 4.3.The penalization technique used in this section can be applied to higher-order el-liptic, and even parabolic, problems and the Navier–Stokes equation (cf. R. Dziriand J.-P. Zolesio [6]) provided that the relaxation is acceptable in the modelingof the physical phenomenon. The continuity of the classical homogeneous Dirichletproblem for the Laplacian will be given in section 6 under capacity conditions.

4.4 Relaxed Dirichlet Boundary Value ProblemIn this section we consider the relaxation (4.6) of the homogeneous Dirichlet bound-ary value problem for the Laplacian to measurable subsets of a bounded open holdallD for the space H1

⋄ (Ω; D).


⋄ (Ω; D) be the solution of the relaxed Dirichletproblem (4.1). Then the function

χΩ ,→ y(χΩ) : X(D)→ H10 (D)-strong (4.19)

is continuous for all Lp(D)-topology, 1 ≤ p <∞, on X(D).


Proof. Given χΩ ∈ X(D), let χΩn in X(D) be a converging sequence to χΩ and lety(χΩ) ∈ H1

⋄ (Ω; D) and y(χΩn) ∈ H1⋄ (Ωn;D) be the corresponding solutions of (4.1).

Since H1⋄ (Ω; D) ⊂ H1

• (Ω; D), we have the continuity χΩ ,→ y(χΩ) from Theorem 4.3.To complete the proof we need to show that the limiting element y(χΩ) in the proofof Theorem 4.3 is solution of the variational equation (4.1) in the smaller spaceH1

⋄ (Ω; D). Indeed, for each n, (1− χΩn)y(χΩn) = 0 implies (1− χΩ)y(χΩ) = 0 andy(χΩ) ∈ H1

⋄ (Ω; D). Finally, since y(χΩ) is solution of the variational equation (4.7)for all ϕ ∈ H1

• (Ω; D), it still is solution for ϕ in the smaller subspace H1⋄ (Ω; D).

Hence y(χΩ) ∈ H1⋄ (Ω; D) is the solution of (4.1).

5 Continuity of the Homogeneous NeumannBoundary Value Problem

As a fourth generic example, consider the homogeneous Neumann boundary valueproblem of mathematicians for the Laplacian

−∆y + y = f in Ω,∂y

∂n= 0 on ∂Ω, (5.1)

∃y ∈ H1(Ω), ∀ϕ ∈ H1(Ω),∫

Ω∇y ·∇ϕ+ y ϕ dx =

∫

Ωf ϕ dx. (5.2)

The classical homogeneous Neumann boundary value problem

−∆y = f in Ω,∂y

∂n= 0 on ∂Ω, (5.3)

∃y ∈ H1(Ω), ∀ϕ ∈ H1(Ω),∫

Ω∇y ·∇ϕ dx =

∫

Ωf ϕ dx (5.4)

has a unique solution up to a constant if∫

D f dx = 0. One way to get around thenonuniqueness is to introduce the closed subspace

H1a(Ω) def=

ϕ ∈ H1(Ω) :

∫

Ωϕ dx = 0

of functions with zero average. Then there exists a unique y ∈ H1a(Ω) such that

∀ϕ ∈ H1a(Ω),

∫

Ω∇y ·∇ϕ dx =

∫

Ωf ϕ dx. (5.5)

Moreover, the solution of (5.5) can be approximated by the ε-problems

∃yε ∈ H1(Ω), ∀ϕ ∈ H1(Ω),∫

Ω∇yε ·∇ϕ+ ε yε ϕ dx =

∫

Ωf ϕ dx (5.6)

and yε → y in H1a(Ω)-strong. Note that, by assumption on f , yε ∈ H1

a(Ω).One of the key conditions to get the continuity with respect to the domain Ω

is the following density property.

5. Continuity of the Homogeneous Neumann Boundary Value Problem 427

Definition 5.1.An open set Ω is said to have the H1-density property if the set

ϕ|Ω : ϕ ∈ C10 (RN) (5.7)

is dense in H1(Ω).

This property is not verified for a domain Ω with a crack in R2, since elementsof H1(Ω) have different traces on each side of the crack. Recall from section 6.1in Chapter 2 that this condition is verified for a large class of sets Ω that can becharacterized by the geometric segment property that is equivalent to the propertythat the set be locally a C0-epigraph (cf. section 6.2 of Chapter 2). We recallTheorem 6.3 of Chapter 5.

Theorem 5.1. If the open set Ω has the segment property, then the setf |Ω : ∀f ∈ C∞

0 (RN)

of restrictions of functions of C∞0 (RN) to Ω is dense in Wm,p(Ω) for 1 ≤ p < ∞

and m ≥ 1. In particular Ck(Ω) is dense in Wm,p(Ω) for any m ≥ 1 and k ≥ m.

Lemma 5.1. Let D be a bounded open subset of RN, let f ∈ L2(D), and let Ω andthe sequence Ωn be open subsets of D. Denote by yn ∈ H1(Ωn) the solution of(5.1) in Ωn and by y ∈ H1(Ω) the solution of (5.1) in Ω. Assume the following:

(i) χΩn → χΩ in L2(D);

(ii) d!Ωn→ d!Ω in C(D);

(iii) Ω has the H1-density property.

Then

(yn, ∇yn)→ (y, ∇y) in (L2(D)× L2(D))-weak,

where z denotes the extension by zero of a function z from Ω or Ωn to D.

Proof. The proof of the lemma follows the approach of W. Liu and J. E. Rubio [1]in 1992. Since f ∈ L2(D), we readily have the following estimates:

∥yn∥2L2(D) + ∥∇yn∥2L2(D) = ∥yn∥2L2(Ωn) + ∥∇yn∥2L2(Ωn) ≤ ∥f∥2L2(D).

There exist subsequences, still indexed by n, z ∈ L2(D), and Z ∈ L2(D)N suchthat

yn z weakly in L2(RN) and ∇yn Z weakly in L2(RN)N .

By the H1-density property of Ω, the variational equation (5.2) on Ωn can berewritten in the form

∀ϕ ∈ C10 (RN),

∫

RNχΩn

[∇yn ·∇ϕ+ yn ϕ

]dx =

∫

RNχΩnf ϕ dx (5.8)


and we can go to the limit as n→∞:

∀ϕ ∈ C10 (RN),

∫

RNχΩ [Z ·∇ϕ+ z ϕ] dx =

∫

RNχΩf ϕ dx. (5.9)

By definition of the distributional derivative, for all n∫

Ωn

∂iyn ϕn dx = −∫

Ωn

yn ∂iϕn dx, ∀ϕn ∈ D(Ωn).

By the compactivorous property (cf. Theorem 2.4 (iii) in Chapter 6) and the con-vergence d!Ωn

→ d!Ω in C(D), for each ϕ ∈ D(Ω), there exists N such that for alln > N , suppϕ ⊂ Ωn. Hence for all n > N

∫

Ω∂iyn ϕ dx = −

∫

Ωyn ∂iϕ dx, ∀ϕ ∈ D(Ω)

and by going to the limit∫

ΩZi ϕ dx = −

∫

Ωz ∂iϕ dx, ∀ϕ ∈ D(Ω), ⇒ ∇z = Z ∈ L2(Ω)

and from (5.9), z ∈ H1(Ω) is solution of the variational equation

∀ϕ ∈ C10 (RN),

∫

Ω∇z ·∇ϕ+ z ϕ dx =

∫

Ωf ϕ dx.

By the H1-density property of Ω, this extends to all ϕ ∈ H1(Ω), which implies thatz = y. Since all weakly convergent subsequences converge to the same limit, thewhole sequence weakly converges and this completes the proof.

Remark 5.1.We know that the H1-density condition holds for the family of domains satisfyingthe segment property of Definition 6.1 in Chapter 2 and not only for H1-spaces butalso Wm,p-spaces (cf. Theorem 6.3 in Chapter 2), thus extending the property andthe results to a broader class of partial differential equations. We have shown inTheorem 6.5 of Chapter 2 that a domain satisfying the segment property is locallya C0-epigraph in the sense of Definition 5.1 of Chapter 2.

Theorem 5.2. Let D be a bounded open subset of RN, let f ∈ L2(D), and letL(D, r,O,λ) be the compact family of sets in W 1,p(D), 1 ≤ p < ∞, verifying theuniform fat segment property. Then the function

Ω ,→ (y(Ω), ∇y(Ω)) : L(D, r,O,λ)→ (L2(D)× L2(D))-weak,

where z denotes the extension by zero of a function z from Ω to D, is continuous.It is also true when L(D, r,O,λ) is replaced by the family of convex sets

Cb(D) def=bΩ : Ω ⊂ D, ∂Ω = ∅, Ω convex

(5.10)

(cf. Theorem 10.2 in Chapter 7).

6. Elements of Capacity Theory 429

Proof. In L(D, r,O,λ) endowed with the W 1,p(D)-topology, 1 ≤ p < ∞, the threeassumptions of the lemma are verified.

Remark 5.2.Therefore the previous arguments are verified for the compact families of sets ofTheorems 13.1, 14.1, and 14.2 in section 13 of Chapter 7 under the uniform fat seg-ment property and for the conditions on the local graph functions of Theorem 13.2(cf. Theorem 5.1 of Chapter 2). However, the subsets of a bounded holdall verifyinga uniform segment property do not include the very important domains with cracksfor which some specific results have been obtained by D. Bucur and J.-P. Zolesio[2, 4] (and D. Bucur and N. Varchon [1] in dimension 2).

An issue that was not tackled is the nonhomogeneous boundary condition thatinvolves a boundary integral in the variational formulation:

−∆y + y = f in Ω,∂y

∂n= g on ∂Ω, (5.11)

∃y ∈ H1(Ω), ∀ϕ ∈ H1(Ω),∫

Ω∇y ·∇ϕ+ y ϕ dx =

∫

Ωf ϕ dx +

∫

Γg ϕ dΓ. (5.12)

6 Elements of Capacity TheoryRecall the relaxation of the homogeneous Dirichlet boundary value problem for theLaplacian (4.6) in section 4.4 by using the space

H1⋄ (Ω; D) def=

ϕ ∈ H1

0 (D) : (1− χΩ)ϕ = 0 a.e. in D

. (6.1)

The classical solution lies in the closed subspace H10 (Ω; D) of H1

⋄ (Ω; D), and weknow that they are not equal when Ω is not stable with respect to D. The differencebetween the two spaces is that a function in H1

0 (Ω; D) is zero in D\Ω as an H10 (D)-

function and not just almost everywhere as an L2(D)-function. Some informationis lost in the relaxation, and this requires the finer notion of quasi-everywhere thatwill characterize the set over which an H1

0 (D)-function is zero.This section introduces the basic elements and results from capacity theory

that will play an essential role in establishing the continuity of the classical ho-mogeneous Dirichlet problem (4.5) with respect to its underlying domain. Thecharacterization of the space H1

0 (Ω; D) in terms of capacity will be given in Theo-rem 6.2. By introducing capacity constraints on sets, it will be possible to constructlarger compact families that include domains with cracks.

6.1 Definition and Basic Properties

For p > N the elements of W 1,p(RN) can be redefined on sets of measure zero tobe continuous functions (cf. R. A. Adams [1, Thm. 5.4, p. 97]):

W 1,p(RN)→ C1,λ(RN), 0 < λ ≤ 1−N/p. (6.2)


But for p ≤ N this is no longer the case. We first recall the definition of (1, p)-capacity with respect to RN and with respect to an open subset G of RN alongwith a number of basic technical results.

Definition 6.1.Let p, 1 < p <∞, be a real number and N ≥ 1 be an integer.

(i) (L. I. Hedberg [2, p. 239]) The capacity is defined as follows: for a compactsubset K ⊂ RN

Cap1,p(K) def= inf∫

RN|∇ϕ|p dx : ϕ ∈ C∞

0 (RN), ϕ > 1 on K

;

for an open subset G ⊂ RN

Cap1,p(G) def= supCap1,p(K) : ∀K ⊂ G, K compact

;

for an arbitrary subset E ⊂ D

Cap1,p(E) def= infCap1,p(G) : ∀G ⊃ E, G open

.

In what follows, we shall use the simpler notation Capp as in L. C. Evansand R. F. Gariepy [1, pp. 147] instead of Cap1,p.

(ii) (L. C. Evans and R. F. Gariepy [1, pp. 160] and J. Heinonen, T. Kilpe-lainen, and O. Martio [1, p. 87]) A function f is said to be quasi-continuouson D ⊂ RN if, for each ε > 0, there exists an open set Gε such thatCap1,p(Gε) < ε and f is continuous on D\Gε.

(iii) A set E in RN is said to be quasi-open if, for all ε > 0, there exists an openset Gε such that Cap1,p(Gε) < ε and E ∪Gε is open.

In the above definitions the capacity is defined with respect to RN. InJ. Heinonen, T. Kilpelainen, and O. Martio [1, Chap. 2, p. 27], the capacityis defined relative to an open subset D of RN instead of the whole space RN.

Definition 6.2.Let N ≥ 1 be an integer, let p, 1 < p <∞, and let D be a fixed open subset of RN.

(i) The (1, p)-capacity is defined as follows: for a compact subset K ⊂ D

Cap1,p(K, D) def= inf∫

D|∇ϕ|p dx : ϕ ∈ C∞

0 (D), ϕ > 1 on K

;

for an open subset G ⊂ D

Cap1,p(G, D) def= supCap1,p(K, D) : ∀K ⊂ G, K compact

;

for an arbitrary subset E ⊂ D

Cap1,p(E, D) def= infCap1,p(G, D) : ∀G open such that E ⊂ G ⊂ D

.


(ii) A function f is said to be quasi-continuous in D if, for all ε > 0, there existsan open set Gε ⊂ D such that capD(Gε) < ε and f is continuous on D\Gε.

(iii) A set E in D is said to be quasi-open if, for all ε > 0, there exists an open setGε ⊂ D such that capD(Gε) < ε and E ∪Gε is open.

Clearly, by definition,

∀E ⊂ D, Cap1,p(E) ≤ Cap1,p(E, D). (6.3)

The number Cap1,p(E, D) ∈ [0,∞] is called the (variational) (p, mN )-capacity ofthe condenser (E, G), where mN is the N -dimensional Lebesgue measure.

It can easily be shown that for a quasi-open set there exists a decreasingsequence Ωn of open sets such that Ωn ⊃ E and capD(Ωn\E) goes to zero as ngoes to infinity.

We say that a property holds quasi-everywhere (q.e.) in D if it holds in thecomplement D\E of a set E of zero capacity. A set of zero capacity has zeromeasure, but the converse is not true. The capacity is a countably subadditive setfunction, but it is not additive even for disjoint sets. Hence the union of a countablenumber of sets of zero capacity has zero capacity.

6.2 Quasi-continuous Representative and H1-Functions

The following theorem is from L. I. Hedberg [2] (sect. 2, p. 241 and Thm. 1.1,p. 237, and footnote, p. 238 referring to T. Wolff and the paper by L. I. Hed-berg and Th. H. Wolff [1]). See also J. Heinonen, T. Kilpelainen, andO. Martio [1, Thm. 4.4, p. 89]

Theorem 6.1. Let p, 1 < p < ∞, be a real number, N ≥ 1 be an integer, andD ⊂ RN be open.

(i) Every f in W 1,p(D) has a quasi-continuous representative: there exists aquasi-continuous function f1 defined on D such that f1 = f almost every-where in D (hence f1 is a representative of f in W 1,p(D) ). Any two quasi-continuous representatives f1 and f2 of f ∈W 1,p(D) such that f1 = f2 almosteverywhere are equal quasi-everywhere in D.

(ii) Let f ∈ W 1,p(RN). Let K ⊂ RN be closed, and suppose that f |K = 0 (tracedefined quasi-everywhere). Then f |!K ∈W 1,p

0 (!K).

The following key lemma completes the picture (cf. J. Heinonen, T. Kilpe-lainen, and O. Martio [1, Thm. 4.5, p. 90]). See also L. I. Hedberg [2, p. 241].

Lemma 6.1. (i) Let Ω be an open subset of RN, let p, 1 ≤ p <∞, and consideran element u of W 1.p(Ω). Then u ∈W 1.p

0 (Ω) if and only if there exists a quasi-continuous representative u1 of u in RN such that u1 = 0 quasi-everywhere in!Ω and u1 = u almost everywhere in Ω.


(ii) Let Ω and D be two bounded open subsets of RN such that Ω ⊂ D and consideran element u of W 1.p

0 (D). Then u|Ω ∈ W 1.p0 (Ω) if and only if there exists a

quasi-continuous representative u1 of u such that u1 = 0 quasi-everywhere inD\Ω and u1 = u almost everywhere in Ω.

(iii) Let Ω and D be two bounded open subsets of RN such that Ω ⊂ D and consideran element u of W 1.p(D). Then u|Ω ∈ W 1.p

0 (Ω) if and only if there exists aquasi-continuous representative u1 of u such that u1 = 0 quasi-everywhere inD\Ω and u1 = u almost everywhere in Ω.

A function ϕ in W 1,p(D) is said to be zero quasi-everywhere in a subset E of Dif there exists a quasi-continuous representative of ϕ which is zero quasi-everywherein E. This makes sense since any two quasi-continuous representatives of an elementϕ of W 1,p(D) are equal quasi-everywhere. For any ϕ ∈W 1,p

0 (D) and t ∈ R, the setx ∈ D : ϕ(x) > t is quasi-open. Moreover, the subspace W 1,p

0 (Ω; D) introducedin section 2.5.3 in Chapter 2 can now be characterized by the capacity.

Theorem 6.2. Given two bounded open subsets Ω and D of RN,

W 1,p0 (Ω; D) =

ϕ ∈W 1,p

0 (D) : (1− χΩ)ϕ = 0 q.e. in D

, if Ω ⊂ D, (6.4)

W 1,p0 (Ω; D) =

ϕ ∈W 1,p(D) : (1− χΩ)ϕ = 0 q.e. in D

, if Ω ⊂ D. (6.5)

The characterization of W 1,p0 (Ω; D) requires the notion of capacity in a very

essential way. It cannot be obtained by saying that the function and its derivativesare zero almost everywhere in D\Ω.

Recall the definition of the other extension H1⋄ (Ω; D) of H1(Ω) to a measurable

set D containing Ω (cf. Chapter 5, section 3.5, Theorem 3.6, identity (3.27)). Itsdefinition readily extends from H1 to W 1,p as follows:

W 1,p⋄ (Ω; D) def=

ψ ∈W 1,p

0 (D) : (1− χΩ)ψ = 0 a.e. in D

.

By definition W 1,p0 (Ω; D) ⊂W 1,p

⋄ (Ω; D), but the two spaces are generally not equal,as can be seen from the following example. Denote by Br, r > 0, the open ballof radius r > 0 in R2. Define Ω = B2\∂B1 and D = B3. The circular crack∂B1 in Ω has zero measure but nonzero capacity. Since ∂B1 has zero measure,W 1,p

⋄ (Ω; D) contains functions ψ ∈ W 1,p0 (B2) whose restrictions to B2 are not zero

on the circle ∂B1 and hence do not belong to W 1,p0 (Ω). Yet for Lipschitzian domains

and, more generally, for domains Ω that are stable with respect to D in the senseof Definition 4.1, the two spaces are indeed equal.

6.3 Transport of Sets of Zero CapacityAny Lipschitz continuous transformation of RN which has a Lipschitz continuousinverse transports sets of zero capacity onto sets of zero capacity.

Lemma 6.2. Let D be an open subset of RN and let T be an invertible transfor-mation of D such that both T and T−1 are Lipschitz continuous. For any E ⊂ D

Cap1,2(E, D) = 0 ⇐⇒ Cap1,2(T (E), D) = 0.


Proof. (a) E = K, K compact in D. Observe that, in the definition of the capacity,it is not necessary to choose the functions ϕ in C∞

0 (D). They can be chosen in alarger space as long as

∫

D|∇ϕ|2 dx <∞ and ϕ > 1 on K.

So the capacity is also given by

Cap1,2(K, D) def= inf∫

D|∇ϕ|2 dx : ϕ ∈ H1

0 (D) ∩ C(D), ϕ > 1 on K

.

But the space H10 (D) ∩ C(D) is stable under the action of T :

ϕ ∈ H10 (D) ∩ C(D) ⇐⇒ T ϕ ∈ H1

0 (D) ∩ C(D).

Consider the matrix

A(x) def= |det(DT (x))| ∗DT (x)( ∗DT (x))−1.

By assumption on T , the elements of the matrix A belong to L∞(D) and

∃α > 0, αI ≤ A(x) ≤ α−1I a.e. in D.

Define E(K) = ϕ ∈ H10 (D) ∩ C(D) : ϕ > 1 on K. For any ϕ ∈ E(T (K)),

ϕ T ∈ E(K) and∫

D|∇ϕ|2 dx =

∫

DA∇(ϕ T ) ·∇(ϕ T ) dx ≥ α

∫

D|∇(ϕ T )|2 dx.

Let K be a compact subset such that Cap1,2(K, D) = 0. For each ε > 0 thereexists ϕ ∈ E(K) such that

∫

D|∇ϕ|2 dx ≤ ε ⇒ ∀ε,

∫

D|∇(ϕ T )|2 dx ≤ ε/α or ϕ T ∈ E(T−1(K))

⇒ ∀ε, Cap1,2(T−1(K), D) ≤ ε/α ⇒ Cap1,2(T

−1(K), D) = 0

and we can repeat the proof with T−1 in place of T .(b) E = G, G open. Let

Cap1,2(G, D) = supCap1,2(K, D) : ∀K ⊂ G, K compact

= 0.

Therefore capD(K) = 0, which implies that Cap1,2(T (K), D) = 0 and hence

Cap1,2(T (G), D) = supCap1,2(K

′, D) : ∀K ′ ⊂ T (G), K ′ compact

= supCap1,2(T (K), D) : ∀K ⊂ G, K compact

= 0.

(c) General case. Let

Cap1,2(E, D) def= inf capD(G) : ∀G ⊃ E, G open .


Therefore, for all ε > 0, there exists G ⊃ E open such that

Cap1,2(E, D) ≤ Cap1,2(G, D) ≤ ε ⇒ Cap1,2(T (E), D) ≤ Cap1,2(T (G), D)

since T (E) ⊂ T (G) is open. By definition of Cap1,2(G, D) we have

∀K ⊂ G, Cap1,2(K, D) ≤ ε ⇒ Cap1,2(T (K), D) ≤ αε⇒ Cap1,2(T (G), D) = supCap1,2(T (K), D) : K ⊂ G ≤ αε.

Finally, for all ε > 0, Cap1,2(T (E), D) ≤ αε and hence Cap1,2(T (E), D) = 0.

7 Crack-Free Sets and Some ApplicationsIn this section we show that the result for the second question raised in the intro-duction to this chapter is true for measurable sets in the following large family.

7.1 Definitions and PropertiesDefinition 7.1. (i) The interior boundary of a set Ω is defined as

∂iΩdef= ∂Ω\∂Ω. (7.1)

The exterior boundary of a set Ω is defined as

∂eΩ def= ∂Ω\∂!Ω. (7.2)

(ii) A subset Ω of RN is said to be crack-free if ∂iΩ = ∅; its complement !Ω issaid to be crack-free if ∂eΩ = ∅.

The intuitive terminology crack-free arises from the following identities:

∂iΩ = ∂Ω\∂Ω = int Ω\int Ω, ∂eΩ = ∂Ω\∂!Ω = int !Ω\int !Ω. (7.3)

A crack-free set Ω does not have internal cracks that would disappear after thecleaning operation with respect to the closure of the set2. It is verified for sets that

2Since ∂Ω = Ω ∩ !Ω ⊂ Ω ∩ !Ω = ∂Ω, the definition of a crack-free set Ω is equivalent to theproperty ∂Ω = ∂Ω. In 1994 A. Henrot [2] introduced the terminology Caratheodory set for sucha set that was later adopted by D. Tiba [3], P. Neittaanmaki, J. Sprekels, and D. Tiba [1],and A. Henrot and M. Pierre [4]. However, this terminology does not seem to be standard. Forinstance, in the literature on polynomial approximations in the complex plane C, a Caratheodoryset is defined as follows.

Definition 7.2 (O. Dovgoshey [1] or D. Gaier [1]).A bounded subset Ω of C is said to be a Caratheodory set if the boundary of Ω coincides withthe boundary of the unbounded component of the complement of Ω. A Caratheodory domain is aCaratheodory set if, in addition, Ω is simply connected.

In V. A. Martirosian and S. E. Mkrtchyan [1], Ω is further assumed to be measurable. Thisdefinition excludes not only interior cracks but also bounded holes inside the set Ω as can be seenfrom the example of the annulus Ω = x ∈ R2 : 1 < |x| < 2 in R2. So, it is more restrictive than∂Ω = ∂Ω. In order to avoid any ambiguity, we choose to keep the intuitive terminology crack-free.

7. Crack-Free Sets and Some Applications 435

are locally the epigraph of a C0-function. Yet, it is also verified for sets that arenot locally the epigraph of a C0-function.

The following equivalent characterization of a crack-free set in terms of theoriented distance function was used as its definition in M. C. Delfour and J.-P. Zolesio [43].

Theorem 7.1. A subset Ω of RN is crack-free if and only if bΩ = bΩ. Its comple-ment !Ω is crack-free if and only if b!Ω = b!Ω (or, equivalently, bΩ = bint Ω).

Proof. If ∂Ω = ∂Ω, then |bΩ| = d∂Ω = d∂Ω = |bΩ|. Hence, dΩ + d!Ω = |bΩ| =|bΩ| = dΩ + d!Ω implies d!Ω = d!Ω and bΩ = bΩ. Conversely, if bΩ = bΩ, thend∂Ω = |bΩ| = |bΩ| = d∂Ω and ∂Ω = ∂Ω since both sets are closed.

We also have the following equivalences:

b!Ω = bint !Ω ⇐⇒ bΩ = bΩ ⇐⇒ !Ω = !Ω ⇐⇒ ∂Ω = ∂Ω

⇐⇒ int !Ω = !Ω ⇐⇒ ∂int !Ω = ∂Ω ⇐⇒ int Ω = int Ω(7.4)

and

bΩ = bint Ω ⇐⇒ b!Ω = b!Ω ⇐⇒ !!Ω = Ω ⇐⇒ ∂!Ω = ∂Ω

⇐⇒ int Ω = Ω ⇐⇒ ∂int Ω = ∂Ω ⇐⇒ int !Ω = int !Ω.(7.5)

Remark 7.1.An open set Ω is crack-free if and only if Ω = int Ω. Indeed, by definition, Ω is crack-free if and only if !Ω = !Ω. But since Ω is open !Ω = !Ω and Ω = !!Ω = int Ω.

We give the following general technical theorem and its corollary.

Theorem 7.2 (M. C. Delfour and J.-P. Zolesio [43]). Let K be a compactsubset of RN and let v ∈W 1,p(RN). Then v = 0 almost everywhere on !K impliesthat v has a quasi-continuous representative v∗ such that v∗|int K ∈W 1,p

0 (intK).

Corollary 1. Let Ω be a bounded set in RN and let v ∈ W 1,p(RN). Then v = 0almost everywhere on !Ω implies that v has a quasi-continuous representative v∗

such that v∗|int Ω ∈W 1,p0 (int Ω).

Proof. Choose the quasi-continuous representative

v∗(x) def= limδ0

1|B(x, δ)|

∫

B(x,δ)v dy (7.6)

of v and let Gε be the associated family of open subsets of (1, p)-capacity lessthan ε of Definition 6.1 (ii).

(a) If there exists ε > 0 such that Gε ⊂ intK, then !K ⊂ !Gε and v∗ iscontinuous in !K. But, by assumption, v = 0 almost everywhere in the open set!K. Hence, by definition of v∗, v∗ = 0 everywhere in the open subset !K ⊂ !K,and, by continuity, v∗ = 0 everywhere in !K.


(b) If for all ε > 0, Gε ∩ !K = ∅, then either x ∈ ∪ε>0Gε ∩ !K or x ∈∩ε>0Gε ∩ ∂K, since !K = int !K ∪ ∂K = !K ∪ ∂K. In the first case, v∗(x) = 0since x ∈ !K. In the second case, Capp(∩ε>0Gε ∩ ∂K) ≤ Capp(Gε) < ε and∩ε>0Gε ∩ ∂K has zero (1, p)-capacity. From (a) and (b), v∗ = 0 quasi-everywherein !K. Therefore by Theorem 5 in L. I. Hedberg and Th. H. Wolff [1], whichgeneralizes Theorem 1.1 in L. I. Hedberg [2], v∗|int K ∈W 1,p

0 (intK).

Remark 7.2.In L. I. Hedberg [2, 3] a compact set K is said to be (1, p)-stable if v = 0 quasi-everywhere on !K implies that v|int K ∈ W 1,p

0 (intK). A sufficient condition (cf.Theorem 6.4 in L. I. Hedberg [2]) for the (1, p)-stability of K is that !K be(1, p)-thick (1, p) quasi-everywhere on ∂K. In the proof of Theorem 7.2 we haveshown that, under the weaker sufficient condition v = 0 almost everywhere on !K,v = 0 quasi-everywhere on !K. However, this condition would not be sufficient forthe (2, p)-stability as shown in the example of L. I. Hedberg [3] (cf. Thm. 3.14,p. 100).

Remark 7.3.There is also a result in dimension N = 2 in Theorem 7 of L. I. Hedberg andTh. H. Wolff [1]: for p, 2 ≤ p < ∞, and q, q−1 + p−1 = 1, K is (1, q)-stable if and only if Capq(∂K ∩ e1,q(!K)) = 0, where e1,q(!K) = x ∈ RN :!K is (1, q)-thin at x. It says that the part of ∂K where !K is thin has zero (1, q)-capacity. However, in the equivalence of parts (b) and (d) of Theorem 7, there seemsto be a typo: the (1, q)-stability is defined as follows: “K is (1, q)-stable if v = 0 q.e.on !K implies that v ∈W 1,q

0 (!K).” It should probably read “v ∈W 1,q0 (intK).”

We now turn to our specific problem. Given a measurable subset Ω of abounded open holdall D ⊂ RN, define the closed subspace

W 1,p⋄ (Ω; D) def=

ψ ∈W 1,p

0 (D) : (1− χΩ)ψ = 0 a.e.

(7.7)

of W 1,p0 (D). This space provides a relaxation of the Dirichlet boundary value prob-

lem to domains that are only measurable. The next theorem will show that in factW 1,p

⋄ (Ω; D) can be embedded in W 1,p0 (int Ω). Furthermore, for crack-free measur-

able subsets Ω of D, W 1,p⋄ (Ω; D) coincides with W 1,p

0 (int Ω). This will yield thecontinuity of the homogeneous Dirichlet problem with respect to the open domainsΩ in the family L(D, r,O,λ) of subsets of D verifying a uniform fat segment prop-erty.

Theorem 7.3. Let D be bounded open in RN.

(i) For any measurable subset Ω ⊂ D such that int Ω = ∅,

v ∈W 1,p⋄ (Ω; D) =⇒

∣∣∣∣∣∃ a quasi-continuous representative v∗

such that v∗|int Ω ∈W 1,p0 (int Ω)

(7.8)

and W 1,p0 (int Ω;D) ⊂W 1,p

⋄ (Ω; D) ⊂W 1,p0 (int Ω; D).


(ii) For any measurable subset Ω ⊂ D such that int Ω = ∅,∣∣∣∣∣v ∈W 1,p

⋄ (Ω; D)and Ω crack-free

=⇒

∣∣∣∣∣∃ a quasi-continuous representative v∗

such that v∗|int Ω ∈W 1,p0 (int Ω)

(7.9)

and W 1,p0 (int Ω;D) = W 1,p

⋄ (Ω; D) = W 1,p0 (int Ω; D).

Proof. (i) From Theorem 7.2 with K = Ω and v, the extension by zero of v fromD to RN. Given ϕ ∈ W 1,p

0 (int Ω), the zero extension e0ϕ ∈ W 1,p0 (int Ω;D) and

(1− χint Ω)ϕ = 0 almost everywhere in D. This implies that (1− χΩ)ϕ = 0 almosteverywhere in D and ϕ ∈ W 1,p

⋄ (Ω; D). (ii) From (i) and the equivalences (7.4):int Ω = int Ω.

Remark 7.4.The condition that v ∈ W 1,p

⋄ (Ω; D) is weaker than the capacitary condition ofD. Bucur and J.-P. Zolesio [12] and hence its conclusion that v∗ ∈W 1,p

0 (int Ω)is also weaker. It gives information only about the function on ∂Ω rather than onthe whole ∂Ω. The “crack-free” property of Definition 7.1 is purely geometric andeven set theoretic. It is different from and probably stronger than the capacitarycondition of D. Bucur and J.-P. Zolesio [12]. Instead of forcing !Ω to be“uniformly thick” at quasi all points of the boundary ∂Ω in order to preserve thezero Dirichlet condition on the whole boundary, it erases the “interior boundary”∂Ω\∂Ω. Its advantage is that it does not require an a priori knowledge of thedifferential operator and does not necessitate the Maximum Principle or harmonicfunctions; its disadvantage is that it cannot handle problems where the trace of thefunction on ∂Ω\∂Ω is an important feature of the problem at hand. Yet, it shouldreadily be extendable to Sobolev spaces on C1,1-submanifolds of codimension 1 todeal with the Laplace–Beltrami or shell equations.

7.2 Continuity and Optimization over L(D, r, O, λ)7.2.1 Continuity of the Classical Homogeneous Dirichlet Boundary Condition

In light of Theorem 4.1, it remains to find families of sets with properties thatwould enable us to sharpen part (i) or that would verify the stability conditionH1

0 (Ω; D) = H1⋄ (Ω; D). In the second case, the family of subsets of a bounded

holdall D that satisfy the uniform fat segment property meets all the requirements.

Theorem 7.4. Let D be a bounded open nonempty subset in RN and Ω be opensubsets of D. Denote by y(Ω) ∈ H1

0 (Ω; D) the solution of the Dirichlet problem (4.5)on Ω. The map

Ω ,→ y(Ω) : L(D, r,O,λ)→ H10 (D)-strong (7.10)

is continuous with respect to any of the topologies on L(D, r,O,λ) of Theorem 13.1in Chapter 7, that is, dΩ in C(D) and W 1,p(D), d!Ω in C(D) and W 1,p(D),bΩ, C(D) in C(D) and W 1,p(D), χΩ in Lp(D), and χ!Ω in Lp(D).


Proof. Since all topologies are equivalent on L(D, r,O,λ), consider a domain Ω anda sequence Ωn such that d!Ωn

→ d!Ω in W 1,p(D) for some p, 1 ≤ p < ∞. ByTheorem 6.1 (ii) of Chapter 2, all sets in L(D, r,O,λ) are crack-free since

int !Ω = ∅, !Ω = int !Ω = !Ω ⇒ bΩ = bΩ.

By Theorem 7.3 (ii), H1⋄ (Ω; D) = H1

0 (Ω; D). Finally, from part (ii) of Theorem 4.1,we get the continuity of the solutions of the classical Dirichlet problem.

7.2.2 Minimization/Maximization of the First Eigenvalue

Consider the first eigenvalue λA(Ω) introduced in (2.1)

λA(Ω) def= inf0 =ϕ∈H1

0 (Ω;D)


D |ϕ|2 dx(7.11)

for the Laplace equation on the open Ω of D. It was shown in Theorem 2.2 thatλA(Ω) is an upper semicontinuous function of Ω and that there is a solution tothe maximization problem with respect to Ω. We now turn to the minimizationof λA(Ω) with respect to Ω that requires the lower semicontinuity and hence thecontinuity of λA(Ω) with respect to Ω. To do that we choose the smaller familyL(D, r,O,λ) of sets verifying a uniform fat segment property.

Theorem 7.5. Let D be a bounded open domain in RN and A be a matrix functionsatisfying assumption (2.2).

(i) The mapping

d!Ω ,→ λA(Ω) : L(D, r,O,λ)→ R∪+∞ (7.12)

is continuous with respect to any of the topologies on L(D, r,O,λ) of Theo-rem 13.1 in Chapter 7, that is, dΩ in C(D) and W 1,p(D), d!Ω in C(D) andW 1,p(D), bΩ, C(D) in C(D) and W 1,p(D), χΩ in Lp(D), and χ!Ω in Lp(D).

(ii) The minimization and maximization problems

infΩ∈L(D,r,O,λ)

λ(Ω) and supΩ∈L(D,r,O,λ)

λ(Ω) (7.13)

have solutions in L(D, r,O,λ). Moreover,

0 < λA(D) ≤ infΩ∈L(D,r,O,λ)

λ(Ω) ≤ supΩ∈L(D,r,O,λ)

λ(Ω) ≤ β λI(O). (7.14)

Proof. (i) Continuity. Go back to the proof of Theorem 2.2. Given Ω ∈ L(D, r,O,λ),consider a sequence Ωn in L(D, r,O,λ) such that d!Ωn

→ d!Ω in C(D). LetλA(Ωn) and un ∈ H1

0 (Ωn;D), ∥un∥L2(D) = 1, be the eigenvalue and an eigenvectorassociated with Ωn.

By definition of L(D, r,O,λ), for all x ∈ ∂Ω, there exists Ax ∈ O(N) such thatfor all y ∈ B(x, r) ∩ Ω, y + AxO ⊂ Ω. Fix x0 ∈ ∂Ω. There exist z ∈ B(x0, r) ∩ Ω


and ε > 0 such that B(z, ε) ⊂ B(x0, r) ∩ Ω. As a result B(z, ε) + AxO ⊂ Ω andz + AxO ⊂ Ω. By the compactivorous property, there exists N such that, for alln > N , z + AxO ⊂ Ωn and

∀n > N, λn = λA(Ωn) ≤ λA(z + AxO) ≤ βλI(z + AxO) = βλI(O)

since the eigenvalue is independent of a translation and a rotation of the set. Thismeans that the sequence λn of eigenvalues is bounded. By coercivity

α∥∇un2L2(D) ≤ λn ∥un∥2L2(D) = λn

and un is bounded in H10 (D). There exist λ0 and u0 ∈ H1

⋄ (Ω; D) and subse-quences, still indexed by n, such that

λn → λ0, ∥un∥L2(D) = 1, un u0 in H10 (D)-weak ⇒ ∥u0∥L2(D) = 1.

But for all Ω ∈ L(D, r,O,λ) we have H1⋄ (Ω; D) = H1

0 (Ω; D) and u0 ∈ H10 (Ω; D).

For any ϕ ∈ D(Ω), K = suppϕ ⊂ Ω is compact and there exists N(K) suchthat for all n ≥ N(K), suppϕ ⊂ Ωn. Therefore for all n ≥ N(K)

un ∈ H10 (Ωn;D),

∫

DA∇un ·∇ϕ dx = λn

∫

Dun ϕ dx.

By letting n→∞

∀ϕ ∈ D(Ω),∫

DA∇u ·∇ϕ dx = λ0

∫

Duϕ dx,

and, by density of D(Ω) in H10 (Ω; D), the variational equation is verified in H1

0 (Ω; D).Hence, u0 ∈ H1

0 (Ω; D), ∥u∥L2(Ω) = 1, satisfies the variational equation

∀ϕ ∈ H10 (Ω; D),

∫

DA∇u0 ·∇ϕ dx = λ0

∫

Duϕ dx.

Since λA(Ω) is the smallest eigenvalue with respect to H10 (Ω; D), λA(Ω) ≤ λ0. To

show that λ0 = λA(Ω), it remains to prove that λ0 ≤ λA(Ω). For each n

∀ϕn ∈ H10 (Ωn;D), λA(Ωn) ≤

∫D A∇ϕn ·∇ϕn dx

∥ϕn∥2L2(D).

By the compactivorous property for each ϕ ∈ D(Ω), ϕ = 0, there exists N suchthat suppϕ ⊂ Ωn and ϕ ∈ D(Ωn). Therefore for all n > N

λA(Ωn) ≤∫

D A∇ϕ ·∇ϕ dx

∥ϕ∥2L2(D)

and by going to the limit

∀ϕ ∈ D(Ω), λ0 ≤∫

D A∇ϕ ·∇ϕ dx

∥ϕ∥2L2(D.


By density this is true for all ϕ ∈ H10 (Ω; D), ϕ = 0, and λ0 ≤ λA(Ω). As a result

λ0 = λA(Ω). Therefore the whole sequence λA(Ωn) converges to λA(Ω) and wehave the continuity of the map d!Ω ,→ λA(Ω). The strong continuity in H1

0 (D) nowfollows from the convergence of un → u0 in L2(D)-strong:

α∥∇(un − u0)∥2L2(D)

≤∫

DA∇(un − u0) ·∇(un − u0) dx

=∫

DA∇un ·∇un dx +

∫

DA∇u0 ·∇u0 dx− 2

∫

DA∇un ·∇u0 dx

= λn

∫

Dun · un dx + λ0

∫

Du0 · u0 dx− 2

∫

DA∇un ·∇u0 dx

→ 2λ0

∫

Du0 · u0 dx− 2

∫

DA∇u0 ·∇u0 dx = 0.

(ii) This follows from the continuity and the compactness of L(D, r,O,λ) inall the mentioned topologies. The inequalities have been proven in part (i).

Corollary 1. Under the assumptions of Theorem 7.5, the maximization and min-imization problems (7.13) also have solutions in L(D, r,O,λ) under an equality oran inequality constraint of the form

m(Ω) = α, m(Ω) ≤ α, or m(Ω) ≥ α

provided that there exists Ω′ ∈ L(D, r,O,λ) such that

m(Ω′) = α, m(Ω′) ≤ α, or m(Ω′) ≥ α.

Proof. This follows from Theorem 13.1 and its Corollaries 1 and 2 of section 13 inChapter 7, where it is shown that the convergence is strong not only in the Hausdorfftopologies but also for the associated characteristic functions in L1(D), thus makingthe volume function continuous with respect to open domains in L(D, r,ω,λ). Forthe spaces of convex subsets recall from Theorem 8.6 of Chapter 6 and Theorem 10.2of Chapter 7 that the compactness remains true in the W 1,1(D)-topology.

8 Continuity under Capacity ConstraintsWe have established the continuity of the homogeneous Dirichlet problem, that is,the continuity of the function d!Ω ,→ u(Ω) : Cc

d(D)→ H10 (D)-strong for the family

of domains verifying a uniform fat segment property. Domains with cracks lie be-tween families of L(D, r,O,λ) type and Cc

d(D). Topologies using the characteristicfunction χΩ are excluded since they do not see interior boundaries of zero measure.Families that contains sets with cracks must be chosen in such a way that the ho-mogeneous Dirichlet boundary condition on the sequence of solutions u(Ωn) in theconverging sets Ωn is preserved by the solution u(Ω) on the boundary of the limitdomain Ω in section 4. We have to be careful since the relative capacity of the

8. Continuity under Capacity Constraints 441

complement of the domains near the boundary must not vanish. In order to handlethis point, we introduce the following concepts and terminology related to the localcapacity of the complement near the boundary points. The following definition isdue to J. Heinonen, T. Kilpelainen, and O. Martio [1, pp. 114–115].

Definition 8.1. (i) Given r > 0, a set E ⊂ RN, and a point x, the function

capx,r(E) def=Capp(E ∩B(x, r), B(x, 2r))

Capp(B(x, r), B(x, 2r))(8.1)

is called the capacity density function.

(ii) We say that a set E is thick at a point x if

∫ 1

0

[Capp(E ∩B(x, r), B(x, 2r))Capp(B(x, r), B(x, 2r))

]1/(p−1)dr

r=∞. (8.2)

This is called the Wiener criterion.

(iii) We say that an open set Ω satisfies the Wiener density condition if

∀x ∈ ∂Ω,

∫ 1

0

[Capp(!Ω ∩B(x, r), B(x, 2r))

Capp(B(x, r), B(x, 2r))

]1/(p−1)dr

r=∞. (8.3)

This means that the set !Ω is thick at every boundary point x ∈ ∂Ω (that is,!Ω has “sufficient” capacity around each boundary point).

Remark 8.1.If Ω satisfies the (r, c)-capacity density condition, the complement is thick in anypoint of the boundary.

The Wiener criterion can be strengthened by introducing the following condi-tion (J. Heinonen, T. Kilpelainen, and O. Martio [1, p. 127]).

Definition 8.2. (i) Given r > 0, c > 0 , and an open set Ω, !Ω is said to satisfythe (r, c)-capacity density condition if

∀x ∈ ∂Ω, ∀r′, 0 < r′ < r,Capp(!Ω ∩B(x, r′), B(x, 2r′))

Capp(B(x, r′), B(x, 2r′))≥ c. (8.4)

We also say that !Ω is uniformly (r, c)-thick.

(ii) Given r > 0, c > 0 , and an open set Ω, !Ω is said to satisfy the strong(r, c)-capacity density condition if

∀x ∈ ∂Ω, ∀r′, 0 < r′ < r,Capp(!Ω ∩B(x, r′), B(x, 2r′))

Capp(B(x, r′), B(x, 2r′))≥ c. (8.5)

We also say that !Ω is strongly uniformly (r, c)-thick.


(iii) For r, 0 < r < 1, and c > 0 define the following family of open subsets of D:

Oc,r(D) def=

Ω ⊂ D :

Ω satisfies the strong(r, c)-capacity density condition

. (8.6)

Example 8.1.Only a few capacities can be computed explicitly. The computation of the sphericalcondensers Capp(B(x, r), B(x, R)) and Capp(B(x, r), B(x, R)) leads to some simpli-fications in the denominator of the above expression (cf. J. Heinonen, T. Kilpe-lainen, and O. Martio [1, p. 35]). For 0 < r < R <∞

Capp(B(x, r), B(x, R)) = Capp(B(x, r), B(x, R) (8.7)

and for p > 1

Capp(B(x, r), B(x, R)) =

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

ωN−1(

|N − p|p− 1

)p−1 ∣∣∣Rp−Np−1 − r

p−Np−1

∣∣∣1−p

, p = N,

ωN−1(

logR

r

)1−N

, p = N,

where ωN−1 is the volume (Lebesgue measure) of the ball in RN−1. In particular

Capp(B(x, r), B(x, 2r)) = c1(N, p) rN−p, 1 < p,

Capp(B(x, r)) = Capp(B(x, r),RN) = c2(N, p) rN−p, 1 < p < N,

Capp(x0,RN) = c3(N, p) rN−p, p < N,

(8.8)

for constants ci(N, p) that depend only on p and N .

Remark 8.2.Definition 8.2 (i) uses a slightly different definition of the previous capacity densityfunction of Definition 8.1:


Capp(B(x, r), B(x, 2r))

instead of


Capp(B(x, r), B(x, 2r)).

Note that

Capp(B(x, r), B(x, 2r)) = Capp(B(x, r), B(x, 2r)) = c1(N, p) rN−p, 1 < p,

and Capp(E ∩B(x, r), B(x, 2r)) ≤ Capp(E ∩B(x, r), B(x, 2r)) imply that

∀r > 0, ∀x, capx,r(E) ≥ capx,r(E)


and the strong (r, c)-capacity density condition implies the (r, c)-capacity densitycondition. But for an open subset Ω of the bounded open subset D of RN, thefollowing two conditions are equivalent:

∀x ∈ ∂Ω, ∀ 0 < r′ < r,Capp(!Ω ∩B(x, r′), B(x, 2r′))

Capp(B(x, r′), B(x, 2r′))≥ c (8.9)

and

∀x ∈ ∂Ω, ∀0 < r′ < r,Capp(!Ω ∩B(x, r′), B(x, 2r′))

Capp(B(x, r′), B(x, 2r′))≥ c, (8.10)

and also from the Oc,r(D) family viewpoint the two definitions are equivalent. It isclear that (8.9) implies (8.10), but the converse is not obvious.

Recall Theorem 4.1 and the notation un for the solution of (4.5) in H10 (Ωn;D)

and u ∈ H1⋄ (Ω; D) for the weak limit in H1

0 (D) which satisfies (4.8). The followingtheorem gives the main continuity result.

Theorem 8.1. Let D be a bounded open nonempty subset of RN of class C2 forN ≥ 3 and Ω be an open subset of D. Assume that A is a matrix function thatsatisfies the conditions (4.4) and that the elements of A belong to C1(D). Denoteby u(Ω) ∈ H1

0 (Ω; D) the solution of the Dirichlet problem (4.5) on Ω. The map

Ω ,→ u(Ω) : Oc,r(D)→ H10 (D)-strong (8.11)

is continuous with respect to the uniform topology C(D) on Ccd(D).

Proof. When the dimension N of the space is equal to 1, H1(D) ⊂ C(D) andno approximation of f is necessary. When N ≥ 2, we first prove the result forf ∈ Hs(D), s > N/2 − 2 (since for D of class C2 the corresponding solution willbelong to Hs+2(D) ⊂ C(D)), and then, by approximation of f , we prove it forf ∈ H−1(D).

(i) First consider f ∈ Hs(D), s > N/2−2, for N ≥ 2 (s = −1 for N = 1). Theproof will make use of the following results from J. Heinonen, T. Kilpelainen,and O. Martio [1, Chap. 3, Thm. 3.70, p. 80].

Lemma 8.1. Let the assumptions of Theorem 8.1 on the open domain D and thematrix function A be satisfied. Let v be an A-harmonic function; that is, v ∈W 1,p

loc (Ω) is a weak solution of equation

∀ϕ ∈ D(Ω),∫

DA∇v(Ω) ·∇ϕ dx = 0 (8.12)

in the open set Ω. Then there exists a continuous function u on Ω such that v(Ω) = ualmost everywhere.

Note that the continuous representative from Lemma 8.1 is in fact a quasi-continuous H1

0 (Ω) representative. Indeed, let v1 = v almost everywhere and let v1 becontinuous on Ω. We want to show that v1 is a quasi-continuous representative of v.There exists a quasi-continuous representative v2 of v, which is equal to v almost


everywhere. So v1 is continuous, v2 is quasi-continuous, and v1 = v2 almost every-where. Using J. Heinonen, T. Kilpelainen, and O. Martio [1, Thm. 4.12], weget v1 = v2 quasi-everywhere. Always from J. Heinonen, T. Kilpelainen, andO. Martio [1, Thm. 6.27, p. 122] we get the following lemma.

Lemma 8.2. Let the assumptions of Theorem 8.1 on the open domain D and thematrix function A be satisfied. Let Ω belong to Oc,r(D). If θ ∈W 1,p(Ω)∩C(Ω) andif h is an A-harmonic function in Ω such that h− θ ∈W 1,p

0 (Ω), then

∀x0 ∈ ∂Ω, limx→x0

h(x) = θ(x0).

Note that the fact that Ω belongs to Oc,r(D) involves the notion of thicknessin any point of its boundary, which necessarily occurs in the proof of the lemma(cf. Definition 8.1). Returning to Theorem 8.1, it will be sufficient to prove thecontinuity for a subsequence of Ωn. By Theorem 4.1 there exists a subsequenceof Ωn, still indexed by n, such that un weakly converges to u in H1

0 (D), andu ∈ H1

⋄ (Ω; D) satisfies (4.8) in Ω. We now prove that under our assumptions uΩ =u|Ω ∈ H1

0 (Ω), which implies that u = eΩ(uΩ) and u ∈ H10 (Ω; D) since (1− χΩ)u =

(1− χΩ)eΩ(uΩ) = 0 almost everywhere in D. For that purpose we use Lemma 6.1,which says that it is sufficient to prove that u = 0 quasi-everywhere in D\Ω forsome quasi-continuous representative u. From Theorem 7.3 (i) H1

0 (int Ω;D) ⊂H1

⋄ (Ω; D) ⊂ H10 (int Ω; D) and hence u = 0 quasi-everywhere in D\Ω. So it remains

to show that u = 0 quasi-everywhere in ∂Ω ∩D. From the Banach–Saks theorem(cf. I. Ekeland and R. Temam [1]) there exists a sequence of averages

ψndef=

Nn∑

k=n

αnkun, 0 ≤ αn

k ≤ 1 ,Nn∑

k=n

αnk = 1

such that ψn → u in H10 (D). Because of the strong convergence of ψn to u in

H10 (D), we have

ψn(x)→ u(x) q.e. in D

for a subsequence of ψn, still indexed by n. Let G0 be the set of zero capacity onwhich ψn(x) does not converge to u(x). Given x ∈ D\(Ω ∪ G0), we prove that,for all ε > 0, |u(x)| < ε. We have

|u(x)| ≤ |u(x)− ψn(x)| + |ψn(x)|.

There exists Nε,x > 0 such that for all n > Nε,x,

|u(x)− ψn(x)| < ε/2.

It remains to show that there exists N ′ such that for all n ≥ N ′, |un(x)| < ε/2implies ψn(x)| < ε/2. Denote by uD the solution of (4.5) in D. By assumptionon D and f , the solution uD is continuous in D. Subtracting the correspondingequations, we obtain

∀ϕ ∈ H10 (Ωn;D),

∫

DA∇(uD − un) ·∇ϕ dx = 0,

∀ϕ ∈ H10 (Ωn),

∫

Ωn

A∇(uD − uΩn) ·∇ϕ dx = 0.


Define

hndef= uD − un, hn

def= uD|Ωn − uΩn , θndef= uD|Ωn .

Therefore the restriction hn of hn to Ωn is A-harmonic in Ωn and, from Lemma 8.1,continuous in Ωn. Moreover we have the continuity of uΩn in the closure Ωn, anduΩn is zero on the boundary. To show that, we use Lemma 8.2 with hn and θn. Bydefinition, hn−θn = −uΩn belongs to H1

0 (Ωn). From the continuity of uD we obtainthat the continuous extension hn of hn is equal to uD on ∂Ωn. Hence the extensionun of uΩn to the boundary is zero. Using J. Heinonen, T. Kilpelainen, andO. Martio [1, Thm. 6.44], we obtain that if hn is δ-Holderian on ∂Ωn, then thereexists δ1, δ ≤ δ1 < 1, such that it is δ1-Holderian on all Ωn. We have that uD is(s + 2−N/2)-Holderian on D (with s + 2−N/2 > 0), with a constant M , becauseof the assumption on f and D. Finally, we get

∀x, y ∈ ∂Ωn, |hn(x)− hn(y)| = |uD(x)− uD(y)| ≤M |x− y|s−N/2+2.

So there exist δ1 = δ1(N,β/α, c) and

M1,n = 80Mr−2 max1, (diam (Ωn))2 ≤ 80Mr−2 max1, (diam (B))2 = M1

such that

∀x, y ∈ Ωn, |hn(x)− hn(y)| ≤M1,n|x− y|δ1 ≤M1|x− y|δ1 .

By a simple argument we obtain that this inequality holds in D, and hence thereexists δ2, δ1 ≤ δ2 < 1, such that for all x, y ∈ D we have

|un(x)− un(y)| ≤ |hn(x)− hn(y)| + |uD(x)− uD(y)|≤M1|x− y|δ1 + M |x− y| ≤M2|x− y|δ2 .

Choose R > 0 such that M2Rδ2 < ε/2. Because of the Hc-convergence of Ωn to Ωthere exists an integer nR > 0 such that for all n ≥ nR we have (D\Ωn)∩B(x, R) =∅. For xn ∈ (D\Ωn) ∩B(x, R),

|un(x)| = |un(x)− un(xn)| ≤M2|x− xn|δ2 ≤M2Rδ2 ≤ ε/2

because un(xn) = 0. So

∀n > nR, |ψn(x)| =∣∣∣∣

Nn∑

k=n

αnkun(x)

∣∣∣∣ ≤Nn∑

k=n

αnkε

2=ε

2.

Finally, we obtain |u(x)| ≤ ε. Because ε was arbitrary, we have u(x) = 0 quasi-everywhere on D\Ω, which implies that u ∈ H1

0 (Ω; D) and uΩ = u|Ω ∈ H10 (Ω). The

strong convergence of un to u now follows from Theorem 4.1 and u = eΩ(uΩ).(ii) In the next step we prove that the continuity is preserved for f ∈ H−1(D).

The main idea is to use the continuous dependence of the solution uΩ,f ∈ H10 (Ω; D)


with respect to f , which is uniform in Ω. Indeed, let Ω ⊂ D, and let f, g ∈ H−1(D).Then, by a simple subtraction of the equations, we get

∫

D|∇(uΩ,f − uΩ,g)|2 dx = ⟨f − g, uΩ,f − uΩ,g⟩

⇒ ∥uΩ,f − uΩ,g∥H10 (D) ≤ ∥f − g∥H−1(D).

So, let f ∈ H−1(D) and fε = f ∗ ρε for some mollifier ρε. Letting ε → 0 we have∥fε − f∥H−1(D) → 0. Let Ωn ⊂ Oc,r(D) be a sequence that converges in theHc-topology to an open set Ω (d!Ωn

→ d!Ω). Then we have

uΩn,fε

H10 (D)−→ uΩ,fε

because fε ∈ Hs(D) and, from the previous considerations,

uΩn,fε

H10 (D)−→ uΩn,f

uniformly in Ωn. Given δ > 0, we get

∥uΩn,f − uΩ,f∥H10 (D) ≤ ∥uΩn,f − uΩn,fε∥H1

0 (D)

+ ∥uΩn,fε − uΩ,fε∥H10 (D) + ∥uΩ,fε − uΩ,f∥H1

0 (D).

Choose ε sufficiently small such that ∥fε − f∥H−1(D) < δ/4, and for each ε choosenε,δ > 0 such that

∀n > nε,δ, ∥uΩn,fε − uΩ,fε∥H10 (D) < δ/2

⇒ ∀n > nε,δ, ∥uΩn,f − uΩ,f∥H10 (D) ≤ δ.

As δ was arbitrary, the proof is complete.

We can now recover the result of V. Sverak [2] in dimension N = 2 usingthe fact that for a fixed number c > 0 the family

d!Ω : Ω open ⊂ D and #(!Ω) ≤ c

is compact in C(D), where #(!Ω) denotes the number of connected components of!Ω (cf. Theorem 2.5 (iv) in Chapter 6 and Definition 2.2 in Chapter 2).

Theorem 8.2. Let the assumptions of Theorem 8.1 on the matrix function A besatisfied. Let N = 2 and let c > 0 be a positive integer. Define the set

Ocdef= Ω : Ω open ⊂ D and #(!Ω) ≤ c.

Then the set Oc is compact and the map

Ω ,→ u(Ω) : Oc → H10 (D)-strong

is continuous for the C(D)-topology on Ccd(D).

9. Compact Families Oc,r(D) and Lc,r(O, D) 447

A proof of the result of V. Sverak [2] is given in D. Bucur [1] as a conse-quence of Theorem 8.1. The main idea of the proof is to consider f ≥ 0 (because ofthe decomposition of f = f+ − f−) and a sequence Ωn ⊂ Oc such that Ωn con-verges in the Hc-topology to an open set Ω, and uΩn u in H1

0 (D). He constructsextensions Ω+

n of the domains Ωn with the following properties:

d!Ω+n−→ d!Ω+ , cap(Ω+\Ω) = 0, Ω+

n ⊂ Oc,r(D),

where c and r are suitable constants. In fact it can be proved that Ω+n satisfy this

capacity density condition, because in an open connected set any two points arelinked by a continuous curve that lies in the set, and in the bidimensional case acurve has a positive capacity. From the previous theorem we get uΩ+

n uΩ+ . We

easily obtain that uΩ+ = uΩ ≥ u ≥ 0, which will imply that u = 0 quasi-everywhereon !Ω, and this concludes the proof. For details, see D. Bucur [1].

9 Compact Families Oc,r(D) and Lc,r(O, D)We first prove the compactness of the family Oc,r(D) for the topology of uniformconvergence of the distance function of the complement in D. Then, by analogy withthe “fat segment property,” we specialize to a “thick set property” that generalizesthe flat cone property of D. Bucur and J.-P. Zolesio [10].

9.1 Compact Family Oc,r(D)Theorem 9.1. Let D ⊂ RN be bounded open, let c > 0, and let r > 0.

(i) The subset

d!Ω : Ω ∈ Oc,r(D) (9.1)

of Ccd(D) is compact for the uniform topology of C(D).

(ii) Given a sequence of open subsets Ωn and Ω in Oc,r(D), and the solutionsu(Ωn) ∈ H1

0 (Ωn;D) and u(Ω) ∈ H10 (Ω; D) of the Dirichlet problem (4.5) on

Ωn and Ω, then

d!Ωn→ d!Ω in C(D) ⇒ u(Ωn)→ u(Ω) in H1(D)-strong. (9.2)

We shall need the following technical lemma.

Lemma 9.1.3 Let D, D = ∅, be a bounded open subset of RN. Consider a sequenceΩn, !Ωn = ∅, of open subsets of D. Assume that

d!Ωn→ d!Ω0

in C0(D)

for some open Ω0 ⊂ D, ∂Ω0 = ∅.3Part (ii) was proved by D. Bucur and J.-P. Zolesio [10, Lem. 3.1, p. 686] by a contradiction

argument. We give a direct proof using the function bΩ.


(i) There exists N such that

∀n > N, ∂Ωn = ∅.

(ii) There exists a subsequence Ωnk, ∂Ωnk = ∅, of Ωn : n > N such that forall ε > 0, there exists K such that

∀k > K, ∂Ω0 ⊂ Uε(∂Ωnk) def= x ∈ RN : d∂Ωnk(x) < ε

and

∀x0 ∈ ∂Ω0, ∃ xnk, xnk ∈ ∂Ωnk such that xnk → x0.

Proof. (i) By the boundedness assumption on D, !Ωn = ∅ and !Ω0 = ∅. Since∂Ω0 = ∅, Ω0 = ∅, and Ω0 = ∅,

∃x0 ∈ Ω0 such that ddef= d!Ω0

(x0) > 0.

There exists N such that

∀n > N, ∥d!Ωn− d!Ω0

∥C0(D) < d/2

⇒ d!Ωn(x0) ≥ d!Ω0

(x0)− |d!Ωn(x0)− d!Ω0

(x0)| > d− d/2 > 0∀n > N, Ωn = ∅ ⇒ ∂Ωn = ∅.

(ii) Since, for n > N , ∂Ωn = ∅, the subsequence bΩn : n > N belongs tothe compact family Cb(D). Therefore there exist a subsequence, denoted by bΩn,and Ω ⊂ D, ∂Ω = ∅, such that bΩn → bΩ. Hence

d!Ωn→ d!Ω and d∂Ωn → d∂Ω

⇒ !Ω0 = !Ω ⇒ Ω0 = !!Ω = int Ω ⇒ ∂Ω0 ⊂ ∂Ω.

Finally, for all ε > 0, there exists N ′ > N such that

∀n > N ′, ∥d∂Ωn − d∂Ω∥C0(D) < ε ⇒ ∀x ∈ ∂Ω0, d∂Ωn(x) < ε

⇒ ∀n > N ′, ∂Ω0 ⊂ Uε(∂Ωn) def= x ∈ RN : d∂Ωn(x) < ε.

As a result, for all x ∈ ∂Ω0, there exists xn ∈ ∂Ωxn such that

|xn − x| = d∂Ωn(x)→ d∂Ω(x) = 0.

Proof of Theorem 9.1. (i) It is sufficient to prove that for all c > 0 and all r, 0 <r < 1, Oc,r(D) is closed in C(D) for the uniform convergence topology. Considera Cauchy sequence d!Ωn

in Oc,r(D). There exists an open set Ω ⊂ D suchthat d!Ωn

→ d!Ω in C(D). If ∂Ω = ∅, then !Ω ∩ Ω = ∅, !Ω ⊂ !Ω ⊂ !Ω,and Ω = ∅ ∈ Oc,r(D). If ∂Ω = ∅, by Lemma 9.1, there exist K ′ ≥ K and asubsequence !Ωnkk>K′ such that

∀k > K ′, ∥d!Ωnk− d!Ω∥C(D) ≤ ε (⇒ !Ωnk ⊂ (!Ω)ε = x ∈ RN : d!Ω(x) ≤ ε)


and, for each x ∈ ∂Ω, there exists xnk ∈ ∂Ωnk such that

∀k > K ′, |xnk − x| ≤ ε.

Define the translations

τk(y) def= y + xnk − x, k > K ′.

Then, for all ρ > 0, τk(B(xnk , ρ)) = B(x, ρ) and

∥dτk(!Ωnk) − d!Ωnk

∥C(D) ≤ |xnk − x| ≤ ε

⇒ τk(!Ωnk) ⊂ (!Ωnk)ε ⊂ ((!Ω)ε)ε ⊂ (!Ω)2ε.

The implication ((!Ω)ε)ε ⊂ (!Ω)2ε follows from

∀y ∈ (!Ω)ε, d!Ω(z) ≤ d!Ω(y) + |z − y| ≤ ε+ |z − y| ⇒ d!Ω(z) ≤ ε+ d(!Ω)ε(z)

⇒ ∀z ∈ ((!Ω)ε)ε, d!Ω(z) ≤ 2ε ⇒ ((!Ω)ε)ε ⊂ (!Ω)2ε.

From the identities τk(!Ωnk) ⊂ (!Ω)2ε and τk(B(xnk , ρ)) = B(x, ρ), for all r′,0 < r′ < r,

c ≤Capp(!Ωnk ∩B(xnk , r′), B(xnk , 2r′))

Capp(B(xnk , r′), B(xnk , 2r′))

=Capp(τk(!Ωnk ∩B(xnk , r′)), τk(B(xnk , 2r′)))

Capp(τk(B(xnk , r′)), τk(B(xnk , 2r′)))

=Capp(τk(!Ωnk) ∩B(x, r′), B(x, 2r′))

Capp(B(x, r′), B(x, 2r′))≤

Capp((!Ω)2ε ∩B(x, r′), B(x, 2r′))Capp(B(x, r′), B(x, 2r′))

,

since Capp is invariant with respect to the translation that is a special case of anaffine isometry. Now (!Ω)2ε is monotone decreasing as ε → 0 and (!Ω)2ε !Ω =∩ε>0(!Ω)2ε. From L. C. Evans and R. F. Gariepy [1, Thm. 2 (ix), p. 151],

limε→0

Capp((!Ω)2ε ∩B(x, r′), B(x, 2r′)) = Capp(!Ω ∩B(x, r′), B(x, 2r′))

⇒ ∀x ∈ ∂Ω, ∀r′, 0 < r′ < r, c ≤Capp(!Ω ∩B(x, r′), B(x, 2r′))

Capp(B(x, r′), B(x, 2r′))

and Ω ∈ Oc,r(D). We conclude that Oc,r(D) is compact as a closed subset of thecompact set Cc

d(D) in the topology of uniform convergence on D.(ii) From part (i) and Theorem 8.1.


9.2 Compact Family Lc,r(O, D) and Thick Set PropertyBy analogy with the “fat segment property,” we introduce a “thickness property.”

Definition 9.1.Given r > 0 and c > 0, let O be a bounded subset of RN such that 0 ∈ O\O andO satisfies the strong (r, c)-capacity density condition in 0:

∀r′, 0 < r′ < r,Capp(O ∩B(0, r′), B(0, 2r′))

Capp(B(0, r′), B(0, 2r′))≥ c. (9.3)

(i) An open set Ω is said to satisfy the (r, c, O)-capacity density condition if

∀x ∈ ∂Ω, ∃Ax ∈ O(N) such that x + AxO ⊂ !Ω. (9.4)

(ii) For r, 0 < r < 1, and c > 0, define the following family of open subsets of D:

Lc,r(O, D) def= Ω ⊂ D : Ω satisfies the (r, c,O)-capacity density condition. (9.5)

As a special case of O we have the flat cone property introduced by D. Bucurand J.-P. Zolesio [1, 5] in 1993: given an aperture ω, 0 < ω < π/2, a height λ > 0,and a unit normal ν,

Oω,λ,νdef=

y ∈ RN :1

tanω∣∣PH(y)

∣∣ < y · eN < λ

∩ ν⊥, (9.6)

where PH is the orthogonal projection onto the hyperplane H = eN⊥ orthogonalto the direction eN (cf. Figure 2.6 in Chapter 2). The intersection with the hyper-plane ν⊥ makes the cone flat. This cone satisfies the strong (r, c)-capacity densitycondition in 0 of Definition 9.1 from the scaling property of this cone.

Theorem 9.2. Given r > 0, Oω,λ,ν associated with an aperture ω, 0 < ω < π/2, aheight λ > 0, and a unit normal ν through identity (9.6), and

cdef=

Capp(Oω,λ,ν ∩B(0, r), B(0, 2r))Capp(B(0, r), B(0, 2r))

, (9.7)

then 0 < c <∞, for all r′, 0 < r′ ≤ r,

∀r′, 0 < r′ < r,Capp(Oω,λ,ν ∩B(0, r′), B(0, 2r′))

Capp(B(0, r′), B(0, 2r′))≥ c, (9.8)

and Oω,λ,ν satisfies the strong (r, c)-capacity density condition in 0.

Proof. Since Oω,λ,ν is a flat cone in the hyperplane ν⊥, the capacity of its inter-section with B(0, r), Capp(Oω,λ,ν ∩B(0, r), B(0, 2r)), is strictly positive and finite.


From L. C. Evans and R. F. Gariepy [1, Thm. 2, p. 151],

Capp(B(0, r′), B(0, 2r′)) =(

r′

r

)N−p

Capp(B(0, r), B(0, 2r)),

Capp(Oω,λ,ν ∩B(0, r′), B(0, 2r′)) =(

r′

r

)N−p

Capp((r/r′Oω,λ,ν) ∩B(0, r), B(0, 2r)),

Capp(Oω,λ,ν ∩B(0, r′), B(0, 2r′))Capp(B(0, r′), B(0, 2r′))

=Capp((r/r′Oω,λ,ν) ∩B(0, r), B(0, 2r))

Capp(B(0, r), B(0, 2r)). (9.9)

Now,

r

r′ Oω,λ,ν =r

r′

y ∈ RN :

1tanω

∣∣PH(y)∣∣ < y · eN < λ

∩ ν⊥

=

r

r′ y ∈ RN :1

tanω

∣∣∣PH

( r

r′ y)∣∣∣ <

r

r′ y · eN < λr

r′

∩ ν⊥

=

z ∈ RN :1

tanω∣∣PH(z)

∣∣ < z · eN < λr

r′

∩ ν⊥

⊃

y ∈ RN :1

tanω∣∣PH(y)

∣∣ < y · eN < λ

∩ ν⊥ = Oω,λ,ν

since r/r′ ≥ 1. Finally from (9.9)

Capp(Oω,λ,ν ∩B(0, r′), B(0, 2r′))Capp(B(0, r′), B(0, 2r′))

=Capp(r/r′Oω,λ,ν ∩B(0, r), B(0, 2r))

Capp(B(0, r), B(0, 2r))

≥Capp(Oω,λ,ν ∩B(0, r), B(0, 2r))

Capp(B(0, r), B(0, 2r))= c.

Therefore, Oω,λ,ν satisfies the strong (r, c)-capacity density condition in 0.

Lemma 9.2. (i) For all c > 0 and r > 0,

Lc,r(O, D) ⊂ Oc,r(D),

where Oc,r(D) is the family of open subsets Ω of D satisfying the strong (r, c)-capacity density condition of Definition 8.2 (iii).

(ii) Given a sequence of open subsets Ωn and Ω in Lc,r(O, D), and the solutionsu(Ωn) ∈ H1

0 (Ωn;D) and u(Ω) ∈ H10 (Ω; D) of the Dirichlet problem (4.5) on

Ωn and Ω, then

d!Ωn→ d!Ω in C(D) ⇒ u(Ωn)→ u(Ω) in H1(D)-strong. (9.10)

Proof. (i) By Definition 9.1 (i) for an open set Ω satisfying the following (r, c,O)-capacity density condition: for all x ∈ ∂Ω and all r′, 0 < r′ < r, x + AxO ⊂ !Ωand

Capp(!Ω ∩B(x, r′), B(x, 2r′))Capp(B(x, r′), B(x, 2r′))

≥Capp([x + AxO] ∩B(x, r′), B(x, 2r′))

Capp(B(x, r′), B(x, 2r′)). (9.11)


From L. C. Evans and R. F. Gariepy [1, Thm. 2, p. 151] since y ,→ x + Axy isan affine isometry and for all ρ > 0, AxB(x, ρ) = B(x, ρ),

Capp([x + AxO] ∩B(x, r′), B(x, 2r′))Capp(B(x, r′), B(x, 2r′))

=Capp(O ∩B(0, r′), B(0, 2r′))

Capp(B(0, r′), B(0, 2r′))≥ c (9.12)

⇒ ∀x ∈ ∂Ω, ∀r′, 0 < r′ < r,Capp(!Ω ∩B(x, r′), B(x, 2r′))

Capp(B(x, r′), B(x, 2r′))≥ c, (9.13)

and Ω satisfies the strong (r, c)-capacity density condition.(ii) From part (i) and Theorem 8.1.

Theorem 9.3. Let D be a bounded open subset of RN and c > 0 and r > 0 beconstants. The subset

d!Ω : Ω ∈ Lc,r(O, D) (9.14)

of Ccd(D) is compact for the uniform topology of C(D).

Proof. Consider an arbitrary sequence d!Ωn, Ωn ∈ Lc,r(O, D). By compactness

of Ccd(D), there exist an open subset Ω0 of D and a subsequence of d!Ωn

, stilldenoted by d!Ωn

, that converges to d!Ω0in Cc

d(D). If ∂Ω0 = ∅, then Ω ∈Lc,r(O, D). If ∂Ω0 = ∅, then, by Lemma 9.1 (ii), for each x ∈ ∂Ω0, there exist asubsequence Ωnk of Ωn and a sequence xk, xk ∈ ∂Ωnk , that converges to x.Since Ωnk ⊂ Lc,r(O, D), by Definition 9.1 of Lc,r(O, D), for each k

∃Ak ∈ O(N) such that xk + AkO ⊂ !Ωnk ⇒ dxk+AkO ≥ d!Ωnk.

But since Ak ∈ O(N),

dxk+AkO(y) = dO(A−1k (y − xk)) = dO( ∗Ak(y − xk)).

There exist a subsequence of Ak, still denoted by Ak, and A ∈ O(N) such thatAk → A and since xk → x, dO(A−1

k (y − xk)) → dO(A−1(y − x)) = dx+AO(y).Finally, since d!Ωnk

→ d!Ω,

dxk+AkO ≥ d!Ωnk⇒ dx+AO ≥ d!Ω ⇒ ∀x ∈ ∂Ω, x + AO ⊂ !Ω

and Ω ∈ Lc,r(O, D).

9.3 Maximizing the Eigenvalue λA(Ω)A first example of extremal domain follows directly from Theorem 9.1.

Theorem 9.4. Let D be a bounded open domain in RN and A be a matrix functionsatisfying assumption (2.2). Given an open E, ∅ = E ⊂ D, consider the compactfamilies Cc

d(E;D) ∩Oc,r(D) and Ccd,loc(E;D) ∩Oc,r(D), where

Ccd(E;D) def= d!Ω : E ⊂ Ω open ⊂ D , (9.15)

Ccd,loc(E;D) def=

d!Ω :


(9.16)


are the compact families defined in Theorem 2.6 (i) of Chapter 6. Then the maxi-mization problems

supλA(Ω) : ∀Ω ∈ Oc,r(D) such that E ⊂ Ω ⊂ D

, (9.17)

sup

λA(Ω) : ∀Ω ∈ Oc,r(D) such that


(9.18)

have solutions.4

Proof. From Theorem 2.6 (i) in Chapter 6, Ccd(E;D) and Cc

d,loc(E;D) are com-pact in C(D). Since sets of that family are nonempty, λA(Ω) < +∞ and, fromTheorem 2.2, the function d!Ω ,→ λA(Ω) : Cc

d(D) ⊂ C0(D) → R∪+∞ is uppersemicontinuous and the function d!Ω ,→ λA(Ω) : Cc

d(E;D) ∩Oc,r(D)→ R is uppersemicontinuous. So there exist maximizers in the compact set Cc

d(E;D)∩Oc,r(D).The same applies for Cc

d,loc(E;D) ∩Oc,r(D).

9.4 State Constrained Minimization ProblemsWe now give the following general existence theorem.

Theorem 9.5. Let the assumptions of Theorem 8.1 on the open domain D and thematrix function A be satisfied. Let uΩ be the solution of (4.5) for Ω in Oc,r(D). If his continuously defined from H1

0 (D) into R, then J(Ω) = h(eΩ(uΩ)) is continuouslydefined from Oc,r(D) into R and reaches its extremal values on that set.

Example 9.1.Consider the following shape function for which we can get the existence of minimiz-ing domains even if the assumptions of Theorem 9.5 on h are not satisfied. Givenα > 0,

J(Ω) def=12

∫

Ω|∇uΩ − z|2dx + α

1∫Ω |uΩ|2dx

, (9.19)

where uΩ is the solution of (4.5) and z ∈ L2(D;RN). From Theorem 2.3 in Chap-ter 2, u

def= eΩ(uΩ) and ∇uΩ are zero almost everywhere in D\Ω, and the functioncan be rewritten as

J(Ω) =12

∫

D|∇u− z|2 dx + α

1∫D |u|2 dx

− 12

∫

D\Ω|z|2 dx. (9.20)

The last term is not of the form h(uΩ) for some h, but is simply of the form h(Ω).It turns out that J is not continuous but only lower semicontinuous for the Hc-topology from Theorem 4.1 of Chapter 6 applied to the last term. Hence it can beminimized.

The minimization problem of Example 9.1 can now be formulated as followson D. Let D be a bounded open nonempty domain in RN, f be an element of

4In some sense the maximizing solution is an Oc,r(D)-approximation of the set E.


H−1(D), and B be a ball containing D. Let µ be a positive measure on D anda be a positive constant such that 0 < a < µ(D) < +∞. Let uΩ be the solutionof (4.5), and let u = eΩ(uΩ), its extension by zero in H1

0 (D). For J(Ω) defined in(9.19) consider the following problem:

minJ(Ω) : Ω ∈ Oc,r(B), Ω ⊂ D, µ(Ω) ≤ a. (9.21)

Theorem 9.6. Let the assumptions of Theorem 8.1 on the open domain D and thematrix function A be satisfied. For any constants α > 0, c > 0 and r, 0 < r < 1,problem (9.21) has at least one solution.

Proof. It is sufficient to notice that the first term in (9.20) is continuous underthe assumptions of Theorem 9.5. The second term is lower semicontinuous fromTheorem 4.1 of Chapter 6 (with the measure of density |z|2, that is, dµ = z|2 dz).To complete the proof, recall that if Ωn ⊂ D and Ωn

Hc

−→Ω, then Ω ⊂ D. As α > 0,a minimizing sequence cannot converge to ∅. In fact, for any admissible domainΩ0 and any optimal solution uΩ0 ,

∫

Ω|uΩ0 |2dx ≥ α

J(Ω0).

9.5 Examples with a Constraint on the GradientLet D be a fixed bounded smooth open domain in RN, let f in H−1(D), and letg in L2(D). For any open subset Ω of D, let uΩ be the solution of the Dirichletproblem (4.5) in H1

0 (Ω). Given α > 0, M > 0, and an open subset E of D, considerthe following minimization problem:

inf J(Ω) : Ω open , E ⊂ Ω ⊂ D, ess sup |∇uΩ| ≤M , (9.22)

J(Ω) def= m(Ω) + α

∫

Ω|uΩ − g|2 dx.

It is understood that if |∇u| is not in L∞(D), then the ess sup |∇u| is +∞. In afirst step it is easy to check that the problem

inf

m(Ω) + α

∫

Ω|uΩ − g|2 dx

: ess sup |∇uΩ| ≤M, E ⊂ Ω, Ω ∈ Oc,r(D) (9.23)

has minimizing solutions. Let Ωn be a minimizing sequence. By assumption thereexist Ω and a subsequence, still indexed by n, such that Ωn → Ω in the Hc-topology(d!Ωn

→ d!Ω in C(D)) and

ess sup |∇uΩn | ≤M.

Then |∇un| converges in L2(D) to |∇u| from the previous sections. For any ϕ ∈L2

+(D) we have ∫

D(|∇un|−M) ϕ dx ≤ 0,


and then, in the limit, we get the same inequality with u. Then, the function beinglower semicontinuous with respect to that topology, we get the existence.

Lemma 9.3. For any f ∈ H−1(D) and any r > 0 and c > 0, problem (9.23) hassolutions in Oc,r(D)

In fact, the presence of the constraint on the gradient of u is helpful for gettingthe continuity of the application Ω ,→ u = eΩ(uΩ), and we directly get the existenceof solutions to the original problem (9.22). In the proof of Theorem 8.1 we onlyneeded the equicontinuity of the family of solutions un = eΩn(uΩn) and the factthat if x ∈ !Ω, then u(x) = 0 for a quasi-continuous representative. If Ω ∈ Oc,r(D),those two assumptions are readily satisfied. Notice that the boundedness of thegradients in (9.22) implies the equicontinuity of any minimizing sequence un.So, in order to obtain a continuity result with constraints on the boundednessof the gradient, we only have to notice that if u ∈ H1

0 (Ω), ess sup |∇u| ≤ M .Then u ∈ W 1,∞(Ω) and u is Lipschitzian with the constant M , or more exactly,there exists a Lipschitzian function almost everywhere equal to u, which is also aquasi-continuous representative of u. To obtain that u(x) = 0 in any point of thecomplement of Ω, we have to introduce the following capacity constraints on Ω: werequire that Ω be capacity extended (see D. Bucur [1]), i.e., that Ω = Ω∗, where

Ω∗ def=x ∈ RN : ∃εx > 0, such that capD(B(x, εx) ∩ !Ω) = 0

. (9.24)

Indeed, let u be continuous on D, u = 0 quasi-everywhere on !Ω and Ω = Ω∗. Thenfor all x ∈ !Ω, for all ε > 0 we have capD(B(x, εx) ∩ !Ω) > 0 and there exists apoint xε in B(x, εx) ∩ !Ω, where u(xε) = 0. By continuity we get u(x) = 0. Werecall from D. Bucur [1] the main properties of the capacity extension.

Proposition 9.1. For any open Ω ⊂ D, the set Ω∗ is open,

Ω ⊂ Ω∗, capD(Ω∗\Ω) = 0, and (Ω∗)∗ = Ω∗.

As capD(Ω∗\Ω) = 0 we get eΩ(uΩ) = e∗Ω(uΩ∗) and any minimizing sequence

can be made up only of capacity-extended domains.

Theorem 9.7. Given f, g ∈ L2(D), α > 0, and M > 0, problem (9.22) hasminimizing solutions.

Proof. From the boundedness of the gradient we obtain that uΩn are uniformlyLipschitz continuous, and we can apply the same arguments as in Theorem 8.1,avoiding the capacity conditions.

Consider the penalized version of problem (9.22). Given M > 0 and β > 0,define

Jβ(Ω) def= J(Ω) + β supD

ess (|∇uΩ|−M)+ (9.25)

and consider the existence of solutions to the following problem:

inf Jβ(Ω) : E ⊂ Ω open ⊂ D . (9.26)


Theorem 9.8. Given f and g in L2(D), α > 0, β > 0, and M ≥ 0, problem (9.26)has minimizing solutions.

Proof. Let Ωn be a minimizing sequence for the function Jβ , Ωn = Ω∗n. We have

ess sup |∇uΩn | ≤ maxM,β−1J(Ω1)

= M ′

for all n ≥ 1. Then, from the previous considerations we have the strong conver-gence in H1

0 (D) of un = eΩn(uΩn) to u = eΩ(uΩ) for some subsequence Ωn thatconverges to Ω in the Hc-topology. Hence Ω is a minimizing domain. To completethe proof note that the map

Ω ,→ supD

ess (|∇u|−M)+

is not lower semicontinuous from the Hc-topology to R. Nevertheless, if Ωn is aminimizing sequence for problem (9.26), the required property is satisfied. We canassume that the sequence is chosen such that

supD

ess |∇un|→ c = lim infn→∞

(supD

ess |∇un|).

For any ε > 0 there exists nε > 0 such that for n ≥ nε, supD ess |∇uΩn | ≤ c + ε.Because of the H1

0 (D)-strong convergence of un to u, we get

∀ε > 0, supD

ess |∇u| ≤ c + ε

⇒ supD

ess |∇u| ≤ lim inf(supD

ess |∇u|).

Remark 9.1.We can change the L∞-norm for a differentiable one; that is, we need

∥∇u∥1,p ≤M, ∀p > N.

By the continuous inclusion W 1,p(D) ⊂W ε,∞(D), ε small, the result still holds.

Chapter 9

Shape and TangentialDifferential Calculuses

1 IntroductionIn Chapters 3, 5, 6, and 7, we have constructed nonlinear and nonconvex completemetric spaces of geometries. The spaces F(Θ) and X(D) have group structures. Forthe groups F(Θ), Θ ⊂ C0,1(RN,RN), it is possible to construct C1-paths in F(Θ)and we have shown that the tangent space is Θ in all points of F(Θ) in Chapter 4. Itis more difficult to fully characterize tangent spaces for the spaces of characteristicfunctions or distance functions. Yet, paths constructed from velocity fields can alsobe used to obtain C1-paths in those spaces.

In the absence of sharper results, we shall concentrate on the notion of semi-derivatives or derivatives of a shape functional (cf. section 3 of Chapter 4) withrespect to a velocity field. This point of view was used in Chapter 4 to give anequivalent characterization of the continuity of a shape functional with respect toCourant metrics in terms of the continuity along continuous paths generated byvelocities. Moreover the velocity approach readily extends to the constrained case.

As we pointed out in section 3.2 of Chapter 4, two types of semiderivativesare of interest: the weaker Gateaux style semiderivative (3.6) and the strongerHadamard style semiderivative of Definition 3.2 for which the chain rule is available.This chapter has been structured along those general directions.

In section 2 we give a self-contained review of semiderivatives and derivativesin topological vector spaces in order to prepare the ground for shape derivatives.

Section 3 gives the definitions and the main properties of first-order shapesemiderivatives and derivatives of shape functionals (Eulerian, Hadamard, Gateaux,and Frechet). For the Hadamard semidifferentiability we get the Courant metriccontinuity of the shape functional as for Banach spaces. A complete structuretheorem is given for the shape gradient in section 3.4.

Before going to second-order derivatives, the main elements of the shape calcu-lus are introduced in section 4 and the basic formulae for domain and boundaryintegrals are given in section 4.1 and section 4.2. Their application is illustrated ina series of examples in section 4.3.

457

458 Chapter 9. Shape and Tangential Differential Calculuses

The final expressions of the gradients in the examples of section 4 alwayslead to a domain and a boundary expression. The boundary expression containsfundamental properties of the gradients, and the natural way to untangle some of theresulting terms is to use the tangential calculus. Section 5 gives the main elementsof that calculus for a C2-submanifold of RN of codimension 1, including Stokes’sand Green’s formulae in section 5.5 and the relationship between tangential andcovariant derivatives in section 5.6. This is applied to the derivative of the integralof the square of the normal derivative of section 4.3.3.

Section 6 extends definitions and structure theorems to second-order deriva-tives. In order to develop a better feeling for the abstract definitions the second-order derivative of the domain integral is computed in section 6.1 using the combinedstrengths of the shape and tangential calculuses. A basic formula for the second-order semiderivative of the domain integral is given in section 6.2. This is completedwith structure theorems in the nonautonomous case in section 6.3 and the auton-omous case in section 6.4. The shape Hessian is decomposed into a symmetricalterm plus the gradient acting on the first half of the Lie bracket in section 6.5. Thissymmetrical part is itself decomposed into a symmetrical part that depends only onthe normal component of the velocity field and a symmetrical term made up of thegradient acting on a generic group of terms that occurs in all examples consideredin section 6.

2 Review of Differentiation in Topological VectorSpaces

In this section we review some elements of semiderivatives and derivatives in topo-logical vector spaces that will be useful for defining shape semiderivatives and deriva-tives. In that context we begin with the weaker notion of Gateaux semiderivative.Its simplicity makes it useful for computing a semiderivative. Yet, the chain rule forthe derivative of the composition of such functions does not hold and the functionitself need not be continuous. The more interesting notion is the stronger one ofHadamard semiderivative that makes the function continuous and builds the chainrule in its definition. It also readily extends to the tangential semiderivative onsmooth submanifolds. In the Euclidean space the Hadamard derivative coincideswith the total differential or Frechet derivative, but it is both weaker and moregeneral in infinite-dimensional vector spaces where the existence of a metric on thespace is not required. The chain rule is the central ingredient of a good differentialor semidifferentiable calculus.

2.1 Definitions of Semiderivatives and DerivativesThe notion of semiderivative corresponds to the intuitive notion of differential thatwas formalized by Newton and Leibniz, who also introduced the notation in 1684.

Definition 2.1.Let f be a real-valued function defined in a neighborhood of a point x of a topologicalvector space E.

2. Review of Differentiation in Topological Vector Spaces 459

(i) We say that f has a Gateaux semiderivative1 at a point x ∈ U in the directionv ∈ E if the following limit exists:

limε0

f(x + εv)− f(x)ε

; (2.1)

when it exists it will be denoted by df(x; v).

(ii) We say that f has a Hadamard semiderivative2 at x ∈ U in the directionv ∈ E if the following limit exists:

limε0w→v

f(x + εw)− f(x)ε

; (2.2)

when it exists it will be denoted by dHf(x; v).

The above definitions extend from a real-valued function to a function f into anothertopological vector space F .

Remark 2.1.Our definition of Hadamard semiderivative is simple and general, but it somewhathides the original definition of J. Hadamard [2] that builds into the definition thechain rule for the derivative of the composition of functions and the continuity ofthe function, as we shall see in Theorem 2.5. It is readily seen that Definition 2.1 (i)is equivalent to

limε0

∀ϕ:[0,τ ]→E such thatϕ(0+)=x,ϕ′(0+)=v

f(ϕ(ε))− f(x)ε

, (2.3)

where the limit is independent of the choice of path ϕ : [0, τ ] → E such thatlimε0 ϕ(ε) = x and limε0 (ϕ(ε)−ϕ(0))/ε = v. Since we are using paths, this def-inition extends to paths on manifolds M with v in the tangent space to M in x.

It is clear that if dHf(x; v) exists, the semiderivative df(x; v) exists and isequal to dHf(x; v), but the converse is not true without additional assumptions, as

1In his original work (published after his death in 1914) R. Gateaux [2] introduces as differen-tial the directional derivative (ε goes to 0 not only by positive values but also by negative valuesyielding the property df(x − v) = −df(x; v)), but he does not a priori assume that this differentialis linear with respect to the direction. So he cannot get the differential of the norm |x|, but hisdefinition of differential is very close to our version.

2In 1923 J. Hadamard [2] introduces a notion of differential that builds in the chain rule forthe composition of functions. M. Frechet [1] shows in 1937 that “the definition of the totaldifferential of Stolz-Young is equivalent to the definition of J. Hadamard [2] (in finite dimension).However, when the latter is extended to functionals (functions of functions - infinite dimensionalspaces), it is more general than the one of the author (M. Frechet) and necessarily verifies the chainrule for the differentiation of the composition of functions.” In the same paper M. Frechet [1,p. 239] weakens the notion of differential by dropping the linearity with respect to the direction.Again this amounts to letting ε go to zero by positive and negative values in our definition. Asin the case of Gateaux his definition is very close to our semiderivative version of the derivativeof Hadamard that can be found in J.-P. Penot [1, p. 250] in 1978 and earlier in A. Bastiani [1,Def. 3.1, p. 17] in 1964 as a seemingly well-known notion.


can be seen from the following example. The Gateaux semiderivative is generallyneither linear nor continuous with respect to the direction.

Example 2.1.Consider the following function f : R2 → R:

f(x, y) =

⎧⎨

⎩

x5

(y − x2)2 + x8 , if (x, y) = (0, 0),

0, if (x, y) = (0, 0).(2.4)

By definition, df((0, 0); (0, 0)) is always 0, but here dHf((0, 0); (0, 0)) does not exist.Choose the sequences

tn =1n 0 and wn =

(1n

,1n3

)→ (0, 0) as n→ +∞

and consider the quotient

qndef=

f((0, 0) + tnwn)− f(0, 0)tn

.

As n goes to infinity,

qn =( 1

n2 )51n ( 1

n2 )8= n4 → +∞

and dHf((0, 0); (0, 0)) does not exist. The function f is Gateaux semidifferentiableat (0, 0) in all directions v = (v1, v2) in R2 and

df(0, 0); (v1, v2)) =

v1, if v2 = 0,

0, if v2 = 0,(2.5)

but it is neither linear nor continuous with respect to v (vn = (1, 1/n)→ (1, 0)).

The Hadamard semiderivative is also generally not linear in v.

Example 2.2.Let E be a normed vector space. The Hadamard semiderivative of the norm f(x) =|x|E at x = 0 is given by

∀v ∈ E, dHf(0; v) = |v|E , (2.6)

which is continuous but not linear in v.

We shall also need the following notions of (full) derivatives.

Definition 2.2.Let f be a real-valued function defined in a neighborhood of a point x of a topologicalvector space E.

(i) The function f has a Gateaux derivative at x if

∀v ∈ E, df(x; v) exists andv )→ df(x; v) : E → R is linear and continuous. (2.7)

Whenever it exists the linear map (2.7) will be denoted by ∇f : E → E′.


(ii) The function f has a Hadamard derivative at x if

∀v ∈ E, dHf(x; v) def= limt0w→v

f(x + tw)− f(x)t

exists and

v )→ dHf(x; v) : E → R is linear and continuous.

The above definitions extend from a real-valued function to a function f into anormed vector space F .

The next example is a Gateaux differentiable function in 0 that is not contin-uous in 0 and not Hadamard differentiable in 0.

Example 2.3.Consider the function f : R2 → R:

f(x, y) def=

⎧⎨

⎩

x6

(y − x2)2 + x8 , if (x, y) = (0, 0),

0, if (x, y) = (0, 0).(2.8)

It is readily seen that f has a Gateaux semiderivative at (0, 0) in all directions,

∀v ∈ R2, df((0, 0); v

)= 0, (2.9)

and that v )→ df((0, 0); v

)= 0 is trivially linear and continuous with respect to

v. But it is not Hadamard semidifferentiable in (0, 0) in the direction (0, 0) by thesame argument as in Example 2.1. Similarly, for ε > 0 by choosing the directions

w(ε) = (1, ε)→ v = (1, 0) as ε→ 0,

thenf(εw(ε)

)− f

((0, 0)

)

ε=

1ε3→ +∞

and dHf((0, 0); (1, 0)) does not exist. Finally, f is not continuous in (0, 0). To seethis follow the path (ε, ε2) as ε goes to 0:

∣∣f(ε, ε2)− f(0, 0)∣∣ =

∣∣∣∣ε6

ε8

∣∣∣∣ =1ε2→ +∞ as ε→ 0.

2.2 Derivatives in Normed Vector SpacesWe now specialize to normed vector spaces. In a normed vector space E, we havethe strong (norm) topology and the weak topology. As a result we have two notionsof Hadamard semiderivatives,

dsHf(x; v) def= lim

t0w→v in E-strong

f(x + tw)− f(x)t

, (2.10)

dwHf(x; v) def= lim

t0wv in E-weak

f(x + tw)− f(x)t

, (2.11)


and two notions of derivatives, where w → v denotes the strong convergence andw v denotes the weak convergence. By definition, the weak Hadamard semide-rivative is a stronger condition than the strong Hadamard semiderivative:

dwHf(x; v) exists ⇒ ds

Hf(x; v) exists. (2.12)

The two notions coincide when E is finite-dimensional.

Definition 2.3.Let f be a real-valued function defined in a neighborhood of a point x of a normedvector space E.

(i) The function f has a strong Hadamard derivative in x if

∀v ∈ E, dsHf(x; v) def= lim

t0w→v in E-strong

f(x + tw)− f(x)t

exists and

v )→ dsHf(x; v) : E → R is linear and continuous.

(ii) The function f has a weak Hadamard derivative in x if

∀v ∈ E, dwHf(x; v) def= lim

t0wv in E-weak

f(x + tw)− f(x)t

exists and

v )→ dwHf(x; v) : E → R is linear and continuous.

(iii) The function f has a Frechet derivative in x if there exists a continuous linearmapping L(x) : E → R such that

lim|h|E→0

f(x + h)− f(x)− L(x)h|h|E

→ 0.

The above definitions extend from a real-valued function to a function f into anothernormed vector space F .

If f has a Frechet derivative in x, then it has a Gateaux derivative at x. Indeedfor all v = 0 and t 0, h = tv → 0 in norm, and

f(x + tv)− f(x)t

− L(x)v = |v| f(x + tv)− f(x)− L(x)tv|tv|E

→ 0

⇒ df(x; v) = L(x)v and v )→ df(x; v) : E → R is linear and continuous.

By definition ⟨∇f(x), v⟩E = L(x)v, where ⟨·, ·⟩E denotes the duality pairing betweenE′ and E.

The following theorem makes explicit the connections and equivalences be-tween the three notions of derivatives.

Theorem 2.1. Let E be a normed vector space and f : V (x) → R be a functiondefined in a neighborhood of a point x ∈ E.


(i) If f has a Frechet derivative in x, then f has a weak Hadamard derivativein x and dw

Hf(x; v) = ⟨∇f(x), v⟩E = L(x)v. If f has a weak Hadamardderivative in x, then f has a strong Hadamard derivative in x. If f has astrong Hadamard derivative in x, then f has a Gateaux derivative in x.

(ii) If E is a reflexive Banach space, f has a weak Hadamard derivative in x ifand only if f has a Frechet derivative in x.

For a function f , the four notions are well ordered:

Frechet derivative ⇒ weak Hadamard derivative ⇒ strong Hadamard derivative

and the last one implies that f has a Gateaux derivative in x. In view of Theo-rem 2.1, the first three notions are equivalent in finite dimension.

Theorem 2.2. When E is finite-dimensional, the strong Hadamard derivative,the weak Hadamard derivative, and the Frechet derivative of Definition 2.3 areequivalent and equal.

Proof of Theorem 2.1. (i) Given t > 0 and w ∈ E, consider the quotient

q(t, w) def=f(x + tw)− f(x)

t

and for h ∈ E the quotient

Q(h) def=

⎧⎨

⎩

f(x + h)− f(x)− L(x)h|h|E

, h = 0,

0, h = 0.

Then

q(t, w) = Q(h(t, w)) |w|E + L(x)w.

Given v ∈ E, as w v and t 0, h(t, w) def= tw → 0 in the E-norm. HenceQ(h(t, w))→ 0 since f has a Frechet derivative in x and L(x)w → L(x)v as w vby continuity. Therefore

limwvt0

q(t, w) = L(x)v

and dwHf(x; v) exists and v )→ dw

Hf(x; v) = L(x)v : E → R is linear and continuouswith respect to v. The second statement follows from (2.12) and the last one bydefinition of the Gateaux and Hadamard derivatives.

(ii) Define the element L(x) of E′ as

∀v ∈ E, L(x)v def= ⟨∇f(x), v⟩E = dwHf(x; v).


Denote by Q the limsup of |Q(h)| as |h| → 0, which can possibly be infinite. Lethn be a sequence in E such that |hn|→ 0 and |Q(hn)|→ Q. For h = 0

Q(h) = Q

(|h| h

|h|

).

Since E is a reflexive Banach space there exist a subsequence of hn, still denotedby hn, and v ∈ E such that

wndef=

hn

|hn| v ∈ E and |Q(hn)|→ Q.

By letting tn = |hn|,

Q(hn) =f(x + tnwn)− f(x)

tn− ⟨∇f(x), wn⟩E

= q(tn, wn)− ⟨∇f(x), wn⟩E → dwHf(x; v)− ⟨∇f(x), v⟩E = 0.

Therefore Q = 0 and f is Frechet differentiable in x.

We complete this section with a classical sufficient condition for the Frechetdifferentiability.

Theorem 2.3. Given a normed vector space E and a map f : V (x)→ R, definedin a neighborhood V (x) of x ∈ E, assume that

(i) for all y ∈ V (x), f has a Gateaux derivative ∇f(y), and

(ii) the map ∇f : V (x)→ E′-strong is continuous in x.

Then f has a Frechet derivative in x.

Proof. There exists ρ > 0 such that the open ball B(x, ρ) is contained in V (x). Forh ∈ E, 0 < |h| < ρ, consider the quotient

Q(h) def=f(x + h)− f(x)− ⟨∇f(x), h⟩E

|h|E.

There exists α(h), 0 < α(h) < 1, such that

f(x + h)− f(x) = ⟨∇f(x + α(h)h), h⟩E

⇒ Q(h) =⟨∇f(x + α(h)h)−∇f(x),

h

|h|E

⟩

E

⇒ |Q(h)| ≤ |∇f(x + α(h)h)−∇f(x)|E′ → 0 as h→ 0.

This shows that f is Frechet differentiable in x.


2.3 Locally Lipschitz FunctionsThe following theorem gives a sufficient condition for the equivalence of Gateauxand Hadamard semiderivatives (resp., Gateaux and Hadamard derivatives).

Theorem 2.4. Let E be a normed vector space and f : E → R be a function thatis uniformly Lipschitzian in a neighborhood U of x in E, that is,

∃c(x) > 0, ∀y, z ∈ U, |f(y)− f(z)| ≤ c(x) |y − z|E . (2.13)

(i) If df(x; v) exists, then dsHf(x; v) exists and ds

Hf(x; v) = df(x; v). Moreover,if df(x; v) exists for all v ∈ E, then

∀v2, v1 ∈ E, |dsHf(x; v2)− ds

Hf(x; v1)| ≤ c |v2 − v1|E .

(ii) If f has a Gateaux derivative at x, then

∀v, dsHf(x; v) exists and

v )→ dsHf(x; v) : E → R is linear and continuous

and f has a strong Hadamard derivative in x.

Proof. (i) There exist ε > 0 and a neighborhood W of v in E such that

∀w ∈W, ∀ε, 0 < ε ≤ ε, x + εw ∈ U and x + εv ∈ U.

Then1ε

[f(x + εw)− f(x)

]=

1ε

[f(x + εw)− f(x + εv)

]+

1ε

[f(x + εv)− f(x)

]

and ∣∣∣∣1ε

[f(x + εw)− f(x)

]− df(x; v)

∣∣∣∣

≤∣∣∣∣1ε

[f(x + εv)− f(x)

]− df(x; v)

∣∣∣∣+ c(x)|w − v|E .

So as ε→ 0 and w → v, dsHf(x; v) = df(x; v).

(ii) By definition, from part (i).

2.4 Chain Rule for SemiderivativesThe family of Hadamard semidifferentiable functions is extremely interesting sincethis property is stable under composition. They form a sufficiently large family ofnondifferentiable functions that include continuous convex functions as we shall seein section 2.5.

Theorem 2.5. Let E and F be two topological vector spaces and let h be the com-position of two mappings f and g

h(x) = f(g(x)

)(2.14)


in a neighborhood U of a point x in E, where

g : U ⊂ E → F and f : g(U)→ R . (2.15)

Assume that

(i) g has a Gateaux (resp., Hadamard) semiderivative at x in the direction v, and

(ii) dHf(g(x); dg(x; v)) exists,

where the Hadamard semiderivative is taken in the general sense of Definition 2.1 (ii)with respect to the topologies of E and F .3 Then

dh(x; v) = dHf(g(x); dg(x; v))(resp., dHh(x; v) = dHf(g(x); dHg(x; v))

).

(2.16)

Proof. (a) For ε > 0 small enough let

m(ε) =g(x + εv)− g(x)

ε− dg(x; v) in F.

By assumption m(ε) → 0 as ε → 0. By the definition of dh(x; v) we want to findthe limit of the differential quotient

d(ε) =f(g(x + εv)

)− f

(g(x)

)

ε,

which can be rewritten as

d(ε) =f(g(x) + ε

(dg(x; v) + m(ε)

))− f

(g(x)

)

ε,

wheredg(x; v) + m(ε)→ dg(x; v) as ε→ 0.

So by the definition of dHf ,

limε0

d(ε) = dHf(g(x); dg(x; v)

).

(b) When g is Hadamard semidifferentiable we replace m(ε) and d(ε) by

m(ε, w) =g(x + εw)− g(x)

ε− dg(x; v)

andd(ε, w) =

1ε

[f(g(x) + ε

(dg(x; v) + m(ε, w)

))− f

(g(x)

)]

and proceed as in part (a).3For normed vector spaces this theorem gives two sets of results: one for the strong topology

and one for the weak topology.


In general we cannot improve the semiderivative of h by improving the semi-derivative of g when f is not Hadamard semidifferentiable.

Example 2.4.Consider the composition of the function f : R2 → R in Example 2.3 and the map

g : R→ R2, g(x) = (x, x2). (2.17)

The map f is Gateaux, but not Hadamard semidifferentiable, and the map g isinfinitely differentiable. However, the composition

h(x) = f(g(x)

)= 1/x2 (2.18)

is not even Gateaux semidifferentiable in 0.

We reiterate that the Hadamard semidifferentiability is a key property for the“chain rule.” In general the composition h = f g of two maps will fail to have asemiderivative unless f has a Hadamard semiderivative, even if the map g : E → Fis Frechet differentiable at the point x.

2.5 Semiderivatives of Convex FunctionsFinally the class of functions that are Hadamard semidifferentiable is quite largesince it contains the classical continuously differentiable functions and the convexcontinuous functions.

Theorem 2.6. Let f : U ⊂ E → R be a function defined in an open convex subsetof a topological vector space E.

(i) f is convex in U if and only if

∀x ∈ U, ∀v ∈ E, df(x; v) exists, (2.19)∀x ∈ U, ∀v ∈ E, df(x; v) + df(x;−v) ≥ 0, (2.20)∀x, y ∈ U, f(y) ≥ f(x) + df(x; y − x). (2.21)

(ii) If E is a normed vector space and if f is convex in U and continuous in x ∈ U ,there exists a neighborhood V (x) of x such that

∀y ∈ V (x), ∀v ∈ E, dsHf(y; v) exists. (2.22)

Remark 2.2.In a normed vector space E, the norm nE(x) def= |x|E is a convex and Lipschitzianfunction in E. Hence for all x, v ∈ E, dHnE(x; v) exists and dHnE(0; v) = |v|E .

Proof. Part (ii) is a consequence of part (i) and the fact that a continuous convexfunction at x is locally Lipschitzian in a neighborhood of x (cf. I. Ekeland andR. Temam, [1, pp. 11–12]) by direct application of Theorem 2.4. We give the proof


of part (i) for completeness. Assume that f is convex in U . Let θ ∈ ]0, 1]. Givenx ∈ U and v ∈ E

∃α, 0 < α < 1, such that x− αv ∈ U,

∃θ0, 0 < θ0 < 1, such that ∀θ ∈]0, θ0], x + θv ∈ U.

For this fixed α, we show that

∀θ ∈ ]0, θ0[ ,f(x)− f(x− αv)

α≤ f(x + θv)− f(x)

θ. (2.23)

This follows from the identity

x =α

α+ θ(x + θv) +

θ

α+ θ(x− αv)

and the convexity of f ,

f(x) ≤ α

α+ θf(x + θv) +

θ

α+ θf(x− αv).

This can be rewritten asθ

θ + α

[f(x)− f(x− αv)

]≤ α

θ + α

[f(x + θv)− f(x)

]

and yields (2.23). Define

ϕ(θ) =f(x + θv)− f(x)

θ, 0 < θ < θ0.

Now we show that ϕ is a monotone increasing function of θ > 0. For all θ1 and θ2,0 < θ1 < θ2 < θ0:

f(x + θ1v)− f(x) = f

(θ1θ2

(x + θ2v) +(

1− θ1θ2

)x

)− f(x)

≤ θ1θ2

f(x + θ2v) +(

1− θ1θ2

)f(x)− f(x),

which implies that ϕ(θ1) ≤ ϕ(θ2). As ϕ is a monotone increasing function which isbounded below, its limit exists as θ goes to 0. By definition it is equal to df(x; v).Now that we know that df(x; v) exists in all directions v, go back to inequality(2.23) and let α go to zero to get (2.20).

Conversely, apply inequality (2.19) twice to x and x + θ(y − x) and to y andx + θ(y − x) with θ ∈ [0, 1]:

f(x) ≥ f(x + θ(y − x)) + df(x + θ(y − x);−θ(y − x)),f(y) ≥ f(x + θ(y − x)) + df(x + θ(y − x); (1− θ)(y − x)).

Multiply the first inequality by 1− θ and the second by θ and add up each side:

(1− θ)f(x) + θf(y) ≥ f(x + θ(y − x)) + (1− θ)θdf(x + θ(y − x);−(y − x))+ θ(1− θ)df(x + θ(y − x); y − x).

By using property (2.20), we get

f(θy + (1− θ)x) ≤ θf(y) + (1− θ)f(x)

and the convexity of f in U .


2.6 Hadamard Semiderivative and Velocity MethodIn section 3 of Chapter 4 we have drawn an analogy between Gateaux and Hadamardsemiderivatives on one hand and shape semiderivatives obtained by a perturbationof the identity and the velocity method on the other hand. The next theoremrelates the Hadamard semiderivative and the semiderivative obtained by the velocitymethod for real functions defined on RN.

Theorem 2.7. Let f : NX → R be a real function defined in a neighborhood NX

of a point X in RN. Then f is Hadamard semidifferentiable at (X, v) if and onlyif there exists τ > 0 such that for all velocity fields V : [0, τ ] → R satisfying thefollowing assumptions:

(a) (V1) ∀x ∈ RN, V (·, x) ∈ C([0, τ ];RN);

(b) (V2) ∃c > 0, ∀x, y ∈ RN, ∥V (·, y)− V (·, x)∥C([0,τ ];RN) ≤ c|y − x|;

(c) the limit

df(X; v) def= lim∀V, V (0)=v

t0

f(Tt(V )(X)

)− f(X)

t(2.24)

exists and is independent of the choice of V satisfying (a) and (b), where

Tt(V )(X) def= x(t;X),dx

dt(t;X) = V

(t, x(t;X)

), 0 < t < τ, x(0;X) = X.

Proof. (⇒) Let V be a vector field satisfying conditions (a), (b), and (c). Define

w(t) def=1t

∫ t

0V(s, x(s)

)ds, 0 < t ≤ τ.

It is continuous on ]0, τ ] and

w(t)− v =1t

∫ t

0

[V(s, x(s)

)− V (0, X)

]ds.

Therefore

|w(t)− v| ≤ cmax[0,t]

|x(s)−X| + max[0,t]

|V (s, X)− V (0, X)|

⇒ limt0

w(t) = v.

So

limt0

f(x(t;X)

)− f(X)

t= lim

w(t)→vt0

f(X + tw(t)

)− f(X)

t= dHf(X; v),


since f is Hadamard semidifferentiable at (X, v) and the limit depends only onV (0) = v.

(⇐) Conversely saying that the dHf(x; v) exists is equivalent to saying thatfor the sequence tn = 2−n and any sequence wn → v the limit

f(x + tnwn)− f(x)tn

exists and depends only on v. We now associate with the sequence wn a velocityfield V satisfying (a) to (c) such that V (tn, X) = wn and Ttn(V )(X) = X + tnwn.Then from properties (2.24) and V (o) = v we conclude that dHf(x; v) exists. Sett0 = 1, tn = 2−n, and xn = X + tnwn and observe that tn − tn+1 = −2−(n+1).Define for m ≥ 1 the following Cm-interpolation in (0, 1]: for t in [tn+1, tn]

Tt(X) def= xn + p

(t− tn

tn+1 − tn

)(xn+1 − xn)

+ q0

(t− tn

tn+1 − tn

)(tn+1 − tn)wn

+ q1

(t− tn

tn+1 − tn

)(tn+1 − tn)wn+1, T0(X) def= X,

where p, q0, q1 ∈ P 2m+1[0, 1] are polynomials of order 2m + 1 on [0, 1] such thatp(0) = 0, p(1) = 1, and p(ℓ)(0) = 0 = p(ℓ)(1), 1 ≤ ℓ ≤ m, q0(0) = 0 = q0(1),q′0(0) = 1, q′

0(1) = 0, and q(ℓ)0 (0) = 0 = q(ℓ)

0 (1), 2 ≤ ℓ ≤ m, and q1(0) = 0 = q1(1),q′1(0) = 0, q′

1(1) = 1, and q(ℓ)1 (0) = 0 = q(ℓ)

1 (1), 2 ≤ ℓ ≤ m. By definitionT (·, X) ∈ Cm((0, 1];RN). Moreover, for 1 ≤ ℓ ≤ m,

Ttn(X) = xn, Ttn+1(X) = xn+1,∂T

∂t(tn, X) = wn,

∂T

∂t(tn+1, X) = wn+1,

∂T

∂t(t, X) = p′

(t− tn

tn+1 − tn

)xn+1 − xn

tn+1 − tn+ q′

0

(t− tn

tn+1 − tn

)wn

+ q′1

(t− tn

tn+1 − tn

)wn+1.

We now show that T satisfies conditions (a) and (b), which are equivalent to con-dition (4.2) in Chapter 4 on V . First observe that Tt(X)−X and ∂T/∂t are bothindependent of X and (Tt − I)(X) ∈ Cm((0, 1];RN). Define

f(t) def= Tt(X)−X ⇒ ∂T

∂t(t, X) =

∂f

∂t.

Hence T and ∂T/∂t clearly satisfy conditions (T1) in (4.8) in Chapter 4. It satis-fies condition (T2) since T−1

t (Y ) = Y − f(t), and, a fortiori, T−1t satisfies condi-

tions (T3). Therefore, the velocity field

V (t, x) def=∂T

∂t(t, T−1

t (x)) =∂f

∂t(t)

3. First-Order Shape Semiderivatives and Derivatives 471

satisfies conditions (V) given by (4.2) of Chapter 4. Thus properties (a) and (b)are satisfied. It remains to show that ∂f/∂t(t) → v as t → 0. It is easy to checkthat by construction4 −p′(τ) + q′

0(τ) + q′1(τ) = 1 in [0, 1] and that on each interval

[tn+1, tn]

xn+1 − xn

tn+1 − tn= wn+1 − 2wn,

∂f

∂t(t) = p′

(t− tn

tn+1 − tn

)(wn+1 − 2wn) + q′

0

(t− tn

tn+1 − tn

)wn

+ q′1

(t− tn

tn+1 − tn

)wn+1,

∂f

∂t(t)− v = p′

(t− tn

tn+1 − tn

)[(wn+1 − v)− 2(wn − v)]

+ q′0

(t− tn

tn+1 − tn

)(wn − v) + q′

1

(t− tn

tn+1 − tn

)(wn+1 − v).

The left-hand side now clearly converges since the three polynomials are boundedby a constant independent of n. This is sufficient to prove the theorem.

3 First-Order Shape Semiderivativesand Derivatives

3.1 Eulerian and Hadamard Semiderivatives

Go back to Definition 3.1 of a shape functional in section 3.1 of Chapter 4 andthe discussion of Gateaux and Hadamard semiderivatives in section 3.2 of the samechapter, where we have shown that results based on perturbations of the identityor asymptotic developments can be readily recovered by direct application of thevelocity method with nonautonomous velocity fields V (t, x). In view of this, wegive all the basic definitions, constructions, and theorems within the context of thevelocity method and emphasize the Hadamard semiderivative, which is importantin situations where the shape is parametrized and the chain rule is necessary.

We give the definitions in the constrained case under assumptions (5.5) inChapter 4 that can be specialized to the unconstrained case (D = RN).5 Considera velocity field

V : [0, τ ]×D → RN (3.2)

4By interpolation of the function g(τ) = τ − 1, g(τ) = g(0)p(τ) + g′(0)q0(τ) + g′(1)q1(τ) =−p(τ) + q0(τ) + q1(τ), and necessarily 1 = g′(τ) = −p′(τ) + q′

0(τ) + q′1(τ).

5In the unconstrained case, everything reduces to the conditions (V) on the velocity field givenby (4.2) in Chapter 4: there exists τ = τ(V ) > 0 such that

(V)∀x ∈ RN, V (·, x) ∈ C

([0, τ ];RN

),

∃c > 0, ∀x, y ∈ RN, ∥V (·, y) − V (·, x)∥C([0,τ ];RN) ≤ c|y − x|.(3.1)


verifying condition (V1D) and the equivalent form (V2C) of condition (V2D)6 forsome τ = τ(V ) > 0:

(V1D)∀x ∈ D, V (·, x) ∈ C

([0, τ ];RN), ∃c > 0,

∀x, y ∈ D, ∥V (·, y)− V (·, x)∥C([0,τ ];RN) ≤ c|y − x|,(V2C) ∀t ∈ [0, τ ], ∀x ∈ D, V (t, x) ∈ LD(x) = −CD(x) ∩ CD(x),

(3.3)

where LD(x) = CD(x) ∩ −CD(x) is the linear tangent space to D at the pointx ∈ ∂D and CD(x) is Clarke’s tangent cone to D at x (Theorem 5.2 in Chapter 4).We shall refer to the set of conditions (3.2)–(3.3) as condition (VD).

Introduce the following linear subspace of Lip (D,RN):

Lip L(D,RN) def=θ ∈ Lip (D,RN) : ∀X ∈ ∂D, θ(X) ∈ LD(X)

. (3.4)

Under the action of a velocity field V satisfying conditions (3.2)–(3.3), a set Ω inD is transformed into a new domain,

Ωt(V ) def= Tt(V )(Ω) = Tt(V )(X) : ∀X ∈ Ω , (3.5)

also contained in D. The transformation Tt : D → D associated with the velocityfield x )→ V (t)(x) def= V (t, x) : D → RN is given by

Tt(X) def= x(t, X), t ≥ 0, X ∈ D, (3.6)

where x(·, X) is solution of the differential equation

dx

dt(t, X) = V (t, x(t, X)), t ≥ 0, x(t, X) = X. (3.7)

Recall Definition 3.1 of section 3 in Chapter 4 of a shape functional.

Definition 3.1.Given a nonempty subset D of RN, consider the set P(D) = Ω : Ω ⊂ D of subsetsof D. The set D will be referred to as the underlying holdall or universe. A shapefunctional is a map

J : A→ E (3.8)

from some admissible family A of sets in P(D) into a topological space E.

For instance, A could be the set X (Ω) = F (Ω) : ∀F ∈ F(Θ).

Definition 3.2.Let J be a real-valued shape functional.

6We replace condition (V2D) in (5.5) of Chapter 4

(V2D) ∀x ∈ D, ∀t ∈ [0, τ ], ±V (t, x) ∈ TD

(x)

by its equivalent form (V2C) by using Theorem 5.2 in Chapter 4.


(i) Let V be a velocity field satisfying conditions (3.2)–(3.3). J has an Euleriansemiderivative at Ω in the direction V if the limit

limt0

J(Ωt(V )

)− J(Ω)

t(3.9)

exists. It will be denoted by dJ(Ω; V ). For simplicity, when V (t, x) = θ(x),θ ∈ Lip L(D,RN) an autonomous velocity field, we shall write dJ(Ω; θ).

(ii) Let Θ be a topological vector subspace of Lip L(D,RN) satisfying conditions(3.2)–(3.3). J has a Hadamard semiderivative at Ω in the direction θ ∈ Θ if

limV ∈C0([0,τ ];Θ)

V (0)=θt0


(3.10)

exists, depends only on θ, and is independent of the choice of V satisfyingconditions (3.2)–(3.3). It will be denoted by dHJ(Ω; θ). If dHJ(Ω; θ) exists,J has an Eulerian semiderivative at Ω in the direction V (t, x) = θ(x) and

dHJ(Ω; θ) = dJ(Ω; V (0)) = dJ(Ω; V ).

(iii) Let Θ be a topological vector subspace of Lip L(D,RN) satisfying conditions(3.2)–(3.3). J has a Hadamard derivative at Ω with respect to Θ if it has aHadamard semiderivative at Ω in all directions θ ∈ Θ and the map

θ )→ dJ(Ω; θ) : Θ→ R (3.11)

is linear and continuous. The map (3.11) will be denoted G(Ω) and referredto as the gradient of J in the topological dual Θ′ of Θ.

The definition of an Eulerian semiderivative is quite general. For instance, itreadily applies to shape functionals defined on closed submanifolds D of RN. Itincludes cases where dJ(Ω; V ) is dependent not only on V (0) but also on V (t) in aneighborhood of t = 0. We shall see that this will not occur under some continuityassumption on the map V )→ dJ(Ω; V ). When dJ(Ω; V ) depends only on V (0), theanalysis can be specialized to autonomous vector fields V and the semiderivativecan be related to the gradient of J .

Example 3.1.For any measurable subset Ω of RN, consider the volume function

J(Ω) =∫

Ωdx. (3.12)

For Ω with finite volume and V in C([0, τ ];C10 (RN,RN)), consider the transforma-

tions Tt(Ω) of Ω:

J(Tt(Ω)) =∫

Tt(Ω)dx =

∫

Ω|det(DTt)| dx =

∫

Ωdet(DTt) dx


for t small since det(DT0) = det I = 1. This yields the Eulerian semiderivative

dJ(Ω; V ) =∫

Ωdiv V (0) dx. (3.13)

By definition it depends only on V (0) and hence J has a Hadamard semiderivative

dHJ(Ω; V (0)) =∫

Ωdiv V (0) dx (3.14)

and even a gradient G(Ω) for Θ = C10 (RN,RN).

The following simple continuity condition can also be used to obtain theHadamard semidifferentiability.

Theorem 3.1. Let Θ be a Banach subspace of Lip L(D,RN) satisfying conditions(3.2)–(3.3), J be a real-valued shape functional, and Ω be a subset of RN.

(i) Given θ ∈ Θ, if

∀V ∈ C([0, τ ]; Θ) such that V (0) = θ, dJ(Ω; V ) exists, (3.15)

and if the map

V )→ dJ(Ω; V ) : C([0, τ ]; Θ)→ R (3.16)

is continuous for the subspace of V ’s such that V (0) = θ, then J is Hadamardsemidifferentiable at Ω in the direction θ with respect to Θ and

∀V ∈ C([0, τ ]; Θ), V (0) = θ, dJ(Ω; V ) = dHJ(Ω; θ

). (3.17)

(ii) If for all V in C([0, τ ]; Θ), dJ(Ω; V ) exists and the map

V )→ dJ(Ω; V ) : C([0, τ ]; Θ)→ R (3.18)

is continuous, then J is Hadamard semidifferentiable at Ω in the directionV (0) with respect to Θ and

∀V ∈ C([0, τ ]; Θ), dJ(Ω; V ) = dJ(Ω; V (0)

)= dHJ

(Ω; V (0)

). (3.19)

Proof. Given V in C([0, τ ]; Θ) such that V (0) = θ, construct the sequence

Vn(t) def=

V (t), 0 ≤ t ≤ τ/n,

V(τn

), τ/n < t ≤ τ,

for all integers n ≥ 1.

Note that for each n ≥ 1, dJ(Ω; Vn) = dJ(Ω; V ) since for t < τ/n, Tt(Vn) =Tt(V ). By continuity of V , Vn converges in C([0, τ ]; Θ) to the autonomous fieldV (t) = V (0):

∀n, ∀V ∈ C([0, τ ]; Θ),∀ε > 0, ∃δ > 0, ∀t′, |t′| < δ, ∥V (t′)− V (0)∥Θ < ε.


Hence

∀ε > 0, ∃N > 0, ∀n ≥ N,τ

n< δ

⇒ supt∈[0, τ

n ]∥V (t)− V (0)∥Θ < ε ⇒ Vn → V .

Hence by continuity of the map (3.18)

dJ(Ω; Vn)→ dJ(Ω; V

)⇒ dJ(Ω; V ) = dJ

(Ω; V

).

But since dJ(Ω; V

)is dependent only on V (0), for all W ∈ Θ such that W (0) =

V (0) we have the identity dJ(Ω; W ) = dJ(Ω; V

). Therefore, J has a Hadamard

semiderivative at Ω in the direction V (0) and

dHJ(Ω; V (0)) = dJ(Ω; V

)= dJ(Ω; V ).

This concludes the proof of the theorem.

For simplicity, the last theorem was given only for a Banach space Θ. Thisincludes the spaces Ck

0 (RN), Ck(RN), C0,1(RN), and Bk(RN,RN) considered inChapters 3 and 4. However, its conclusions are not limited to Banach spaces. Otherconstructions can be used as illustrated below in the unconstrained case. Considervelocity fields in

−→V m,k def= lim−→K

V m,k

K : ∀K compact in RN

, (3.20)

where for m ≥ 0,

V m,kK

def= Cm([0, τ ];Dk(K,RN) ∩ Lip(RN,RN)

)(3.21)

and lim−→ denotes the inductive limit set with respect to K endowed with its naturalinductive limit topology. For autonomous fields, this construction reduces to

−→V k def= lim−→K

V k

K : ∀K compact in RN , (3.22)

V kK

def=

D0(K,RN) ∩ Lip (K,RN), k = 0,

Dk(K,RN), 1 ≤ k ≤ ∞.(3.23)

For k ≥ 1,−→V k = Dk(RN,RN). In all cases the conditions (3.1) of the unconstrained

case D = RN are verified.

Theorem 3.2. Let J be a real-valued shape functional, Ω be a subset of RN, andk ≥ 0 be an integer.

(i) Given θ ∈ −→V k, assume that

∀V ∈−→V 0,k, V (0) = θ, dJ(Ω; V ) exists (3.24)


and that the map

V )→ dJ(Ω; V ) :−→V 0,k → R (3.25)

is continuous for all V ’s such that V (0) = θ. Then J is Hadamard semidif-ferentiable in Ω in the direction θ with respect to Vk and

∀V ∈ −→V 0,k, V (0) = θ, dHJ(Ω; θ

)= dJ(Ω; V ) = dJ

(Ω; V (0)

). (3.26)

(ii) Assume that for all V ∈ −→V 0,k, dJ(Ω; V ) exists and that the map

V )→ dJ(Ω; V ) :−→V 0,k → R (3.27)

is continuous. Then J is Hadamard semidifferentiable in Ω in the directionV (0) with respect to Vk and

∀V ∈ −→V 0,k, dHJ(Ω; V (0)

)= dJ(Ω; V ) = dJ

(Ω; V (0)

). (3.28)

Proof. It is sufficient to prove the theorem for any compact subset K of RN andhence only for velocities in V0,k

K = C([0, τ ];VkK), where Vk

K is a Banach space con-tained in Ck

0 (RN,RN). So the theorem follows from Theorem 3.1.

3.2 Hadamard Semidifferentiability and Courant MetricContinuity

We now relate the Hadamard semidifferentiability of a shape functional with theCourant metric continuity studied in section 6 of Chapter 4. Recall the genericmetric spaces F(Θ) of Micheletti with metric d and the quotient group F(Θ)/Gwith the Courant metric dG in (2.31) of Chapter 3.

Theorem 3.3. Let Θ be equal to Ck+10 (RN,RN), Ck+1(RN,RN), or Ck,1(RN,RN),

k ≥ 0. Assume that G = G(Ω) for Ω closed or open and crack-free. If a shapefunctional J is Hadamard semidifferentiable for all θ ∈ Θ, then it is continuouswith respect to the Courant metric dG(Ω).

Proof. By definition of the Hadamard semiderivative and Theorem 6.1 for Θ =Ck+1

0 (RN,RN), Theorem 6.2 for Θ = Ck+1(RN,RN), and Theorem 6.3 for Θ =Ck,1(RN,RN) of Chapter 4.

3.3 Perturbations of the Identity and Gateaux and FrechetDerivatives

In the unconstrained case (D = RN), we have introduced in sections 3 and 4.2 ofChapter 4 a notion of directional semiderivative associated with first- and second-order perturbations of the identity. Even if this approach does not naturally extendto the constrained case, it is interesting to compare the definitions and results withthose associated with the velocity method of Definition 3.2.


Recall the generic metric spaces F(Θ) of Micheletti with metric d and thequotient group F(Θ)/G with the Courant metric dG in (2.31) in Chapter 3. Assumethat G = G(Ω) for Ω closed or open and crack-free and that there exist a ball Bε ofradius ε > 0 in Θ and a constant c > 0 such that

∀f ∈ Bε, ∥[I + f ]−1 − I∥Θ ≤ c ∥f∥Θ. (3.29)

Except for C0(RN,RN) and B0(RN,RN), this is true in all Banach spaces Θ ⊂C0,1(RN,RN) considered in Chapters 3 and 4 (cf. Theorem 2.14 in section 2.5.2and Theorem 2.17 in section 2.6.2 of Chapter 3). Hence the maps

f )→ [I + f ] )→ [I + f ] : Bε ⊂ Θ→ F(Θ)→ F(Θ)/G(Ω)

are well-defined and continuous in f = 0 since

dG(I, I + f) ≤ d(I, I + f) ≤ ∥f∥Θ + ∥[I + f ]−1 − I∥Θ ≤ (1 + c) ∥f∥Θ.

For a shape functional J , the map

[I + f ] )→ JΩ(f) def= J([I + f ](Ω)) : F(Θ)/G(Ω)→ R

is well-defined since J is invariant on G(Ω) and the map

f )→ JΩ(f) def= J([I + f ](Ω)) : Θ→ R

is continuous in f = 0 if J is continuous in Ω for the Courant metric on F(Θ)/G(Ω).The following definitions are now the extension of the standard definitions of

section 2 for a function JΩ(f) defined on the ball Bε in the normed vector space Θto topological vector spaces. They parallel the ones of Definition 3.2.

Definition 3.3.Let J be a real-valued shape functional and Θ be a topological vector subspace ofLip (RN,RN). For f ∈ Θ, denote [I + f ](Ω) by Ωf .

(i) JΩ is said to have a Gateaux semiderivative at f in the direction θ ∈ Θ if thefollowing limit exists and is finite:

dJΩ(f ; θ) def= limt0

J([I + f + tθ](Ω)

)− J([I + f ](Ω))

t. (3.30)

(ii) JΩ is said to be Gateaux differentiable at f if it has a Gateaux semiderivativeat f in all directions θ ∈ Θ and the map

θ )→ dJΩ(f ; θ) : Θ→ R (3.31)

is linear and continuous. The map (3.31) is denoted ∇JΩ(f) and referred toas the gradient of JΩ in the topological dual Θ′ of Θ.


(iii) If, in addition, Θ is a normed vector space, we say that J is Frechet differ-entiable at f if J is Gateaux differentiable at f and

lim∥θ∥Θ→0

|J([I + f + θ](Ω))− J([I + f ](Ω))− ⟨∇JΩ(f), θ⟩Θ|∥θ∥Θ

= 0. (3.32)

The semiderivatives of J and JΩ are related.

Theorem 3.4. Let J be a real-valued shape functional, let f ∈ Θ, and let I + f ∈F(Θ).

(i) Assume that JΩ has a Gateaux semiderivative at f in the direction θ ∈ Θ;then J has an Eulerian semiderivative at Ωf in the direction V f

θ and

dJΩ(f ; θ) = dJ(Ωf ;V fθ ), V f

θ (t) def= θ [I + f + tθ]−1 (3.33)

for t sufficiently small.

(ii) If J has a Hadamard semiderivative at Ωf in the direction θ [I + f ]−1, thenJΩ has a Gateaux semiderivative at f in the direction θ and

dJΩ(f ; θ) = dHJ(Ωf ; θ [I + f ]−1). (3.34)

Conversely, if JΩ has a Gateaux semiderivative at f in the direction θ [I +f ],then J has a Hadamard semiderivative at Ωf in the direction θ. If eitherdJΩ(f ; θ) or dHJ(Ωf ; θ) is linear and continuous with respect to all θ in Θ,so is the other and

∀θ ∈ Θ,⟨∇JΩ(f), θ⟩Θ = ⟨G(Ωf ), θ [I + f ]−1⟩Θ,

⟨G(Ωf ), θ⟩Θ = ⟨∇JΩ(f), θ [I + f ]⟩Θ.(3.35)

Proof. By definition

dJΩ(f ; θ) = limt0

J([I + f + tθ](Ω)

)− J([I + f ](Ω))

t

= limt0

J(Tt([I + f ](Ω))

)− J([I + f ](Ω))

t= dJ([I + f ](Ω);V f

θ )

for the family of transformations

T ft

def= [I + f + tθ] [I + f ]−1,

and from Theorem 4.1 in section 4.1 of Chapter 4, T ft corresponds to the velocity

field

V fθ (t) def=

∂T ft

∂t (T f

t )−1 = θ [I + f + tθ]−1.

Identity (3.34) now follows from the fact that J has a Hadamard semiderivative atΩf . The other properties readily follow from the definitions.


We have standard sufficient conditions for the Frechet differentiability fromTheorem 2.3.

Theorem 3.5. Let J be a real-valued shape functional. Let Ω be a subset of RN

and Θ be Ck+10 (RN,RN), Ck+1(RN,RN), Ck,1(RN), or Bk+1(RN,RN), k ≥ 0. If

JΩ is Gateaux differentiable for all f in Bε and the map

f )→ ∇JΩ(f) : Bε → Θ′

is continuous in f = 0, then JΩ is Frechet differentiable in f = 0.

3.4 Shape Gradient and Structure TheoremIn view of the previous discussion we now specialize to autonomous vector fields Vto further study the properties and the structure of dJ(Ω; V ). For simplicity wealso specialize to the unconstrained case in RN. The constrained case yields similarresults but is technically more involved (cf. M. C. Delfour and J.-P. Zole-sio [14] for D open in RN).

The choice of a shape gradient depends on the choice of the topological vectorsubspace Θ of Lip (RN,RN). We choose to work in the classical framework of theTheory of Distributions (cf. L. Schwartz [3]) with Θ = D(RN,RN), the spaceof all infinitely differentiable transformations θ of RN with compact support. Forthese velocity fields V , conditions (3.1) are satisfied.

Definition 3.4.Let J be a real-valued shape functional. Let Ω be a subset of RN.

(i) The function J is said to be shape differentiable at Ω if it is differentiable atΩ for all θ in D(RN,RN).

(ii) The map (3.11) defines a vector distribution G(Ω) in D(RN,RN)′, which willbe referred to as the shape gradient of J at Ω.

(iii) When, for some finite k ≥ 0, G(Ω) is continuous for the Dk(RN,RN)-topology,we say that the shape gradient G(Ω) is of order k.

The next theorem gives additional properties of shape differentiable functions.

Notation 3.1.Associate with a subset A of RN and an integer k ≥ 0 the set

LkA

def=V ∈ Dk(RN,RN) : ∀x ∈ A, V (x) ∈ LA(x)

,

where LA(x) = −CA(x)∩CA(x) and CA(x) is given by (5.27) in Chapter 4.

Theorem 3.6 (Structure theorem). Let J be a real-valued shape functional. As-sume that J has a shape gradient G(Ω) for some Ω ⊂ RN with boundary Γ.

(i) The support of the shape gradient G(Ω) is contained in Γ.


(ii) If Ω is open or closed in RN and the shape gradient is of order k for somek ≥ 0, then there exists [G(Ω)] in (Dk/Lk

Ω)′ such that for all V in Dk def=Dk(RN,RN)

dJ(Ω; V ) = ⟨[G(Ω)], qLV ⟩Dk/LkΩ, (3.36)

where qL : Dk → Dk/LkΩ is the canonical quotient surjection. Moreover

G(Ω) = (qL)∗[G(Ω)], (3.37)

where (qL)∗ denotes the transpose of the linear map qL.

Proof. (i) Any V in D such that V = 0 on Γ satisfies assumptions (V 1Ω) and (V 2Ω)in (5.5) (with Ω in place of D) and V ∈ L∞

Ω . Since V = 0 on Γ, Ts(Γ) = Γ.Moreover, by Theorem 5.1 (i) of Chapter 4, Ts : Ω → Ω is a homeomorphism andTs(Ω) = Ω and (Ω)s = Ts(Ω) = Ω since Ω = Ω∪ Γ and Ω∩ Γ = ∅. Thus when Ω isclosed (Ω = Ω) or open (Ω = int Ω),

∀s ≥ 0, Ts(Ω) = Ω ⇒ J(Ωs) = J(Ω) ⇒ dJ(Ω; V ) = 0.

(ii) It is sufficient to prove that dJ(Ω; V ) = 0 for all V in LkΩ. The other

statements follow by standard arguments and the fact that LkΩ is a closed linear

subspace of Dk. From part (i) we know that the result is true for all V in L∞Ω and

hence by a density argument for all V in LkΩ.

Remark 3.1.When the boundary Γ of Ω is compact and J is shape differentiable at Ω, the distri-bution G(Ω) is of finite order. Once this is known, the conclusions of Theorem 3.6(ii) apply with k equal to the order of G(Ω). Hence G(Ω) will belong to a Hilbertspace H−s(RN) for some s ≥ 0.

The quotient space is very much related to a trace on the boundary Γ, andwhen the boundary Γ is sufficiently smooth we can indeed make that identification.

Corollary 1. Assume that the assumptions of Theorem 3.6 are satisfied for anopen domain Ω, that the order of G(Ω) is k ≥ 0, and that the boundary Γ of Ω isCk+1. Then for all x in Γ, LΩ(x) is an (N − 1)-dimensional hyperplane to Ω atx and there exists a unique outward unit normal n(x) which belongs to Ck(Γ;RN).As a result, the kernel of the map

V )→ γΓ(V ) · n : Dk(RN,RN)→ Ck(Γ) (3.38)

coincides with LkΩ, where γΓ : Dk(RN,RN) → Ck(Γ,RN) is the trace of V on Γ.

Moreover, the map pL(V )

qL(V ) )→ pL

(qL(V )

) def= γΓ(V ) · n : Dk/LkΩ → Ck(Γ) (3.39)

is a well-defined isomorphism. In particular, there exists a scalar distribution g(Γ)in RN with support in Γ such that g(Γ) ∈ Ck(Γ)′ and for all V in Dk(RN,RN)

dJ(Ω; V ) = ⟨g(Γ), γΓ(V ) · n⟩Ck(Γ) (3.40)


and

G(Ω) = ∗(qL)[G(Ω)], [G(Ω)] = ∗(pL)g(Γ). (3.41)

When g(Γ) ∈ L1(Γ)

dJ(Ω; V ) =∫

Γg V · n dΓ and G = γ∗

Γ(g n), (3.42)

where γΓ is the trace operator on Γ.

Proof. The surjectivity of (3.39) is a consequence of the fact that Γ is compactand that for a Ck+1 boundary, k ≥ 0, it is always possible to construct an exten-sion N of the unit normal n on Γ which belongs to Dk(RN,RN) with support ina neighborhood of Γ. Then for any v in Ck(Γ), there exists also an extension v inDk(RN) with support in a neighborhood of Γ, and the vector V = vN belongs toDk(RN,RN) and coincides with vn on Γ.

Remark 3.2.In 1907, J. Hadamard [1] used displacements along the normal to the boundaryΓ of a C∞-domain (as in section 3.3.1 of Chapter 4) to compute the derivativeof the first eigenvalue of the clamped plate. Theorem 3.6 and its corollary aregeneralizations to arbitrary shape functionals of that property to open or closeddomains with an arbitrary boundary. The structure theorem for shape functionalson open domains with a Ck+1-boundary is due to J.-P. Zolesio [12] in 1979 andnot to Hadamard even if the formula (3.42) is often called the Hadamard formula.

Example 3.2.For any measurable subset Ω of RN, consider the volume shape function (3.12) ofExample 3.1:

J(Ω) =∫

Ωdx.

For Ω with finite volume and V in D1(RN,RN), we have seen in Example 3.1 that

dJ(Ω; V ) =∫

Ωdiv V dx =

∫

RNχΩ div V dx, (3.43)

and this is formula (3.40) with k = 1. For an open domain Ω with a C1 compactboundary Γ,

dJ(Ω; V ) =∫

ΓV · n dΓ, (3.44)

which is also continuous with respect to V in D0(RN,RN). Here the smoothness ofthe boundary decreases the order of the distribution G(Ω). This raises the questionof the characterization of the family of all subsets Ω of RN for which the map

V )→∫

Ωdiv V dx : D1(RN,RN)→ R (3.45)

can be continuously extended to D0(RN,RN). But this is the family of locally finiteperimeter sets: sets Ω whose characteristic function belongs to BVloc(RN).


4 Elements of Shape CalculusIn this section we recall a number of basic formulae from J.-P. Zolesio [7, 8] forthe derivative of domain and boundary integrals. The reader is also referred to thecompanion book of J. Soko!lowski and J.-P. Zolesio [9] for the computation ofshape derivatives associated with a wide range of partial differential equations.

In this section, a more modern treatment of the derivative of boundary inte-grals is given that emphasizes a new approach to the tangential calculus on a C2

submanifold of RN of codimension 1. This development took place in the contextof the theory of shells, where a simple and self-contained intrinsic differential calcu-lus has been developed by using the oriented distance function of Chapter 7. Itcompletely avoids the use of local maps and bases and Christoffel’s symbols. Inthis section a new, considerably simplified proof of the formula for the derivative ofboundary integrals is presented by using the oriented distance function.

4.1 Basic Formula for Domain IntegralsThe simplest examples of domain functions are given by volume integrals over abounded open domain Ω in RN. They use a basic formula in connection withthe family of transformations Tt : 0 ≤ t ≤ τ. Assume that condition (3.1)is satisfied by the velocity field V (t) : 0 ≤ t ≤ τ. Further assume that V ∈C0([0, τ ];C1

loc(RN,RN)) and that τ > 0 is such that the Jacobian Jt is strictly

positive:

∀t ∈ [0, τ ], Jt(X) def= det DTt(X) > 0, (DTt)ij = ∂jTi,

where DTt(X) is the Jacobian matrix of the transformation Tt = Tt(V ) associatedwith the velocity vector field V . Given a function ϕ in W 1,1

loc (RN), consider for0 ≤ t ≤ τ the volume integral

J(Ωt(V )) def=∫

Ωt(V )ϕ dx, (4.1)

where Ωt(V ) def= Tt(V )(Ω). By the change of variables formula

J(Ωt(V )) =∫

Ωt(V )ϕ dx =

∫

Ωϕ Tt Jt dx, (4.2)

and the following formulae and results are easy to check.

Theorem 4.1. Let ϕ be a function in W 1,1loc (RN). Assume that the vector field

V = V (t) : 0 ≤ t ≤ τ satisfies condition (V ).

(i) For each t ∈ [0, τ ] the map

ϕ )→ ϕ Tt : W 1,1loc (RN)→W 1,1

loc (RN)

and its inverse are both locally Lipschitzian and

∇(ϕ Tt) = ∗DTt∇ϕ Tt.

4. Elements of Shape Calculus 483

(ii) If V ∈ C0([0, τ ];C1loc(R

N,RN)), then the map

t )→ ϕ Tt : [0, τ ]→W 1,1loc (RN)

is well-defined and for each t

d

dtϕ Tt = (∇ϕ · V (t)) Tt ∈ L1

loc(RN). (4.3)

Hence the function

t )→ ϕ Tt belongs to C1([0, τ ];L1loc(R

N)) ∩ C0([0, τ ];W 1,1loc (RN)).

(iii) If V ∈ C0([0, τ ];C1loc(R

N,RN)), then the map

t )→ Jt : [0, τ ]→ C0loc(R

N)

is differentiable and

dJt

dt= [ div V (t)] Tt Jt ∈ C0

loc(RN). (4.4)

Hence the map t )→ Jt belongs to C1([0, τ ];C0loc(R

N)).

Indeed it is easy to check that

d

dtDTt(X) = DV (t, Tt(X))DTt(X), DT0(X) = I,

d

dtdet DTt(X) = tr DV (t, Tt(X)) det DTt(X)

⇒ d

dtdet DTt(X) = div V (t, Tt(X)) det DTt(X), det DT0(X) = 1,

and (4.4) follows directly by definition of Jt(X).From (4.2), (4.3), and (4.4)

dJ(Ω; V ) =d

dtJ(Ωt(V ))

∣∣∣∣t=0

=∫

Ω∇ϕ · V (0) + ϕ div V (0) dx

⇒ dJ(Ω; V ) =∫

Ωdiv (ϕ V (0)) dx.

If Ω has a Lipschitzian boundary, then by Stokes’s theorem

dJ(Ω; V ) =∫

ΓϕV (0) · n dΓ.

Theorem 4.2. Assume that there exists τ > 0 such that the velocity field V (t)satisfies conditions (V ) and V ∈ C0([0, τ ];C1

loc(RN,RN)). Given a function ϕ ∈


C(0, τ ;W 1,1loc (RN)) ∩ C1(0, τ ;L1

loc(RN)) and a bounded measurable domain Ω with

boundary Γ, the semiderivative of the function

JV (t) def=∫

Ωt(V )ϕ(t) dx (4.5)

at t = 0 is given by

dJV (0) =∫

Ωϕ′(0) + div (ϕ(0) V (0)) dx, (4.6)

where ϕ(0)(x) def= ϕ(0, x) and ϕ′(0)(x) def= ∂ϕ/dt(0, x). If, in addition, Ω is an opendomain with a Lipschitzian boundary Γ, then

dJV (0) =∫

Ωϕ′(0) dx +

∫

Γϕ(0) V (0) · n dx. (4.7)

4.2 Basic Formula for Boundary IntegralsGiven ψ in H2

loc(RN), consider for some bounded open Lipschitzian domain Ω in

RN the shape functional

J(Ω) def=∫

Γψ dΓ. (4.8)

This integral is invariant with respect to a homeomorphism which maps Ω ontoitself (and hence Γ onto itself). Given the velocity field V and t ≥ 0, consider theexpression

J(Ωt(V )) def=∫

Γt(V )ψ dΓt.

Using the change of variables Tt(V ) and the material introduced in (3.13) ofsection 3.2 and (5.30) of section 5 in Chapter 2, this integral can be brought backfrom Γt to Γ:

J(Ωt(V )) def=∫

Γt

ψ dΓt =∫

Γψ Tt ωt dΓ, (4.9)

where the density ωt is given as

ωt = |M(DTt)n|, (4.10)

n is the outward normal field on Γ, and M(DTt) is the cofactor matrix of DTt,that is,

M(DTt) = Jt∗(DTt)−1 ⇒ ωt = Jt | ∗(DTt)−1n|. (4.11)


It can easily be checked from (4.10) and (4.11) that t )→ ωt is differentiable in C0(Γ)and that the limit

ω′ = limt0

1t(ωt − ω) = div V (0)−DV (0)n · n (4.12)

in the C0(Γ)-norm is linear and continuous with respect to V (0) in the C1loc(R

N,RN)Frechet topology. Hence

dJ(Ω; V ) =∫

Γ∇ψ · V (0) + ψ (div V (0)−DV (0)n · n) dΓ. (4.13)

From Corollary 1 to the structure Theorem 3.6, dJ(Ω; V ) depends only on thenormal component vn of the velocity field V (0) on Γ:

vndef= v · n, v

def= V (0)|Γ (4.14)

through Hadamard’s formula (3.40). In view of this property any other velocityfield with the same smoothness and normal component on Γ will yield the samelimit. Given k > 0 consider the tubular neighborhood

Sk(Γ) def=x ∈ RN : |b(x)| < k

(4.15)

of Γ in RN for the oriented distance function b = bΩ associated with Ω. Assumingthat Γ is compact and of class C2, there exists h > 0 such that b ∈ C2(S2h(Γ)). Letϕ ∈ D(RN) be such that ϕ = 1 in Sh(Γ) and ϕ = 0 outside of S2h(Γ). Consider thevelocity field

W (t) def= (V (0) ·∇b)∇bϕ.

Clearly, the normal component of W (0) on Γ coincides with vn. Moreover, in Sh(Γ)

∇ψ · W = ∇ψ ·∇b V (0) ·∇b ⇒ ∇ψ · W |Γ = ∇ψ · n V (0) · n =∂ψ

∂nvn,

DW = V (0) ·∇b D2b +∇b ∗∇(V (0) ·∇b),div W = V (0) ·∇b ∆b +∇b ·∇(V (0) ·∇b),

DW∇b ·∇b =V (0) ·∇b D2b∇b ·∇b +∇(V (0) ·∇b) ·∇b

=∇(V (0) ·∇b) ·∇b,

div W −DW∇b ·∇b = V (0) ·∇b ∆b

⇒ div W −DW∇b ·∇b|Γ = V (0) · n H = H vn

since ∇b|Γ = n, D2b∇b = 0, and H = ∆b is the additive curvature, that is, the sumof the N − 1 curvatures of Γ or N − 1 times the mean curvature H. Finally,

dJ(Ω; V ) =∫

Γ

(∂ψ

∂n+ ψH

)vn dΓ. (4.16)

We have proven the following result.


Theorem 4.3. Let Γ be the boundary of a bounded open subset Ω of RN of class C2

and ψ be an element of C1([0, τ ];H2loc(R

N)). Assume that V ∈ C0([0, τ ];C1loc(R

N,RN)). Consider the function

JV (t) def=∫

Γt(V )ψ(t) dΓt.

Then the derivative of JV (t) with respect to t in t = 0 is given by the expression

dJV (0) =∫

Γψ′(0) +

(∂ψ

∂n+ Hψ

)V (0) · n dΓ

=∫

Γψ′(0) +∇ψ · V (0) + ψ (div V (0)−DV (0)n · n) dΓ,

(4.17)

where ψ′(0)(x) def= ∂ψ/∂t(0, x).

Note that, as in the case of the volume integral, we have two formulae. Hencewe have the following identity:

∫

Γ

(∂ψ

∂n+ Hψ

)V (0) · n dΓ

=∫

Γ∇ψ · V (0) + ψ (div V (0)−DV (0)n · n) dΓ.

(4.18)

4.3 Examples of Shape Derivatives4.3.1 Volume of Ω and Surface Area of Γ

Consider the volume shape functional

J(Ω) =∫

Ωdx.

This shape functional is used as a constraint on the domain in several examples ofshape optimization problems. We get

dJ(Ω; V ) =∫

Ωdiv V (0) dx (4.19)

and, if Γ is Lipschitzian,

dJ(Ω; V ) =∫

ΓV (0) · n dΓ. (4.20)

A sufficient condition on the field V (0) to preserve the volume is div V (0) = 0 in Ωand, if Γ is Lipschitzian, V (0) · n = 0 on Γ.

Consider the (shape) area function

J(Ω) =∫

ΓdΓ.


Assuming that Γ is of class C2 we get from (4.17)

dJ(Ω; V ) =∫

ΓH V (0) · n dΓ, (4.21)

where H = ∆bΩ is the additive curvature. The condition to keep the surface of Γconstant is that V (0) · n be orthogonal (in L2(Γ)) to H.

4.3.2 H1(Ω)-Norm

Given φ and ψ in H2loc(R

N), consider the shape functional

J(Ω) =∫

Ω∇φ ·∇ψ dx.

By using the change of variables Tt(V ), Ωt = Ωt(V ) = Tt(V )(Ω) and∫

Ωt

∇φ ·∇ψ dx =∫

Ω[A(V )(t)∇(φ Tt)] ·∇(ψ Tt) dx, (4.22)

where A(V ) is the following matrix associated with the field V :

A(V )(t) = J(t) (DTt)−1 ∗(DTt)−1 (4.23)

and J(t) = det(DTt). Expression (4.23) is easily obtained from the identity

(∇φ) Tt = ∗(DTt)−1∇(φ Tt). (4.24)

If V ∈ C0([0, τ ];Ckloc(R

N;RN)), k ≥ 1, T and T−1 belong to C1([0, τ ];Ckloc(R

N,RN))and A(V ) to C1([0, τ ], Ck−1

loc (RN;RN2)). Then if φ,ψ belongs to H2

loc(RN), we get

t )→ φ Tt, which is differentiable in H1loc(R

N), with ∂φ/∂t Tt|t=0 = ∇φ · V (0),which belongs to H1

loc(RN). Finally, we obtain

dJ(Ω; V ) =∫

Ω[A′(V )∇φ] ·∇ψ dx

+∫

Ω∇(∇φ · V (0)) ·∇ψ +∇φ ·∇(∇ψ · V (0)) dx,

(4.25)

where A′(V ) is the derivative in the Ck−1loc (RN,RN2

)-norm

A′(V ) def=∂

∂tA(V )(t)|t=0 = div V (0) I − 2ε(V (0)) (4.26)

and ε(V (0)) is the symmetrized Jacobian matrix (the strain tensor associated withthe field V (0) in elasticity)

ε(V (0)) =12[∗DV (0) + DV (0)]. (4.27)

We finally obtain the volume expression

dJ(Ω; V ) =∫

Ω[div V (0) I − 2ε(V (0))]∇φ ·∇ψ+ [∇(∇φ · V (0)) ·∇ψ +∇φ ·∇(∇ψ · V (0))] dx.

(4.28)


When Γ is a C1-submanifold, φ,ψ ∈ H2loc(R

N), we obtain the simpler boundaryexpression by directly using formula (4.17):

dJ(Ω; V ) =∫

Γ∇φ ·∇ψ V (0) · n dΓ. (4.29)

This simple example nicely illustrates the notion of density gradient g. From ex-pression (4.28) it was obvious that the mapping V )→ dJ(Ω; V ) was well-defined,linear, and continuous on C1([0, τ ];C1

loc(RN;RN)). By the structure theorem and

Hadamard’s formula we knew that dJ could be written in the form∫

Γg V (0) · n dΓ.

But from (4.29) we know that g, which is an element of D1(Γ)′, is an element ofW 1/2,1(Γ) given by

g = ∇φ ·∇ψ (traces on Γ). (4.30)

The direct calculation of g from expression (4.24) would have been very fastidious.

4.3.3 Normal Derivative

Let Γ be of class C2 and φ ∈ H2loc(R

N) be given. Consider the following shapefunction:

J(Ω) def=∫

Γ

∣∣∣∣∂φ

∂n

∣∣∣∣2

dΓ =∫

Γ|∇φ · n|2 dΓ.

By the change of variables formula we get, with Ωt = Tt(V )(Ω) and Γt = Tt(V )(Γ),

J(Ωt)def=∫

Γt

|∇φ · nt|2 dΓt =∫

Γ

∗(DTt)−1∇(φ Tt) · (nt Tt)2ωt dΓ, (4.31)

where nt Tt is the transported normal field nt from Γt onto Γ. The derivativecan be obtained by using formula (4.17) of Theorem 4.3 and one of the abovetwo expressions. However, the first expression first requires the construction of anextension Nt of the normal nt in a neighborhood of Γ. In both cases the followingresult will be useful.

Theorem 4.4. Let k ≥ 1 be an integer. Given a velocity field V (t) satisfyingcondition (V ) such that V ∈ C([0, τ ];Ck

loc(RN,RN)), then

nt Tt =∗(DTt)−1n

| ∗(DTt)−1n| =M(DTt)n|M(DTt)n| , (4.32)

where n and nt are the respective outward normals to Ω and Ωt on Γ and Γt andM(DTt) is the cofactor’s matrix of DTt.


Proof. Go back to Definition 3.1 in section 3.1 of Chapter 2. We have shown inthat section that for a domain Ω of class Ck the unit outward normal at any pointy ∈ Γx = Γ ∩ U(x) is given by expression (3.9)

∀y ∈ Γx, n(y) = −∗(Dhx)−1 eN

| ∗(Dhx)−1 eN | (h−1x (y)),

where hx is the local diffeomorphism specified by (3.3):

gx ∈ Ck(U(x), B

), hx = g−1

x ∈ Ck(B,U(x)

).

For Ωt = Ωt(V ) and xtdef= Tt(x), choose the following new neighborhood and local

diffeomorphism:

Utdef= Tt(U(x)), ht

def= Tt hx : B → Ut, gt = h−1t

def= gx T−1t : Ut → B.

On Γt(V )∩Ut the normal in given by the same expression but with ht in place of hx:

nt = −∗(Dht)−1 h−1

t eN

| ∗(Dht)−1 h−1t eN |

.

However,

D(Tt hx) = DTt hx Dhx,

D(Tt hx) (Tt hx)−1 =[DTt Dhx h−1

x

] T−1

t ,∗(DTt hx)−1 (Tt hx)−1eN =

[ ∗(DTt)−1 ∗(Dhx)−1 h−1x eN

] T−1

t .

Therefore, by using the previous expressions for n and nt,

nt =∗(DTt)−1n

| ∗(DTt)−1n| T−1t ⇒ nt Tt =

∗(DTt)−1n

| ∗(DTt)−1n| . (4.33)

The other expression for nt Tt in terms of the cofactor matrix M(DTt) of DTt

readily follows from the identity M(DTt) = J(t) ∗(DTt)−1.

Recalling the expression for ωt

ωt = |M(DTt)n| = J(t) | ∗(DTt)−1n|,

our boundary integral becomes

∫

Γt

|∇φ · nt|2 dΓt =∫

Γ|[A(t)∇(φ Tt)] · n |2 ω−1

t dΓ. (4.34)


Using expression (4.26) of A′(V ) and expression (4.12) of ω′ we get

dJ(Ω; V )

= 2∫

Γ

∂φ

∂n[A′(V )∇φ · n +∇(∇φ · V (0)) · n]−

∣∣∣∣∂φ

∂n

∣∣∣∣2

ω′ dΓ

= 2∫

Γ

∂φ

∂n [div V (0) I −DV (0)− ∗DV (0)] ∇φ · n +∇(∇φ · V (0)) · n

−∣∣∣∣∂φ

∂n

∣∣∣∣2

(div V (0)−DV (0)n · n) dΓ

=∫

Γ2∂φ

∂n

div V (0)

∂φ

∂n−DV (0)∇φ · n + D2φV (0) · n

−∣∣∣∣∂φ

∂n

∣∣∣∣2

(div V (0)−DV (0)n · n) dΓ

=∫

Γ

∣∣∣∣∂φ

∂n

∣∣∣∣2

(div V (0)−DV (0)n · n)

+ 2∂φ

∂n

DV (0)n · n

∂φ

∂n−DV (0)∇φ · n + D2φV (0) · n

dΓ.

This formula can be somewhat simplified by using identity (4.18) with

ψ = |∇φ ·∇b|2 ⇒ ψ|Γ =∣∣∣∣∂φ

∂n

∣∣∣∣2

,

∇ψ = 2∇φ ·∇b ∇(∇φ ·∇b) = 2∇φ ·∇b[D2φ∇b + D2b∇ψ

]

⇒ ∇ψ|Γ = 2∂φ

∂n

[D2φn + D2b∇φ

]

⇒ ∇ψ · V (0)|Γ = 2∂φ

∂n

[D2φn + D2b∇φ

]· V (0)

⇒ ∂ψ

∂n= ∇ψ ·∇b|Γ = 2

∂φ

∂n

[D2φn + D2b∇φ

]·∇b = 2

∂φ

∂nD2φn · n.

We obtain∫

Γ

(2∂φ

∂nD2φn · n + H

∣∣∣∣∂φ

∂n

∣∣∣∣2)

V (0) · n dΓ

=∫

Γ2∂φ

∂n

[D2φn + D2b∇φ

]· V (0) +

∣∣∣∣∂φ

∂n

∣∣∣∣2

(div V (0)−DV (0)n · n) dΓ,

(4.35)

and hence

dJ(Ω; V )

=∫

Γ

(2∂φ

∂nD2φn · n + H

∣∣∣∣∂φ

∂n

∣∣∣∣2)

V (0) · n

+ 2∂φ

∂n

∂φ

∂nDV (0)n · n−∇φ ·

( ∗DV (0)n + D2b V (0))

dΓ.

(4.36)

5. Elements of Tangential Calculus 491

This formula can be more readily obtained from the first expression (4.31) and theextension

Nt =∗(DTt)−1∇b

| ∗(DTt)−1∇b| T−1t (4.37)

of the normal nt. To compute N ′, decompose Nt as follows:

Nt = f(t) T−1t , f(t) =

g(t)√g(t) · g(t)

, g(t) = ∗(DTt)−1∇b,

N ′ = f ′ −Df(0)V (0), g′ = − ∗DV (0)∇b, f(0) = ∇b,

f ′ =g′ |g(0)|− g(0) · g′ g(0)/|g(0)|

|g(0)|2

= g′ − g′ ·∇b∇b = DV (0)∇b ·∇b∇b− ∗DV (0)∇b.

So finally

N ′|Γ = (DV (0)n · n) n− ∗DV (0)n−D2b V (0) (4.38)

and∂

dt|∇φ · Nt|2

∣∣∣∣t=0

= 2∂φ

∂n∇φ · N ′

= 2∂φ

∂n∇φ ·

(DV (0)n · n)n− ∗DV (0)n−D2b V (0)

.

The first part of the integral (4.36) depends explicitly on the normal component ofV (0). Yet we know from the structure theorem that for this function, the shapederivative depends only on the normal component of V (0). To make this explicit,it is necessary to introduce some elements of tangential calculus.

5 Elements of Tangential CalculusIn this section some basic elements of the differential calculus on a C2-submanifoldof codimension 1 are introduced. The approach avoids local bases and coordinatesby using the intrinsic tangential derivatives.

In the classical theory of shells (see, for instance, Ph. D. Ciarlet [1]), themidsurface ω is defined as the image of a flat smooth bounded connected domainU in R2 via a C2-immersion ϕ : U → R3. When U is sufficiently smooth and thethickness sufficiently small, the associated tubular neighborhood Sh(ω) of thickness2h is a Lipschitzian domain that is identified with a thin shell of thickness 2h aroundω. Local bases are generated by the derivatives of the immersion ϕ and tangentialderivatives can be defined in the surface ω.

Such hypersurfaces ω in RN can be viewed as a subset of the boundary Γ of anopen subset Ω of RN. In that case, the gradient and Hessian matrix of the associ-ated oriented distance function bΩ to the underlying set Ω completely describes thenormal and the N fundamental forms of ω, and a fairly complete intrinsic theory

Sh(ω)


of Sobolev spaces on C1,1-hypersurfaces is available in M. C. Delfour [7]. M. C.Delfour [8] showed that all smooth hypersurfaces and the hypersurfaces with littleregularity of S. Anicic, H. Le Dret, and A. Raoult [1] are C1,1 hypersurfacesthat can be described by the oriented distance function bΩ or the signed distancefunction bω. Such an approach has been used successfully in the theory of thin andasymptotic shells with a C1,1-midsurface.

Let Ω be an open domain of class C2 in RN with compact boundary Γ. There-fore, there exists h > 0 such that b = bΩ ∈ C2(S2h(Γ)). The projection of a point xonto Γ is given by

p(x) def= x− b(x)∇b(x),

and the orthogonal projection operator of a vector onto the tangent plane Tp(x)Γ isgiven by

P (x) def= I −∇b(x) ∗∇b(x).

Notice that, as a transformation of Tp(x)Γ,

P (x) : Tp(x)Γ→ Tp(x)Γ, P (x) = I −∇b(x) ∗∇b(x)

is the identity transformation on Tp(x)Γ. In fact P (x) coincides with the first fun-damental form. Similarly D2b can be considered as a transformation of Tp(x) sinceD2b(x)n(x) = D2b(x)∇b(x) = 0. We shall show in section 5.6 that

D2b(x) : Tp(x) → Tp(x)

coincides with the second fundamental form of Γ. Similarly D2b(x)2 coincides withthe third fundamental form. Finally

Dp(x) = I −∇b(x) ∗∇b(x)− b D2b(x) and Dp|Γ = P.

5.1 Intrinsic Definition of the Tangential GradientThe classical way to define the tangential gradient of a scalar function f : Γ→ Ris through an appropriately smooth extension F of f in a neighborhood of Γ us-ing the fact that the resulting expression on Γ is independent of the choice of theextension F . In this section an equivalent direct intrinsic definition is given interms of the extension f p of the function f . This is the basis of a simple differ-ential calculus on Γ which uses the Euclidean differential calculus in the ambientneighborhood of Γ.

Given f ∈ C1(Γ), let F ∈ C1(S2h(Γ)) be a C1-extension of f . Define

g(F ) def= ∇F |Γ −∂F

∂nn on Γ.

This is the orthogonal projection P (x)∇F (x) of ∇F (x) onto the tangent plane TxΓto Γ at x. To get something intrinsic, g(F ) must be independent of the choice ofF . It is sufficient to show that g(F ) = 0 for f = 0. But F = f = 0 on Γ and thetangential component of ∇F is 0 on Γ, and

∇F |Γ =∂F

∂nn ⇒ g(F ) = ∇F |Γ −

∂F

∂nn = 0.

first fundamental form.


Definition 5.1 (General extension).Assume that Γ is compact and that there exists h > 0 such that bΩ ∈ C2(S2h(Γ)).Given an extension F ∈ C1(S2h(Γ)) of f ∈ C1(Γ), the tangential gradient of f in apoint of Γ is defined as

∇Γfdef= ∇F |Γ −

∂F

∂nn.

The notation is quite natural. The subscript Γ of ∇Γf indicates that thegradient is with respect to the variable x in the submanifold Γ.

Theorem 5.1. Assume that Γ is compact and that there exists h > 0 such thatbΩ ∈ C2(S2h(Γ)) and that f ∈ C1(Γ). Then

(i) ∇Γf = (P ∇F )|Γ and n ·∇Γf = ∇b ·∇Γf = 0;

(ii) ∇(f p) = [I − b D2b]∇Γf p and ∇(f p)|Γ = ∇Γf .

Proof. (i) By definition

P ∇F = (I −∇b ∗∇b)∇F = ∇F −∇F ·∇b∇b,

(P ∇F )|Γ = (∇F )|Γ −∂F

∂nn = ∇Γf.

Moreover

∇b · P ∇F = (I −∇b ∗∇b)∇b ·∇F = 0⇒ (∇b)|Γ · (P ∇F )|Γ = 0 ⇒ n ·∇Γf = 0.

(ii) F = f p is a C1-extension of f and

∇(f p) = ∇(f p p) = Dp∇(f p) p

= [I −∇b ∗∇b− b D2b]∇(f p)|Γ p.

But by definition of ∇Γf

∇(f p)|Γ = ∇Γf +∂(f p)∂n

n

⇒ ∇(f p) = [I −∇b ∗∇b− b D2b](∇Γf +

∂(f p)∂n

n

) p.

Recall, from Theorem 8.4 (i) in Chapter 7, that n p = ∇b so that

∇(f p) = [I −∇b ∗∇b− b D2b](∇Γf p +

∂(f p)∂n

∇b

)

= [I −∇b ∗∇b− b D2b]∇Γf p.

Again∇b ·∇Γf p = n p ·∇Γf p = (n ·∇Γf) p = 0

from part (i) and

∇(f p) = [I − bD2b]∇Γf p ⇒ ∇(f p)|Γ = ∇Γf.

tangential gradient

∇(f p)|Γ = ∇Γf.

def ∂F ∇Γf = ∇F|Γ − ∂n n.


In view of part (ii) of the theorem f p plays the role of a canonical extensionof a map f : Γ → R to a neighborhood S2h(Γ) of Γ and its gradient is tangentto the level sets of b. This suggests the use of the following definition of tangentialgradient, which will be the clue to the tangential differential calculus.

Definition 5.2 (Canonical extension).Under the assumptions of Definition 5.1 on bΩ, associate with f ∈ C1(Γ)

∇Γfdef= ∇(f p)|Γ . (5.1)

Remark 5.1.This definition naturally extends to nonempty sets A such that d2

A belongs toC1,1(S2h(A)), since the projection onto A,

pA(x) = x− 12∇d2

A(x),

is C0,1. They are the sets of positive reach introduced by Federer. They includeconvex sets and submanifolds of codimension larger than or equal to 1.

Theorem 5.2. Under the assumption of Definition 5.1 on bΩ for f ∈ C1(Γ) thefollowing hold:

(i) ∇b ·∇(f p) = 0 in Sh(Γ) and n ·∇Γf = 0 on Γ.

(ii) ∇F |Γ − ∂F∂n n = (P ∇F )|Γ = ∇Γf and ∇(f p) = [I − b D2b]∇Γf p in Γ.

Proof. (i) Consider

∇(f p) ·∇b = ∇(f p p) ·∇b = Dp∇(f p) p ·∇b

= ∇(f p) p · Dp∇b.

ButDp∇b = [I −∇b ∗∇b− b D2b]∇b = 0

and

∇(f p) ·∇b = 0, ∇Γf · n = ∇(f p)|Γ ·∇b|Γ = 0.

(ii) By definitionP ∇F = (I −∇b ∗∇b)∇F.

However, since F p = f p on Γ

∇(F p) = ∇(F p p) = Dp∇(F p) p = Dp∇(f p) p = Dp∇Γf p

and

∇(F p) = Dp∇F p ⇒ Dp∇F p = Dp∇Γf p.

By restricting to Γ

Dp|Γ∇F |Γ = Dp|Γ∇Γf ⇒ P ∇F |Γ = P ∇Γf = ∇Γf


5.2 First-Order DerivativesThe tangential Jacobian matrix of a vector function v ∈ C1(Γ)M , M ≥ 1, is definedin the same way as the gradient:

DΓvdef= D(v p)|Γ or (DΓv)ij = (∇Γvi)j . (5.2)

If ∗v = (v1, . . . , vM ), then∗DΓv = (∇Γv1, . . . ,∇ΓvM ),

where ∇Γvi is a column vector. From the previous theorems about the tangentialgradient we can recover the definition from an extension V ∈ C1(S2h(Γ))M of v:

∗DΓv = (P ∇V1, . . . , P ∇VM )|Γ = (I −∇b ∗∇b) ∗DV |Γ= ∗DV |Γ −∇b ∗(DV ∇b)|Γ

and

DΓv = DV |Γ −DV n ∗n = (DV P )|Γ. (5.3)

Also,∗D(v p) = [I − b D2b] (∇Γv1, . . . ,∇ΓvM ) p = [I − b D2b] ∗DΓv p,

and we have for the extension

D(v p) = DΓv p [I − b D2b]. (5.4)

Note that

D(v p)∇b = 0 and DΓv n = 0. (5.5)

For a vector function v ∈ C1(Γ)N define the tangential divergence as

divΓvdef= div (v p)|Γ, (5.6)

and it is easy to show that

divΓv = div (v p)|Γ = tr D(v p)|Γ = tr DΓv,

divΓv = tr [DV |Γ −DV n ∗n] = div V |Γ −DV n · n.

The tangential linear strain tensor of linear elasticity is given by

εΓ(v) def=12(DΓv + ∗DΓv), (5.7)

ε(v p) =12(D(v p) + ∗D(v p))

= εΓ(v) p− b

2[DΓv p D2b + D2b ∗DΓv p]

(5.8)

⇒ εΓ(v) = ε(v p)|Γ. (5.9)


The tangential Jacobian matrix of the normal n is especially interesting since,from Theorem 8.4 (i) in Chapter 7, n p = ∇b = ∇b p. As a result

DΓ(n) = D2b|Γ = ∗DΓ(n) ⇒ εΓ(n) = D2b|Γ. (5.10)

The tangential vectorial divergence of a matrix or tensor function A is defined as

( ˜div Γ A)idef= divΓ (Ai,·) . (5.11)

5.3 Second-Order Derivatives

Assume that Ω is of class C3. The simplest second-order derivative is the Laplace–Beltrami operator of a function f ∈ C2(Γ), which is defined as

∆Γfdef= div Γ(∇Γf). (5.12)

Recall from Theorem 5.2 the identity

∇(f p) = [I − b D2b]∇Γf p.

Then

div (∇Γf p) = div (∇(f p)) + div (b D2b∇Γf p),

div (∇Γf p) = ∆(f p) + b div (D2b∇Γf p) + (D2b∇Γf p) ·∇b,

div (∇Γf p) = ∆(f p) + b div (D2b∇Γf p),

and by taking restrictions to Γ,

∆Γf = ∆(f p)|Γ.

The tangential Hessian matrix of second-order derivatives is defined as

D2Γf

def= DΓ(∇Γf). (5.13)

Here the curvatures of the submanifold begin to appear. The Hessian matrix is notsymmetrical and does not coincide with the restriction of the Hessian matrix of thecanonical extension. Specifically,

D2(f p) = D(∇(f p)) = D([I − b D2b]∇Γf p)

= D(∇Γf p)−D(b D2b∇Γf p)

= D(∇Γf p)− b D(D2b∇Γf p)−D2b∇Γf p ∗∇b,

D2(f p)|Γ = DΓ(∇Γf)−D2b∇Γf ∗∇b = D2Γf −D2b∇Γf ∗n.


Of course, since D2(f p) is symmetrical we also have

D2(f p) = ∗D(∇(f p)) = ∗D([I − b D2b]∇Γf p)

= ∗D(∇Γf p)− ∗D(b D2b∇Γf p)

= ∗D(∇Γf p)− b ∗D(D2b∇Γf p)−∇b ∗(D2b∇Γf p),

D2(f p)|Γ = ∗DΓ(∇Γf)−∇b ∗(D2b∇Γf) = ∗D2Γf − n ∗(D2b∇Γf).

As a final result we have the following identity:

D2Γf − (D2b∇Γf) ∗n = D2(f p)|Γ = ∗D2

Γf − n ∗(D2b∇Γf), (5.14)

since by definition ∗D2Γf = ∗ (D2

Γf). So the Hessian and its transpose differ by termsthat contain a first-order derivative as is well known in differential geometry. Notethat D2b∇Γf = −∗DΓ(∇Γf)n = −∗D2

Γf n and that we also can write

D2Γf + ∗D2

Γf [n ∗n] = D2(f p)|Γ = ∗D2Γf + [n ∗n] D2

Γf

⇒ PD2Γf = ∗(PD2

Γf).(5.15)

5.4 A Few Useful Formulae and the Chain RuleAssociate with F ∈ C1(S2h(Γ)) and V ∈ C1(S2h(Γ))N

fdef= F |Γ, v

def= V |Γ, vndef= v · n, vΓ

def= v − vn n, (5.16)

where vΓ and vn are the respective tangential part and the normal component of v.In view of the previous definitions the following identities are easy to check:

∇F |Γ = ∇Γf +∂F

∂nn, (5.17)

DV |Γ = DΓv + DV n ∗n, (5.18)div V |Γ = divΓ v + DV n · n, (5.19)

DΓv n = 0, D2b n = 0. (5.20)

Decomposing v into its tangential part and its normal component,

DΓv = DΓvΓ + vnD2b + n ∗∇Γvn, (5.21)divΓ v = divΓ vΓ + ∆b vn = divΓ vΓ + H vn, (5.22)

∇Γvn = ∗DΓv n + D2b vΓ. (5.23)

Given f ∈ C1(Γ) and g ∈ C1(Γ; Γ), consider the canonical extensions f p ∈C1(S2h(Γ)) and g p ∈ C1(S2h(Γ);RN) and the gradient of the composition

∇(f p g p) = ∗D(g p)∇(f p) g p

⇒ ∇Γ(f g) = ∗DΓg∇Γf g, (5.24)

and for a vector-valued function v ∈ C1(Γ;RN),

DΓ(v g) = DΓv g DΓg. (5.25)


5.5 The Stokes and Green FormulaeOne interesting application of the shape calculus in connection with the tangentialcalculus is the tangential Stokes formula. Given v ∈ C1(Γ)N , consider Stokesformula in RN for the vector function v p:

∫

Ωdiv (v p) dx =

∫

Γv · n dΓ.

Given an autonomous velocity field V , differentiate both sides of Stokes’s formulawith respect to t,

∫

Ωt(V )div (v p) dx =

∫

Γt(V )(v p) · nt =

∫

Γt(V )(v p) · Nt dΓt,

where Nt is the extension (4.37) of nt. This gives the new identity∫

Γdiv (v p) V · n dΓ =

∫

Γv · N ′ +

∂

∂n[(v p) ·∇b] + H v · n

V · n dΓ.

Now choose the velocity field V = ∇bψ, where ψ ∈ D(S2h(Γ)) is chosen in such away that ψ = 1 on Sh(Γ). Using expression (4.38) for N ′,

V · n = n · n = 1, div (v p)|Γ = divΓ v,

N ′|Γ = DV n · n n− ∗DV n−D2b V

= D2b∇b ·∇b∇b−D2b∇b−D2b∇b = 0,

∇((v p) ·∇b) = D2b v p + ∗D(v p)∇b,

∂

∂n[(v p) ·∇b] =

[D2b v + ∗DΓv n

]· n = v · D2b∇b + n · DΓv n = 0.

Finally we get the tangential Stokes formula with H = ∆b:

∫

ΓdivΓ v dΓ =

∫

ΓH v · n dΓ. (5.26)

For a function f ∈ C1(Γ) and a vector v ∈ C1(Γ)N , the above formula alsoyields the tangential Green’s formula

∫

Γf divΓv +∇Γf · v dΓ =

∫

ΓH f v · n dΓ. (5.27)

5.6 Relation between Tangential and Covariant DerivativesOne of the simplest examples of a tangential derivative is when Ω is the half-space

H+def=ζ ∈ RN : ζ · eN > 0


for some orthonormal basis e1, . . . , eN in RN and Γ is the boundary of H+ de-noted by

Hdef=ζ ∈ RN : ζ · eN = 0

.

If pH denotes the projection onto H, then pH(ζ) = pH(ζ ′, ζN ) = ζ ′ def= (ζ1, . . . ,ζN−1) ∈ H, and for any function ϕ ∈ C1(H) the tangential gradient

∇Hϕ = ∇(ϕ pH)|H

coincides with the gradient of ϕ in H:

∇Hϕ · eα =∂ϕ

∂ζα, 1 ≤ α ≤ N − 1.

With the notation of section 3.1 of Chapter 2, consider a set Ω locally of classC2 in RN. Its boundary Γ = ∂Ω is an (N − 1)-dimensional submanifold of RN ofclass C2. At each point x ∈ Γ, there is a C2-diffeomorphism hx from the open unitball B onto a neighborhood U(x) of x such that

hx(B0) = Γxdef= Γ ∩ U(x), hx(B+) = Ω ∩ U(x),

where B+ = H+ ∩B and B0 = H ∩B and H+ and H are as defined above for someappropriate orthonormal basis.

For a function f ∈ C1(Γ), f Φ ∈ C1(B0) with Φ def= hx|B0 : B0 → Γ.Observing that f Φ pH = f p Φ pH , we have

∇(f Φ pH) = ∇(f p Φ pH),∇(f Φ pH) = ∗D(Φ pH)∇(f p) Φ pH ,

∇H(f Φ) = ∗DHΦ∇Γf Φ.

The covariant basis associated with Φ is defined as

aαdef=

∂Φ∂ζα

, 1 ≤ α ≤ N − 1, aNdef=

∗(Dhx)−1eN

| ∗(Dhx)−1eN |

∣∣∣∣B0

,

where from identity (3.9) of section 3.1 in Chapter 2 we know that aN Φ−1 is theinward unit normal to Ω, that is,

n = −aN Φ−1. (5.28)

The covariant partial derivatives of f are defined as

fαdef=

∂(f Φ)∂ζα

, 1 ≤ α ≤ N − 1.


But from the previous observations

aα =∂Φ∂ζα

= DHΦeα, fα =∂(f Φ)∂ζα

= eα ·∇H(f Φ),

f Φ pH = f p Φ pH , and

∇(f Φ pH) = ∇(f p Φ pH) = ∗D(Φ pH)∇(f p) Φ pH ,

∇H(f Φ) = ∗DH(Φ)∇Γf Φ,

fα = eα ·∇H(f Φ) = eα · ∗DHΦ∇Γf Φ =DHΦ eα ·∇Γf Φ= aα ·∇Γf Φ

⇒ fα = aα ·∇Γf Φ = eα ·∇H(f Φ).

For a vector function v ∈ C1(Γ;RN)

v,αdef=

∂(v Φ)∂ζα

, 1 ≤ α ≤ N − 1.

Again, v Φ pH = v p Φ pH and

D(v Φ pH) = D(v p Φ pH) = D(v p) Φ pH D(Φ pH),

DH(v Φ) = DΓv Φ DHΦ ⇒ v,α = DH(v Φ)eα = DΓv Φ aα.

The second fundamental form of Γ is defined from the inward normal aN ,

bαβdef= −aα · aN,β ,

where from (5.28) aN = −n Φ, and from (5.10) DΓn = D2b|Γ,

DHaN = −DΓn Φ DHΦ = −D2b Φ DHΦ,

aN,β = −DHaN eβ = −D2b Φ DHΦ eβ = −D2b Φ aβ ,

bαβ = aα · D2b Φ aβ .

Recall that since |∇b| = 1, D2b∇b = 0 and ∇b is an eigenvector for the eigenvalue 0.As a result the eigenvalues of D2b at a point of Γ are the eigenvalues of the secondfundamental form bαβ (the N − 1 principal curvatures) plus 0. In particular,

∆b = tr D2b = H = (N − 1)H,

where H is the mean curvature of Γ and H the additive curvature (cf. section 3.3of Chapter 2).

6. Second-Order Semiderivative and Shape Hessian 501

5.7 Back to the Example of Section 4.3.3Coming back to formula (4.36) for dJ(Ω; V (0)) of the boundary integral of thesquare of the normal derivative in section 4.3.3, the tangential calculus is now usedon the term

∫

Γ2∂φ

∂n

∂φ

∂nDV (0)n · n−∇φ ·

( ∗DV (0)n + D2b V (0))

dΓ,

Tdef=

∂φ

∂nDV (0)n · n−∇φ ·

( ∗DV (0)n + D2b V (0)).

From identity (5.18) with v = V (0)|Γ and identity (5.23),

∗DV (0)n = ∗DΓv n + DV (0)n · n n,

∇φ · ∗DV (0)n = ∇φ · ∗DΓv n +∂φ

∂nDV (0)n · n

⇒ T = −∇φ ·[ ∗DΓv n + D2b vΓ

]= −∇φ ·∇Γvn = −∇Γφ ·∇Γvn.

Therefore by using the tangential Stokes formula (5.26),∫

Γ2∂φ

∂nT dΓ = −

∫

Γ2∂φ

∂n∇Γφ ·∇Γvn dΓ

=∫

Γ−2 divΓ

∂φ

∂nvn∇Γφ

+ 2 divΓ

∂φ

∂n∇Γφ

vn dΓ

=∫

Γ−2H

∂φ

∂nvn∇Γφ · n + 2 divΓ

∂φ

∂n∇Γφ

vn dΓ

=∫

Γ2 divΓ

∂φ

∂n∇Γφ

vn dΓ =

∫

Γ2 divΓ

∂φ

∂n∇Γφ

V (0) · n dΓ.

Substituting into expression (4.36), we finally get the explicit formula in terms ofV (0) · n as predicted by the structure theorem

dJ(Ω; V )

=∫

Γ

2∂φ

∂nD2φn · n + H

∣∣∣∣∂φ

∂n

∣∣∣∣2

+ 2 divΓ

(∂φ

∂n∇Γφ

)V (0) · n dΓ.

(5.29)

6 Second-Order Semiderivative and Shape HessianThe object of this section is to study second-order derivatives and semiderivatives bythe velocity method for smooth and nonsmooth domains and discuss their relation-ship with the method of perturbations of the identity in the unconstrained case. Theanalysis of this section is motivated by first computing the second-order derivativeof a domain integral by using the combined strengths of the shape and tangentialcalculuses in section 6.1. A basic formula for the second-order semiderivative ofthe domain integral is given in section 6.2, which also reveals the general structure


of this derivative. Structure theorems for the second-order Eulerian semiderivatived2J(Ω; V ;W ) of a function J(Ω) for two vector fields V and W are given in sec-tion 6.3 and section 6.4. A first theorem shows that under some natural continuityassumptions,

d2J(Ω; V ;W ) = d2J(Ω; V (0);W (0)

)+ dJ

(Ω; V ′(0)

),

where V ′(0) is the time-partial derivative ∂tV (t, x) at t = 0. As in the study offirst-order Eulerian semiderivatives, this first theorem reduces the study of second-order Eulerian semiderivatives to the autonomous case. So we then specialize tofields in Dk(D,RN) and give the equivalent of Hadamard’s structure theorem forthe term d2J

(Ω; V (0);W (0)

).

This bilinear term is decomposed into a symmetrical term plus the gradientacting on the first half of the Lie bracket [V (0), W (0)] in section 6.5. The symmet-rical part is itself decomposed into a symmetrical part that depends only on thenormal component of the velocity fields and a symmetrical term made up of thegradient acting on a generic group of terms, which occurs in all examples consideredin this section.

6.1 Second-Order Derivative of the Domain IntegralGiven f ∈ C2

loc(RN) and a domain Ω of class C2, consider the function

J(Ωt,s(V, W )) def=∫

Ωt,s(V,W )f dx, (6.1)

where

Ωt,s(V, W ) def= Ts(W )(Ωt(V )) = Ts(W )(Tt(V )(Ω)),

JV,W (t, s) def= J(Ωt,s(V, W ))

for some pair of autonomous velocity fields V and W satisfying condition (3.1) andadditional smoothness conditions as necessary. The objective is to compute

d2JV,Wdef=

∂

∂s

∂

∂tJV,W (t, s)

∣∣∣∣t=0

∣∣∣∣s=0

.

From formula (4.7) in Theorem 4.2, we already know that

∂

∂tJV,W (t, s)

∣∣∣∣t=0

= dJ(Ωs(W );V ) =∫

Γs(W )f V · ns dΓs =

∫

Γs(W )f V · Ns dΓs,

where Ns = Ns(W ) is the extension (4.37) of the normal ns. So we can readily useformula (4.17) from Theorem 4.3:

d2JV,W =∫

Γf V · N ′(W ) +

∂

∂n(f V ·∇b) + H f V · n

W · n dΓ, (6.2)


where N ′(W ) is the derivative of the extension Ns = Ns(W ) given by expression(4.38). This yields

d2JV,W

=∫

Γf V ·

(DWn · n) n− ∗DW −D2b W

+(∂

∂n(f V ·∇b) + H f V · n

)W · n dΓ

=∫

Γf

V ·(DWn · n) n− ∗DWn−D2b W

+

∂

∂n(V ·∇b) W · n

+(∂f

∂n+ H f

)V · n W · n dΓ.

It remains to untangle the following term in the first part of the integral:

Tdef= V ·

(DWn · n) n− ∗DWn−D2b W

+

∂

∂n(V ·∇b) W · n.

Using the notation v = V |Γ and w = W |Γ,

∇(V ·∇b) = ∗DV∇b + D2b V,

∇(V ·∇b) ·∇b = ∗DV∇b ·∇b + D2b V ·∇b = DV∇b ·∇b

⇒ ∂

∂n(V ·∇b) = DV n · n,

V · ∗DWn = V · [ ∗DΓw + n ∗(DWn)] n

= V ·[∇Γwn −D2b w

]+ DWn · n V · n,

V ·[ ∗DWn + D2b W

]= ∇Γwn · vΓ + DWn · n vn.

Finally,

T = DWn · n vn − (vΓ ·∇Γwn + DWn · n vn) + DV n · n wn

= DV n · n wn − vΓ ·∇Γwn,

and by using the tangential Stokes formula (5.26)∫

Γf T dΓ =

∫

Γf DV n · n wn − vΓ ·∇Γwn dΓ

=∫

Γf DV n · n wn − divΓ (fwmvΓ) + divΓ (fvΓ) wn dΓ

=∫

Γf DV n · n + divΓ (fvΓ) wn − fwn vΓ · n dΓ

=∫

Γf (DV n · n + divΓvΓ) +∇Γf · vΓ wn dΓ.


Finally, we get two equivalent expressions:

d2JV,W

=∫

Γ

(∂f

∂n+ H f

)vn wn + f (DV n · n wn − vΓ ·∇Γwn) dΓ

=∫

Γ

(∂f

∂n+ H f

)vn + f (DV n · n + divΓvΓ) +∇Γf · vΓ

wn dΓ.

(6.3)

Since the above expressions involved the composition Ts(W ) Tt(V ), it isexpected that the condition for the symmetry of expression (6.3) will involve theLie bracket [V, W ] = DV W −DWV . Indeed, by using identity (5.23)

DV W · n = (DΓv + DV n ∗n W · n

= w · ∗DΓv n + DV n · n wn

= wΓ ·(∇Γvn −D2b vΓ

)+ DV n · n wn

and substituting in the first expression, we get a symmetrical term plus the firsthalf of the Lie bracket:

d2JV,W

=∫

Γ

(∂f

∂n+ H f

)vn wn + f

(D2b vΓ · wΓ − vΓ ·∇Γwn − wΓ ·∇Γvn

)

+ f DV W · n dΓ.

(6.4)

Thus

d2V,W = d2

W,V ⇐⇒∫

Γf [V, W ] · n dΓ = 0, (6.5)

from which either f [V, W ] · n = 0 on Γ or div (f [V, W ]) = 0 on Ω can be used assufficient conditions.

Example 6.1.Let Ω = (x, y) : x2 + y2 < 1 be the unit ball in R2 with boundary Γ = (x, y) :x2 + y2 = 1. Choose

V (x, y) = (1, 0), W (x, y) = (x2/2, 0), n = (x, y)/√

x2 + y2

⇒ DV W −DWV = (x, 0) ⇒ [DV W −DWV ] · n = x2/√

x2 + y2 = x2.

6.2 Basic Formula for Domain IntegralsWe have proved the following result.

Theorem 6.1. Let f ∈ C2([0, τ ]× [0, τ ];H2loc(R

N)) and let Γ be the boundary of abounded open subset Ω of RN of class C2. Assume that V ∈ C0([0, τ ];C2

loc(RN,RN))


and that W ∈ C0([0, τ ];C1loc(R

N,RN)). Consider the function

JV,W (t, s) def=∫

Ωt,s(V,W )f(t, s) dx. (6.6)

Then the partial derivative of JV,W (t, s) with respect to t in t = 0 is given by

∂

∂tJV,W (t, s)

∣∣∣∣t=0

=∫

Ωs(W )

∂f

∂t(0, s) + div (f(0, s)V (0)) dx

=∫

Ωs(W )

∂f

∂t(0, s) dx +

∫

Γs(W )f(0, s) V (0) · ns dΓs

(6.7)

and the second-order mixed derivative of JV,W (t, s) in (t, s) = (0, 0),

d2JV,Wdef=

∂

∂s

∂

∂tJV,W (t, s)

∣∣∣∣t=0

∣∣∣∣s=0

, (6.8)

is given by the expression

d2JV,W =∫

Ω

∂

∂s

(∂f

∂t

)+ div

(∂f

∂sV (0) +

∂f

∂tW (0)

)

+ div [div (f V (0)) W (0)] dx

=∫

Ω

∂

∂s

(∂f

∂t

)dx +

∫

Γ

(∂f

∂sV (0) +

∂f

∂tW (0)

)· n

+ div (f V (0)) W (0) · n dΓ.

(6.9)

The last term in the second integral can be expressed in terms of v = V (0)|Γ andw = W (0)|Γ as follows:

∫

Γdiv (f V (0)) W (0) · n dΓ

=∫

Γ

(∂f

∂n+ H f

)vn + f (DV n · n + divΓvΓ) +∇Γf · vΓ

wn dΓ

=∫

Γ

(∂f

∂n+ H f

)vn wn + f

(D2bvΓ · wΓ − vΓ ·∇Γwn − wΓ ·∇Γvn

)

+ f DV W · n dΓ.

(6.10)

6.3 Nonautonomous CaseThe framework introduced in sections 4 and 5 of Chapter 4 has reduced the compu-tation of the Eulerian semiderivative of J(Ω) to the computation of the derivativeof the function

j(t) def= J(Ωt(V )

)(6.11)


for a velocity field V ∈ C([0, τ ];Ckloc(R

N;RN)). In t ≥ 0

j′(t) = dJ(Ωt(V );Vt

)(6.12)

since Ts+t(V ) = Ts(Vt) Tt(V ), where Vt(s)def= V (t + s) and Vt(0) = V (t).

This suggests the following definition.

Definition 6.1.Let J be a real-valued shape functional. Let V and W satisfy conditions (V)(conditions (4.2) of Chapter 4) and assume that for all t ∈ [0, τ ], dJ

(Ωt(W ); Vt

)

exists for Ωt(W ) = Tt(W )(Ω). The function J is said to have a second-orderEulerian semiderivative at Ω in the directions (V, W ) if the following limit exists:

limt0

dJ(Ωt(W );Vt)− dJ(Ω; V )t

. (6.13)

When it exists, it is denoted d2J(Ω; V ;W ).

If, for all t, J has a Hadamard semiderivative at Ωt(W ), recall that

dJ(Ωt(W );Vt) =dHJ(Ωt(W );Vt(0))=dHJ(Ωt(W );V (t)) = dJ(Ωt(W );V (t)),

and the above definition reduces to

d2J(Ω; V ;W ) = limt0

dJ(Ωt(W );V (t)

)− dJ

(Ω; V (0)

)

t.

Remark 6.1.This last definition is compatible with the second-order expansion of j(t) with re-spect to t around t = 0:

j(t) ∼= j(0) + tj′(0) +t2

2j′′(0), (6.14)

where

j(0) = J(Ω), j′(0) = dJ(Ω; V ), j′′(0) = d2J(Ω; V ;V ). (6.15)

The next theorem is the analogue of Theorem 3.2 and provides the canon-ical structure of the second-order Eulerian semiderivative (cf. (3.20) to (3.22) insection 3.1 for the definitions of

−→V m,ℓ and Vℓ).

Theorem 6.2. Let J be a real-valued shape functional, Ω be a subset of RN, andm ≥ 0 and ℓ ≥ 0 be two integers. Assume that

(i) ∀V ∈ −→V m+1,ℓ,∀W ∈ −→V m,ℓ, d2J(Ω; V ;W ) exists;


(ii) ∀W ∈ −→V m,ℓ, ∀t ∈ [0, τ ], J has a shape gradient of order ℓ and is Hadamardsemidifferentiable at Ωt(W );

(iii) ∀U ∈ Vℓ, the map

W )→ d2J(Ω; U ;W ) :−→V m,ℓ → R (6.16)

is continuous.

Then for all V in−→V m+1,ℓ and all W in

−→V m,ℓ,

d2J(Ω; V ;W ) = d2J(Ω; V (0);W (0)

)+ dJ

(Ω; V ′(0)

), (6.17)

where

V ′(0)(x) def= limt0

V (t, x)− V (0, x)t

. (6.18)

Proof. The differential quotient (6.13) can be split into the sum of two terms:

dJ(Ωt(W );V (0)

)− dJ

(Ω; V (0)

)

t+

dJ(Ωt(W );V (t)

)− dJ

(Ωt(W );V (0)

)

t. (6.19)

In view of (i) and (iii), for all U in Vℓ,

d2J(Ω; U ;W ) = d2J(Ω; U ;W (0)

)

by the same argument as in the proof of Theorem 3.2 for the gradient. Hence thefirst term converges to

d2J(Ω; V (0);W

)= d2J

(Ω; V (0);W (0)

).

For the second term recall that V belongs to−→V m+1,ℓ and observe that the vector

field

V (t) =V (t)− V (0)

t

belongs to−→V m,ℓ and that V (0) = V ′(0). Thus by linearity of dJ(Ω; V ), the second

term in (6.19) can be written as

dJ

(Ωt(W );

V (t)− V (0)t

)= dJ

(Ωt(W ); V (t)

),

dJ(Ωt(W ); V (t)

)= t

1t

[dJ(Ωt(W ); V (t)

)− dJ

(Ω; V (0)

)]+ dJ

(Ω; V (0)

).


But for any V in−→V m+2,ℓ, V belongs to

−→V m+1,ℓ. Then by assumption (i),

limt0

dJ(Ωt(W ); V (t)

)− dJ

(Ω; V (0)

)

t= d2J(Ω; V ;W ),

which implies that

limt0

dJ(Ωt(W ); V (t)

)= dJ

(Ω; V (0)

)= dJ

(Ω; V ′(0)

).

Now by assumption (ii), the map U )→ dJ(Ω; U) is linear and continuous onDℓ(RN,RN), and the map

V )→ V ′(0) )→ dJ(Ω; V ′(0)

):−→V m+2,ℓ → Vℓ → R

is linear and continuous (hence uniformly continuous) for the topology−→V m+1,ℓ for

all V in the dense subspace−→V m+2,ℓ. Hence it uniquely and continuously extends

to all elements of−→V m+1,ℓ. This completes the proof of the theorem.

This important theorem gives the canonical structure of the second-orderEulerian semiderivative: a first term that depends on V (0) and W (0) and a sec-ond term that is equal to dJ

(Ω; V ′(0)

). When V is autonomous the second term

disappears and the semiderivative coincides with d2J(Ω; V ;W (0)

), which can be

separately studied for autonomous vector fields in Vℓ.We conclude this section with the explicit computation of the second-order

Eulerian semiderivative for a shape function J(Ω) with respect to two velocity fieldsV and W satisfying the conditions of Theorem 6.2 and such that the shape gradientat any t is of the form

dJ(Ωt(W );V (t)) =∫

Γt(W )g(t) V (t) · nt dΓt (6.20)

for some function g(t) ∈ C(Γt(W )). Further assume that the family of functions g(t)has an extension Q ∈ C1([0, τ ];Ck

loc(N(Γ);RN)) to an open neighborhood N(Γ) ofΓ such that ∪ Γt(W ) : 0 ≤ t ≤ τ ⊂ N(Γ). Therefore using the extension Nt(W )of the normal nt on Γt(W ), it amounts to differentiating the expression

j(t) =∫

Γt(W )Q(t)V (t) · Nt(W ) dΓt.

Apply the first formula (4.17) of Theorem 4.3 to get

j′(0) =∫

Γ(Q′

W (0)V (0) + Q(0)V ′(0)) · n + Q(0)V (0) · N ′(W )

+(∂

∂n(Q(0)V (0) ·∇b) + H Q(0)V (0) · n

)W (0) · n dΓ,

where Q′W (0) depends only on W . But the last three terms have already been

computed in several forms. They constitute expression (6.2) in section 6.1, which


yields (6.3) and (6.4) with f = Q(0), V = V (0), and W = W (0). This yields withthe notation v = V (0)|Γ and w = W (0)|Γ

d2J(V ;W ) =∫

ΓQ′

W (0) vn + Q(0)V ′(0) · n +(∂Q(0)∂n

+ H Q(0))

vn wn

+ Q(0)(D2b vΓ · wΓ − vΓ ·∇Γwn − wΓ ·∇Γvn

)

+ Q(0)DV W · n dΓ

=∫

ΓQ′

W (0) vn + Q(0)V ′(0) · n +(∂Q(0)∂n

+ H Q(0))

vn wn

+ Q(0) (DV n · n + divΓvΓ) +∇ΓQ(0) · vΓ wn dΓ.

(6.21)

Remark 6.2.When V is autonomous the term in V ′(0) disappears. In that case the first half of theLie bracket can be eliminated by restarting the computation with V(t) = V T−1

t (W )in place of V since V(0) = V and

V ′(0) = −DV W ⇒∫

ΓQ(0)V ′(0) · n + Q(0)DV W · n dΓ = 0

⇒ d2J(V, W ) = d2J(V, W ) + dJ(Ω; DV W ).

Remark 6.3.Except for the terms that contain the first half of the Lie bracket DV W and V ′(0),the only term that might not be symmetrical in the first expression (6.21) is the onein Q′

W (0). In fact, according to the second expression and our theorem, Q′W (0) =

Q′W (0) depends only on W (0). Now choose autonomous velocity fields V and W .

Furthermore, assume that W is of the form W = wΓ p. Since W · n = 0 on Γ,dJ(Ωt(W );V (0)) = dJ(Ω; V (0)) and necessarily d2J(Ω; V ;wΓ p) = 0. Therefore,d2J(Ω; V ;W ) depends only on wn and hence Q′

W (0) = Q′wn

(0) and the integral∫

ΓQ′

wn(0) vn dΓ

depends only on wn and vn.

The above expressions give valuable information on the structure of the second-order Eulerian derivative. Other expressions can also be obtained. For instance, ifj(t) is transformed into the volume integral

j(t) = dJ(Ωt(W );V (t)) =∫

Γt(W )Q(t)V (t) · nt dΓt =

∫

Ωt(W )div (Q(t)V (t)) dx,

we get from formula (4.6) in Theorem 4.2 the equivalent volume expression

d2J(Ω; V ;W )

=∫

Ωdiv

Q′

W (0)(0)V (0) + Q(0)V ′(0) + div (Q(0)V (0)) W (0)

dx,

which can obviously be transformed into a boundary expression.


6.4 Autonomous CaseDefinition 6.2.Let J be a real-valued shape functional. Let Ω be a subset of RN.

(i) The function J(Ω) is said to be twice shape differentiable at Ω if

∀V,∀W ∈ D(RN,RN), d2J(Ω; V ;W ) exists (6.22)

and the map

(V, W ) )→ d2J(Ω; V ;W ) : D(RN,RN)×D(RN,RN)→ R (6.23)

is bilinear and continuous. We denote by h the map (6.23).

(ii) Denote by H(Ω) the vector distribution in (D(RN,RN)⊗D(RN,RN))′ asso-ciated with h:

d2J(Ω; V ;W ) = ⟨H(Ω), V ⊗W ⟩ = h(V, W ), (6.24)

where V ⊗W is the tensor product of V and W defined as

(V ⊗W )ij(x, y) = Vi(x)Wj(y), 1 ≤ i, j ≤ N, (6.25)

and Vi(x) (resp., Wj(y)) is the ith (resp., jth) component of the vector V(resp., W ) (cf. L. Schwartz’s [1] kernel theorem and M. Gelfand andN. Y. Vilenkin [1]). H(Ω) will be called the shape Hessian of J at Ω.

(iii) When there exists a finite integer ℓ ≥ 0 such that H(Ω) is continuous for theDℓ(RN,RN)⊗Dℓ(RN,RN)-topology, we say that H(Ω) is of order ℓ.

In what follows, the compact notation Dℓ will be used in place of Dℓ(RN,RN).

Theorem 6.3. Let J be a real-valued shape functional and Ω be a subset of RN

with boundary Γ. Assume that J is twice shape differentiable.

(i) The vector distribution H(Ω) has support in Γ× Γ.

(ii) If Ω is an open or closed domain in RN and H(Ω) is of order ℓ ≥ 0, thenthere exists a continuous bilinear form

[h] :(Dℓ/Dℓ

Γ)×(Dℓ/LℓΩ

)→ R (6.26)

such that for all [V ] in Dℓ/DℓΓ and [W ] in Dℓ/LℓΩ,

d2J(Ω; V ;W ) = [h](qD(V ), qL(W )

), (6.27)

where qD : Dℓ → Dℓ/DℓΓ and qL : Dℓ → Dℓ/LℓΩ are the canonical quotient

surjections and

DℓΓ =

V ∈ Dℓ

(RN,RN) : ∂αV = 0 on Γ, ∀α, |α| ≤ ℓ

. (6.28)


Proof. (i) It is sufficient to prove the following two properties:(a) For all V, W ∈ D such that W = 0 in a neighborhood of Γ, d2J(Ω;

V ;W ) = 0.(b) For all V, W ∈ D such that V = 0 in a neighborhood of Γ, d2J(Ω;

V ;W ) = 0.In case (a) the proof is similar to the one in Theorem 3.6 for the gradient and

we prove the stronger result that for W such that W = 0 on Γ,

Ωt(W ) = Ω,∀t ≥ 0 =⇒ dJ(Ωt(W );V

)= dJ(Ω; V )

=⇒ d2J(Ω; V ;W ) = 0.

In case (b) V = 0 in a neighborhood N of Γ and in !K, the complement of thecompact support K of V . So U = !K is a neighborhood of Γ where V = 0. Byconstruction U ∩ K = ∅ and there exists a bounded neighborhood U of K suchthat U ∩ Γ = ∅. Since U is compact and Γ is closed, the minimum distance d fromU to Γ is finite and nonzero. Let

N(Γ) =y ∈ RN : dΓ(y) < d/2

,

wheredΓ(y) = inf|y − x| : x ∈ Γ.

For all X in Γ

Tt(X)−X =∫ t

0W(Ts(X)

)ds = tW (X) +

∫ t

0

[W(Ts(X)

)−W (X)

]ds,

and by condition (3.1) on W ,

|Tt(X)−X| ≤ t|W (X)| + ctmax[0,t]

|Ts(X)−X|,

and it can easily be shown that for t < 1/c

max[0,t]

|Ts(X)−X| <t

1− ct|W (X)|.

ThussupX∈Γ

max[0,t]

|Ts(X)−X| ≤ t

1− ctsupX∈Γ

|W (X)|.

But W is continuous with compact support. Therefore,

supX∈Γ

|W (X)| ≤ supX∈supp W

|W (X)| = ∥W∥C(RN;RN) <∞

and there exists τ > 0 such that

∀s ∈ [0, τ ],s

1− cs∥W∥C <

d

2.

By definition and the previous inequalities

dΓ(Ts(X)

)= inf

Y ∈Γ|Ts(X)− Y | ≤ |Ts(X)−X| <

d

2


for all s in [0, τ ] and all X ∈ Γ. This implies that

∀s ∈ [0, τ ],∀X ∈ Γ, Γs(W ) = Ts(W )(Γ) ⊂ N(Γ).

By construction, V = 0 in N(Γ) since the distance from K to Γ is greater than orequal to d. Therefore,

∀s ∈ [0, τ ], V ∈ L∞Ωs(W )

and as in the proof of Theorem 3.6, dJ(Ωs(W );V ) = 0 and necessarily d2J(Ω;V ;W ) = 0.

(ii) We have already established in (i) that the bilinear form

(V, W ) )→ h(V, W ) : D ×D → R

is zero for all V ∈ D and W ∈ D such that W = 0 on Γ and also zero for all W ∈ Dand V ∈ D for which V = 0 in a neighborhood of Γ. By density all this is stilltrue in Dℓ, and now by the same argument as in the proof of Theorem 3.6 for allV in Dℓ,

[W ] )→ h(V, W ) : Dℓ/Lℓ → R

is well-defined, linear, and continuous. For the first component it is necessary toshow that for all W in

DℓΓ =

V ∈ Dℓ(RN,RN) : ∂αV = 0 on Γ, ∀α, |α| ≤ ℓ

the bilinear form h(V, W ) = 0. We first prove the result for the subspace

A = D(Ω;RN)⊕D(!Ω;RN).

Then by density and continuity the result holds for the Dℓ(RN,RN)-closure A ofA. Finally, we prove that A = Dℓ

Γ. For any V in A, there exist V1 ∈ D(Ω;RN) andV2 ∈ D(!Ω;RN) such that V = V1 + V2. Moreover,

K1 = suppV1 ⊂ Ω and K2 = suppV2 ⊂ !Ω

are compact subsets of the open sets Ω and !Ω, respectively. Hence V1 = 0 (resp.,V2 = 0) in the open neighborhood !K1 (resp., !K2) of Γ and necessarily V =V1 + V2 = 0 in the neighborhood U = !(K1 ∪ K2) of Γ. Hence from part (i)h(V, W ) = 0. By definition of Dℓ

Γ, DℓΓ ⊂ Dℓ(Ω;RN)⊕Dℓ(!Ω;RN). Now A ⊂ Dℓ

Γ,

A = D(Ω;RN)⊕D(!Ω;RN),

and

D(Ω;RN) = Dℓ(Ω;RN), D(!Ω;RN) = Dℓ(!Ω;RN).

By construction each V in A is of the form V = V1 + V2 for

V1 ∈ Dℓ(Ω;RN), K1 = suppV1 compact in Ω,

∀|α| ≤ ℓ, ∂αV1 = 0 on Γ,

V2 ∈ Dℓ(!Ω;RN), K2 = suppV2 compact in !Ω,

∀|α| ≤ ℓ, ∂αV2 = 0 on Γ.


HencesuppV = K1 ∪K2 compact in

[Ω]∪[RN \Ω

]= RN

and V ∈ Dℓ(RN,RN). Moreover,

∀α, |α| ≤ ℓ, ∂αV = ∂αV1 + ∂αV2 = 0 on Γ.

This proves that A ⊂ DℓΓ and hence A = Dℓ

Γ. To complete the proof notice that bycontinuity of V )→ h(V, W ), for all W in Dℓ the map

[V ] )→ h(V, W, ) : Dℓ/DℓΓ → R

is well-defined, linear, and continuous. Finally the map([V ], [W ]

))→ h(V, , W ) : (Dℓ/Lℓ)× (Dℓ/Dℓ

Γ)→ R

is well-defined, bilinear, and continuous.

The next and last result is the extension of the structure Theorem 3.6 tosecond-order Eulerian semiderivatives. We need the result established in the corol-lary to Theorem 3.6. For a domain Ω with a boundary Γ which is Cℓ+1, ℓ ≥ 0,the map

qL(W ) )→ pL

(qL(W )

)= γΓ(W ) · n : Dℓ

Ω/LℓΩ → Cℓ(Γ) (6.29)

is a well-defined isomorphism. This will be used for the V -component. For theW -component we need the following lemma.

Lemma 6.1. Assume that the boundary Γ of Ω is Cℓ+1, ℓ ≥ 0. Then the map

qD(V ) )→ pD

(qD(V )

)= γΓ(V ) : Dℓ/Dℓ

Γ → Cℓ(Γ,RN) (6.30)

is a well-defined isomorphism, where

pD : Dℓ → Dℓ/DℓΓ (6.31)

is the canonical surjection and DℓΓ is given by (6.28).

Proof. The proof follows by standard arguments.

Theorem 6.4. Let J be a real-valued shape functional. Assume that the conditionsof Theorem 6.3 (ii) are satisfied and that the boundary Γ of the open domain Ω isCℓ+1 for ℓ ≥ 0.

(i) The map

(v, w) )→ hD×L(v, w) = [h](p−1D v, p−1

L w): Cℓ(Γ,RN)× Cℓ(Γ)→ R

(6.32)

is bilinear and continuous, and for all V and W in Dℓ(RN,RN)

d2J(Ω; V ;W ) = hD×L

(γΓP (V ),

((γΓW ) · n

)), (6.33)

where P (V ) is a linear combination of derivatives of V up to order ℓ.


(ii) This induces a vector distribution h(Γ⊗ Γ) on Cℓ(Γ,RN)⊗Cℓ(Γ) of order ℓ,

h(Γ⊗ Γ) : Cℓ(Γ,RN)⊗ Cℓ(Γ)→ R, (6.34)

such that for all V and W in Dℓ(RN,RN)

⟨h(Γ⊗ Γ), (γΓV )⊗((γΓW ) · n

)⟩ = d2J(Ω; V ;W ), (6.35)

where (γΓV )⊗((γΓW ) · n

)is defined as the tensor product

((γΓV )⊗

((γΓW ) · n

))

i(x, y) = (γΓVi)(x)

((γΓW ) · n

)(y), x, y ∈ Γ, (6.36)

Vi(x) is the ith component of V (x), and

∀y ∈ Γ,(γΓ(W ) · n

)(y) = (γΓW )(y) · n(y). (6.37)

In the regular case the reader is referred to the early papers of J.-P. Zolesio[24, p. 434] and D. Bucur and J.-P. Zolesio [12].

Remark 6.4.Finally, under the assumptions of Theorems 6.3 and 6.4

d2J(Ω; V ;W )= ⟨h(Γ⊗ Γ), (γΓP (V (0)))⊗ ((γΓW (0)) · n)⟩+ ⟨(g(Γ), (γΓV ′(0)) · n)⟩

(6.38)

for all V in−→V m+1,ℓ and W in

−→V m,ℓ.

Example 6.2.Go back to the example of the domain integral in section 6.1,

J(Ω) def=∫

Ωf dx.

For V ∈ D1(RN,RN)

dJ(Ω; V ) =∫

Γf V · n dΓ =

∫

Ωdiv (fV ) dx.

Using the domain expression with V in D2(RN,RN) and W in D1(RN,RN)

dJ(Ωs(W );V ) =∫

Ωs(W )div (fV ) dx

we readily get

d2J(Ω; V ;W ) =∫

Γdiv (fV ) W · n dΓ =

∫

Ωdiv[div (fV ) W

]dx (6.39)

and if Γ is C1,

d2J(Ω; V ;W ) =∫

Γdiv V W · n dΓ (6.40)

is continuous for pairs

(V, W ) ∈ D1(RN,RN)×D0(RN,RN) or C1(Γ,RN)× C0(Γ,RN).


6.5 Decomposition of d2J(Ω; V (0), W (0))One important observation in the explicit computation of the second-order Euleriansemiderivative of the domain integral (6.1) in section 6.1 was the lack of symmetryand the appearance of the first half of the Lie bracket in (6.4). The same phenome-non was observed in the final form of the basic formula (6.10) for domain integrals(6.6) in Theorem 6.1, and also in section 6.3 for the derivative of the shape gradi-ent (6.21) when it can be represented in the integral form (6.20). In this sectionperturbations of the identity will be used to show that d2J(Ω; V (0), W (0)) can befurther decomposed into a symmetric term plus the gradient applied to the velocityDV (0)W (0):

d2J(Ω; V (0), W (0)) = ⟨d2JΩ(0)V (0), W (0)⟩+ dJ(Ω; DV (0)W (0)).

Furthermore the symmetrical term can be obtained by the velocity method

d2JΩ(0;V (0), W (0)) =d

dtJ(Ωt(W (0);V (0) T−1

t (W (0))∣∣∣∣

t=0.

Theorem 6.5. Let J be a real-valued shape functional and Θ be a Banach subspaceof Lip (RN;RN).

(i) Given f , θ, and ξ in Bε, assume that there exists τ > 0 such that

∀t ∈ [0, τ ], d2JΩ(f + tξ; θ) exists.

Then

d2JΩ(f ; θ; ξ) exists ⇐⇒ d2J(Ωf ;V;Wξ) exists (6.41)

for Ωf = [I + f ](Ω) and the velocity fields

Wξ(t)def= ξ [I + f + tξ]−1 and V(t) def= θ [I + f + tξ]−1. (6.42)

(ii) If f , θ belong to Vℓ+1 and ξ to Vℓ, and V and Wξ satisfy the conditions ofTheorem 6.2, then

d2JΩ(f ; θ; ξ) = d2J(Ωf ; θ [I + f ]−1; ξ [I + f ]−1)

− dJ(Ωf ;D(θ [I + f ]−1) ξ [I + f ]−1),(6.43)

d2J(Ωf ; θ; ξ) = d2JΩ(f ; θ [I + f ]; ξ [I + f ]) + dJ(Ωf ;Dθ ξ). (6.44)

(iii) If V and W satisfy the conditions of Theorem 6.2 and V belongs to−→V m+1,ℓ+1,

then

d2J(Ω; V (0);W (0))

= d2JΩ(0;V (0);W (0)) + dJ(Ω; DV (0)W (0)),(6.45)

d2J(Ω; V ;W )

= d2JΩ(0;V (0);W (0)) + dJ(Ω; V ′(0) + DV (0)W (0)),(6.46)


andd2JΩ(0;V (0);W (0))

=d

dtdJ(Ωt(W (0));V (0) T−1

t (W (0)))∣∣∣∣t=0

.(6.47)

Proof. (i) Assume that d2JΩ(f ; θ; ξ) exists and consider the differential quotient

q(t) def=1t

[dJΩ(f + tξ; θ)− dJΩ(f ; θ)]→ d2JΩ(f ; θ; ξ).

From Theorem 3.4 (ii),

dJΩ(f + tξ; θ) = dJ([I + f + tξ](Ω); θ [I + f + tξ]−1).

Define

Ttdef= [I + f + tξ] [I + f ]−1, V(t) def= θ [I + f + tξ]−1.

From Theorem 4.1 in Chapter 4, Tt = Tt(Wξ) for the velocity field

Wξ(t)def=

∂Tt

∂t T−1

t = ξ [I + f + tξ]−1.

Therefore

dJΩ(f + tξ; θ) = dJ(Tt(Wξ)(Ωf );V(t)),q(t) =

1t

[dJ(Tt(Wξ)(Ωf );V(t))− dJ(Ωf ;V(0))]→ d2(Ωf ;V;Wξ)

since q(t) converges as t goes to 0. The converse is obvious.(ii) This is now a direct consequence of Theorem 6.2 and the fact that

V(t) = θ [I + f ]−1 T−1t (Wξ)

⇒ V ′(0) = −D(θ [I + f ]−1) ξ [I + f ]−1.

(iii) From Theorem 6.2 and part (ii) with f = 0, θ = V (0), and ξ = W (0).

If D2JΩ(f) exists in a neighborhood of f = 0 and if it is continuous at f = 0,then D2JΩ(0) is symmetrical and this completes the decomposition of the shapegradient into a symmetrical operator and the gradient applied to the first half ofthe Lie bracket [V (0), W (0)]:

d2J(Ω; V (0);W (0))

= ⟨D2JΩ(0)V (0), W (0)⟩+ ⟨G(Ω), DV (0)W (0)⟩.

But this is not the end of the story. From the computation of the Hessian ofthe domain integral (6.4) in section 6.1,

d2JV,W

=∫

Γ

(∂f

∂n+ H f

)vn wn + f

(D2b vΓ · wΓ − vΓ ·∇Γwn − wΓ ·∇Γvn

)

+ f DV W · n dΓ,


the result of the computation (6.10) in section 6.2,∫

Γdiv (f V (0)) W (0) · n dΓ

=∫

Γ

(∂f

∂n+ H f

)vn wn + f

(D2bvΓ · wΓ − vΓ ·∇Γwn − wΓ ·∇Γvn

)

+ f DV W · n dΓ,

and expression (6.21) of the derivative of (6.20) in section 6.3,

d2J(V ;W ) =∫

ΓQ′

wn(0) vn + Q(0)V ′(0) · n +

(∂Q(0)∂n

+ H Q(0))

vn wn

+ Q(0)(D2b vΓ · wΓ − vΓ ·∇Γwn − wΓ ·∇Γvn

)

+ Q(0)DV W · n dΓ,

it is readily seen that the symmetrical term ⟨D2JΩ(0)V (0), W (0)⟩ further decom-poses into a symmetrical term which depends only on the normal components vn

and wn of V (0) and W (0) and another symmetrical term, which is boxed in theabove expressions. This last term is the same in all expressions and depends onlyon the trace of G(0) (here of f and Q(0)) on Γ and the group of terms

D2b vΓ · wΓ − vΓ ·∇Γwn − wΓ ·∇Γvn (6.48)

involving vΓ, wΓ, and tangential derivatives of vn and wn on Γ.It would be interesting to further investigate this structure. In the context of

a domain optimization problem, if the shape gradient is zero, both the term (6.48)and the first half of the Lie bracket will be multiplied by zero. Thus they will notcontribute to the Hessian which will reduce to the symmetrical part that dependsonly on vn and wn; that is, for autonomous velocity fields V and W ,

∫

Γ

(∂f

∂n+ H f

)vn wn dΓ,

∫

ΓQ′

wn(0) vn +

(∂Q(0)∂n

+ H Q(0))

vn wn dΓ,

and the term∫

ΓQ′

wn(0) vn dΓ =

∫

ΓQ′

vn(0)wn dΓ

is symmetrical. For earlier results on the structure of the second-order shape de-rivative the reader is referred to J.-P. Zolesio [24, p. 434] and D. Bucur andJ.-P. Zolesio [12]. Of course, the task of establishing similar results in the con-strained case (D = RN) remains to be done.

Chapter 10

Shape Gradients under aState EquationConstraint

1 IntroductionWhen a shape functional depends on the solution of a boundary value problemdefined on the underlying domain, it is said to be constrained. This special typeof constraint is to be distinguished from constraints on the geometry such as thevolume, perimeter, and curvatures. Using the generic terminology of control theorythe solution of the boundary value problem will be called the state and its corre-sponding equation or inequality the state equation or inequality. The domains willbe identified with the controls and the constraints on the domains with the controlconstraints. Additional constraints on the solution of the state equation are usu-ally called state constraints. Such problems have received a considerable amountof attention over the last century, and a rich and abundant literature can be foundin mechanics, control theory, and optimization. Treating the state equation as anequality constraint and using Lagrange multipliers naturally yields a dual variablewhich is solution of the adjoint equation of the linearized state equation. This dualvariable is called the adjoint state.

Of course under appropriate differentiability assumptions with respect to thestate and the control variables, necessary conditions can be obtained within the clas-sical framework of the calculus of variations. Probably one of the most influentialcontributions of the last half of the 20th century to optimal control theory was madeby L. Pontryagin, V. Boltyanskii, R. Gamkrelidze, and E. Mishchenko [1]in the 1960s when they showed that the differentiability with respect to the controlcould be relaxed and replaced by the pointwise maximization with respect to thecontrol variable of the Hamiltonian constructed from the objective function andthe coupled state and adjoint state equations, the so-called Maximum Principle.

One of the important technical advantages of the control theory approach is toavoid the differentiation of the state with respect to the control. This is true not onlyfor the characterization of optimal controls but also to obtain explicit expressions ofthe derivative of an objective function constrained by a state equation with respectto the control. Using a Lagrangian or Hamiltonian formulation, the derivative of

519

Treating the state equation as an equality constraint and using Lagrange multipliers naturally yields a dual variable which is solution of the adjoint equation of the linearized state equation. This dual variable is called the adjoint state.

520 Chapter 10. Shape Gradients under a State Equation Constraint

the constrained objective function with respect to the control is typically equal tothe “partial” derivative of the Lagrangian or the Hamiltonian with respect to thecontrol, where the adjoint variable is a solution of an appropriate adjoint stateequation that is coupled with the state equation.

This chapter will concentrate on two generic examples often encountered inshape optimization. The first one is associated with the so-called compliance prob-lems, where the shape functional is equal to the minimum of a domain-dependentenergy functional. The special feature of such functionals is that the adjoint statecoincides with the state. This obviously leads to considerable simplifications in theanalysis. In that case it will be shown that theorems on the differentiability ofthe minimum of a functional with respect to a real parameter readily give explicitexpressions of the Eulerian semiderivative even when the minimizer is not unique.The second one will deal with shape functionals that can be expressed as the saddlepoint of some appropriate Lagrangian. As in the first example, theorems on thedifferentiability of the saddle point of a functional with respect to a real parame-ter readily give explicit expressions of the Eulerian semiderivative even when thesolution of the saddle point equations is not unique.

Avoiding the differentiation of the state equation with respect to the domainis particularly advantageous in shape problems. Here the state variable (the so-lution of the boundary value problem) lives in a function space (Banach, Hilbert,Sobolev) which depends on the control (underlying variable domain)! Thus thenotion of derivative of the state with respect to the domain is more delicate. Inthis chapter two techniques will be presented to get around this difficulty: functionspace parametrization and function space embedding.

Function space parametrization consists in transporting the functions on vari-able domains back onto the initial domain, where they can be compared. It is relatedto the notion of material derivative in continuum and structural mechanics. Thiswill be illustrated by computing the volume and boundary integral expressions of theEulerian semiderivative of a simple shape functional for the homogeneous Dirichletand Neumann boundary value problems by the theorems on the differentiation of aminimum and a saddle point.

Function space embedding consists in constructing extensions of the domain-dependent boundary value problems to a larger fixed domain. For homogeneousDirichlet boundary conditions, it is sufficient to consider extensions by zero to RN.The nonhomogeneous case requires the introduction of a multiplier to take into ac-count the boundary condition. This technique will be illustrated by computing theboundary integral expression of the Eulerian semiderivative of a simple shape func-tional for the nonhomogeneous Dirichlet boundary value problem by the theoremson the differentiation of a saddle point.

In addition to the generic example, the theorem on the differentiation of aninfimum with respect to a parameter will be applied to the example of the bucklingof columns considered in section 5 of Chapter 5. In section 3 an explicit expressionof the semiderivative of Euler’s buckling load with respect to the cross-sectional areawill be given. A necessary and sufficient condition will also be given to character-ize the maximum Euler’s buckling load with respect to a family of cross-sectionalareas. The theory is further illustrated in section 4 by providing the semiderivative

Function space parametrization

Function space embedding

2. Min Formulation 521

of the first eigenvalue of several boundary value problems over a bounded open do-main: Laplace equation, bi-Laplace equation, linear elasticity. In general the firsteigenvalue is not simple over an arbitrary bounded open domain and the eigenvalueis not differentiable; yet the main theorem provides explicit domain expressionsfor a bounded open domain and boundary expressions of the semiderivatives for asufficiently smooth bounded open domain.

2 Min Formulation2.1 An Illustrative Example and a Shape Variational PrincipleLet Ω be a bounded open domain in RN with a smooth boundary Γ. Let y = y(Ω)be the solution of the Dirichlet problem

−y = f in Ω, y = 0 on Γ, (2.1)

where f is a fixed function in H1(RN). The solution of (2.1) is the minimizingelement in H1

0 (Ω) of the energy functional

E(Ω,ϕ) =∫

Ω

[12|∇ϕ|2 − fϕ

]dx. (2.2)

Introduce the shape function

J(Ω) def= infϕ∈H1

0 (Ω)E(Ω,ϕ) = −

∫

Ω

12|∇y|2 dx. (2.3)

We want to show that

dJ(Ω; V ) = −12

∫

Γ

∣∣∣∣∂y

∂n

∣∣∣∣2

V · n dΓ. (2.4)

This example is the prototype of a free boundary problem1 which can be obtainedfrom the following shape variational principle:

∀V, dJ(Ω; V ) = 0. (2.5)

It yields the extra boundary condition2

∂y

∂n= 0 on Γ. (2.6)

Equations (2.1) and (2.6) characterize a free boundary problem. It is the simplestexample of a large family of problems where the shape function is an extremal ofa natural internal energy (cf. section 7 of Chapter 5). They correspond to the so-called compliance problems in elasticity theory. They also occur in fracture theoryand image segmentation. The first-order variation of this shape functional yieldsthe extra boundary condition which is characteristic of a free boundary problem.

1This usually comes with other constraints, such as a volume constraint on Ω and/or an areaconstraint on Γ.

2With a volume equality constraint we would get that |∂y/∂n|2 is equal to a constant on Γ.


2.2 Function Space ParametrizationTo compute the first-order derivative of J(Ω) we perturb the bounded open domainΩ by a velocity field V , which generates the family of transformations Tt : 0 ≤ t ≤τ of RN and the family of domains Ωt = Tt(Ω) : 0 ≤ t ≤ τ. At t,

J(Ωt) = infϕ∈H1

0 (Ωt)E(Ωt,ϕ) (2.7)

and the minimizing element yt = y(Ωt) is the solution of the Dirichlet problem

−yt = f in Ωt, yt = 0 on Γt, (2.8)

where Γt is the boundary of Ωt. We want to compute the derivative

dj(0) = limt0

j(t)− j(0)t

(2.9)

of the functionj(t) def= J(Ωt). (2.10)

We need a theorem that would give the derivative of an infimum with respect to a pa-rameter t ≥ 0 at t = 0. The difficulty here is that the function space H1(Ωt) dependson the parameter t. To get around this and obtain an infimum with respect to afunction space that is independent of t, we introduce the following parametrization:

H10 (Ωt) =

ϕ T−1

t : ϕ ∈ H10 (Ω)

. (2.11)

Notice that since Tt is a homeomorphism, it transforms the open domain Ω intothe open domain Ωt and sends the boundary Γ of Ω onto the boundary Γt of Ωt.In particular, when V is sufficiently smooth for all ϕ in H1

0 (Ω), ϕ T−1t ∈ H1

0 (Ωt),and conversely, for all ψ in H1

0 (Ωt), ψ Tt ∈ H10 (Ω). This parametrization does not

affect the value of the minimum J(Ωt) but changes the functional E

J(Ωt) = infϕ∈H1

0 (Ω)E(Tt(Ω),ϕ T−1

t ). (2.12)

This parametrization is typical in shape analysis. It amounts to introducing thenew energy functional

E(t,ϕ) = E(Tt(Ω),ϕ T−1t ), ϕ ∈ H1

0 (Ω). (2.13)

Our objective in the next section is to compute the limit (2.9) for

j(t) = infϕ∈H1

0 (Ω)E(t,ϕ). (2.14)

This will be done in section 2.3. Before closing, it is interesting to characterize theminimizing element yt in H1

0 (Ω) of

E(t,ϕ) =∫

Ωt

[12|∇(ϕ T−1

t )|2 − f(ϕ T−1t )

]dx, (2.15)


which is the solution of the following variational equation: find yt ∈ H10 (Ω) such

that for all ϕ ∈ H10 (Ω)

∫

Ωt

∇(yt T−1

t ) ·∇(ϕ T−1t )− f(ϕ T−1

t )

dx = 0. (2.16)

Compare this expression with the characterization of the minimizing element yt ofE(Ωt,ϕ) on H1

0 (Ωt): find yt ∈ H10 (Ωt) such that for all ϕ ∈ H1

0 (Ωt)∫

Ωt

∇yt ·∇ϕ− fϕ dx = 0. (2.17)

It is easy to verify that

yt = yt T−1t and yt = yt Tt. (2.18)

So yt is the solution yt of (2.8) transported back onto the fixed domain Ω by thechange of variables induced by Tt.

In view of this expression, (2.15) can be rewritten on the fixed domain Ω as

E(t,ϕ) =∫

Ω

12

(A(t)∇ϕ) ·∇ϕ− (f Tt)ϕ Jt dx, (2.19)

where for t in [0, τ ] small

DTt = Jacobian matrix of Tt, (2.20)Jt = |det DTt| = detDTt for t ≥ 0 small, (2.21)

A(t) = Jt[DTt]−1 ∗[DTt]−1. (2.22)

With this change of variables yt is now characterized by the variational equation⎧⎨

⎩

yt ∈ H10 (Ω) and ∀ϕ ∈ H1

0 (Ω),∫

ΩA(t)∇yt ·∇ϕ− Jt(f Tt)ϕ dx = 0.

(2.23)

2.3 Differentiability of a Minimum with Respect to a ParameterConsider a functional

G : [0, τ ]×X → R (2.24)

for some τ > 0 and some set X. For each t in [0, τ ] define

g(t) def= infG(t, x) : x ∈ X, (2.25)

X(t) def= x ∈ X : G(t, x) = g(t). (2.26)

The objective is to characterize the limit

dg(0) def= limt0

g(t)− g(0)t

(2.27)

when X(t) is not empty for 0 ≤ t ≤ τ .

So yt is the solution yt of (2.8) transported back onto the fixed domain Ω by the change of variables induced by Tt.


When X(t) = xt is a singleton, 0 ≤ t ≤ τ , and the derivative

x = limt0

xt − x0

t(2.28)

of x is known, then it is easy to obtain dg(0) under appropriate differentiability ofthe functional G with respect to t and x. When x is not readily available or whenthe sets X(t) are not singletons, this direct approach fails or becomes very intricate.In this section we present a theorem that gives an explicit expression for dg(0), thederivative of the min of the functional G with respect to t at t = 0. Its originalityis that the differentiability of xt is replaced by a continuity assumption on the set-valued function and the existence of the partial derivative of the functional G withrespect to the parameter t. In other words this technique does not require a prioriknowledge of the derivative x of the minimizing elements xt with respect to t.

Theorem 2.1. Let X be an arbitrary set, let τ > 0, and let G : [0, τ ]×X → R bea well-defined functional. Assume that the following conditions are satisfied:

(H1) for all t ∈ [0, τ ], X(t) = ∅;

(H2) for all x in⋃

t∈[0,τ ] X(t), ∂tG(t, x) exists everywhere in [0, τ ];

(H3) there exists a topology TX on X such that for any sequence tn ⊂ ]0, τ ],tn → t0 = 0, there exist x0 ∈ X(0) and a subsequence tnk of tn, and foreach k ≥ 1, there exists xnk ∈ X(tnk) such that

(i) xnk → x0 in the TX-topology and

(ii)lim infk→∞t0

∂tG(t, xnk) ≥ ∂tG(0, x0);

(H4) for all x in X(0), the map t -→ ∂tG(t, x) is upper semicontinuous at t = 0.

Then there exists x0 ∈ X(0) such that

dg(0) = limt0

g(t)− g(0)t

= infx∈X(0)

∂tG(0, x) = ∂tG(0, x0). (2.29)

Remark 2.1.In the literature condition (H3) (i) is known as sequential semicontinuity forset-valued functions. When X(0) is a singleton x0 we readily get dg(0) =∂tG(0, x0).

Remark 2.2.This theorem, and in particular the last part of property (2.29), extends a formerresult by B. Lemaire [1, Thm. 2.1, p. 38], where sequential compactness of the setX was assumed. It also completes and extends Theorem 1 in J.-P. Zolesio [7]and M. C. Delfour and J.-P. Zolesio [3].

Theorem2.1. LetXbeanarbitraryset,letτ>0,andletG:[0,τ]×X→Rbea well-defined functional. Assume that the following conditions are satisfied: (H1) for all t ∈ [0,τ], X(t) = ∅;(H2) for all x in t∈[0,τ] X(t), ∂tG(t,x) exists everywhere in [0,τ]; (H3) there exists a topology TX on X such that for any sequence tn ⊂ ]0, τ ],tn → t0 = 0, there exist x0 ∈ X(0) and a subsequence tnk of tn, and foreach k ≥ 1, there exists xnk ∈ X(tnk ) such that (i) xnk → x0 in the TX -topology and(ii) lim inf ∂tG(t, xnk ) ≥ ∂tG(0, x0);k→∞ t0(H4)forallxinX(0),themapt→ ∂tG(t,x)isuppersemicontinuousatt=0. Then there exists x0 ∈ X(0) such thatdg(0) = lim g(t) − g(0) = inf ∂tG(0, x) = ∂tG(0, x0). (2.29) t0 t x∈X(0)


Proof of Theorem 2.1. (i) We first establish upper and lower bounds to the differ-ential quotient

∆(t)t

, ∆(t) def= g(t)− g(0).

Choose arbitrary x0 in X(0) and xt in X(t). Then, by definition,

G(t, xt) = g(t) ≤ G(t, x0),−G(0, xt) ≤ −g(0) = −G(0, x0).

Add the above two inequalities to obtain

G(t, xt)−G(0, xt) ≤ ∆(t) ≤ G(t, x0)−G(0, x0).

By assumption (H2), there exist θt, 0 < θt < 1, and αt, 0 < αt < 1, such that

G(t, xt)−G(0, xt) = t∂tG(θtt, xt),G(t, x0)−G(0, x0) = t∂tG(αtt, x0),

and by dividing by t > 0

∂tG(θtt, xt) ≤∆(t)

t≤ ∂tG(αtt, x0). (2.30)

(ii) Define

dg(0) = lim inft0

∆(t)t

, dg(0) = lim supt0

∆(t)t

.

There exists a sequence tn : 0 < tn ≤ τ, tn → 0, such that

limn→∞

∆(tn)tn

= dg(0).

By assumption (H3), there exist x0 ∈ X(0) and a subsequence tnk of tn andfor each k ≥ 1, there exists xnk ∈ X(tnk) such that xnk → x0 in TX and

lim inft0

k→∞

∂tG(t, xnk) ≥ ∂tG(0, x0).

So, from the first part of the estimate (2.30) for t = tnk ,

∂tG(θtnktnk , xnk) ≤ ∆(tnk)

tnk

and∂tG(0, x0) ≤ lim inf

k→∞∂tG(θtnk

tnk , xnk) ≤ limk→∞

∆(tnk)tnk

= dg(0).

Therefore,∃x0 ∈ X(0), ∂tG(0, x0) ≤ dg(0),


andinf

x∈X(0)∂tG(0, x) ≤ ∂tG(0, x0) ≤ dg(0).

From the second part of (2.30) and assumption (H4) we also obtain

∀x ∈ X(0), ∂tG(0, x) ≥ dg(0),

dg(0) ≤ infx∈X(0)

∂tG(0, x), (2.31)

and necessarily

infx∈X(0)

∂tG(0, x) = dg(0) = dg(0) = infx∈X(0)

∂tG(0, x).

In particular, from (2.3) and (2.31)

∂tG(0, x0) = dg(0) = infx∈X(0)

∂tG(0, x)

and x0 is a minimizing point of ∂tG(0, ·).

2.4 Application of the TheoremOur example has a unique minimizing point yt for t ≥ 0 small. Here X = H1

0 (Ω),X(t) = yt, and it is sufficient to establish the continuity of the map t -→ yt att = 0 for an appropriate topology on H1

0 (Ω).We now check assumptions (H1) to (H4). Assume that V ∈ C0([0, τ ];D2(RN,

RN))

and that f ∈ H1(RN). Choose τ > 0 small enough such that

Jt = |Jt|, 0 ≤ t ≤ τ, (2.32)

and that there exist constants 0 < α < β such that

∀ξ ∈ RN, α|ξ|2 ≤ A(t) ξ · ξ ≤ β|ξ|2, and α ≤ Jt ≤ β. (2.33)

Since the bilinear form associated with (2.23) is coercive, there exists a uniquesolution yt to (2.23) and

∀t ∈ [0, τ ], X(t) = yt = ∅. (2.34)

So assumption (H1) is satisfied. To check (H2) use expression (2.19) and compute

∂tE(t,ϕ) =∫

Ω

12

[A′(t)∇ϕ] ·∇ϕ− [div Vt (f Tt) + Jt∇f · Vt]ϕ

dx, (2.35)

where

Vt(X) = V (t, Tt(X)), DVt(X) = DV (t, Tt(X)), (2.36)A′(t) = (div Vt)I − (DVt)∗ −DVt. (2.37)

By assumptions on V and f , ∂tE(t,ϕ) exists everywhere in [0, τ ] for all ϕ in H10 (Ω)

and assumption (H2) is satisfied.


To check assumption (H3)(i) we first show that yt is bounded in H10 (Ω).

From (2.33)

α∥∇yt∥2L2(Ω) ≤∫

ΩA(t)∇yt ·∇yt dx

=∫

ΩJt(f Tt) yt dx ≤ ∥Jt f Tt∥L2(Ω) ∥yt∥L2(Ω).

(2.38)

By using the norm

∥ϕ∥H10 (Ω) = ∥∇ϕ∥L2(Ω)

and the continuous injection of H10 (Ω) into L2(Ω),

∃c > 0, ∥ϕ∥L2(Ω) ≤ c∥ϕ∥H10 (Ω).

So from (2.38)

∥yt∥H10 (Ω) ≤

c

α∥Jtf Tt∥L2(Ω). (2.39)

But Jt → 1 as t→ 0 and f Tt → f in L2(Ω) by the following lemma.

Lemma 2.1. Assume that V ∈ C0([0, τ ];D1(RN,RN))

satisfies assumptions (V)of Chapter 4 and that f ∈ L2(RN). Then

limt0

f Tt = f and limt0

f T−1t = f in L2(RN). (2.40)

So by (2.39) yt is bounded:

∃c > 0, supt∈[0,τ ]

∥yt∥H10 (Ω) ≤ c. (2.41)

The next step is to prove the continuity by subtracting (2.23) at t > 0 from (2.23)at t = 0: ∫

Ω∇yt ·∇ϕ dx +

∫

Ω(A(t)− I)∇yt ·∇ϕ dx

=∫

Ωfϕ dx +

∫

Ω[Jt(f Tt)− f ]ϕ dx,

∫

Ω∇y ·∇ϕ dx =

∫

Ωfϕ dx.

Subtract and set ϕ = yt − y:∫

Ω|∇(yt − y)|2 dx

= −∫

Ω(A(t)− I)∇yt ·∇(yt − y) + [Jt(f Tt)− f ](yt − y) dx

≤ |A(t)− I|∥∇yt∥L2(Ω)∥∇(yt − y)∥L2(Ω)

+ ∥Jt f Tt − f∥L2(Ω)∥yt − y∥L2(Ω)


and∥yt − y∥H1

0 (Ω) ≤ c |A(t)− I| + ∥Jt f Tt − f∥L2(Ω).

But A(t) − I → 0 and Jt f Tt → f in L2(Ω), and finally yt → y in H10 (Ω). So

assumption (H3)(i) is satisfied for H10 (Ω)-strong.

For assumption (H3)(ii) we have

∂tG(t,ϕ) =∫

Ω

12

[A′(t)∇ϕ] ·∇ϕ− [div Vt(f Tt) + Jt∇f Vt]ϕ

dx

and for ϕ in H10 (Ω), f in H1(RN), and V in C0([0, τ ];D2(RN,RN))

∂tG(t,ϕ)− ∂tG(0, y)

=∫

Ω

12(A′(t)−A′(0))∇ϕ ·∇ϕ

− [div Vt(f Tt)− Jt∇f · Vt − div V (0)f +∇f · V (0)]ϕ

dx

+∫

Ω

12

A′(0)∇ϕ ·∇ϕ−A′(0)∇y ·∇y dx.

As ϕ goes to y in H10 (Ω) and t → 0, the first term converges to zero since ϕ is

bounded and

A′(t)→ A′(0),

div Vt(f Tt)→ div V (0)f in L2(Ω),

Jt∇f · Vt → ∇f · V (0) in L2(Ω).

The second term is continuous with respect to ϕ in H10 (Ω) and goes to zero as ϕ→ y

in H10 (Ω). So assumption (H3)(ii) is satisfied.Finally, (H4) is satisfied since

t -→ ∂tE(t, y) =∫

Ω

12A′(t)∇y ·∇y − [div Vtf +∇f · Vt] y

dx (2.42)

is continuous in [0, τ ]. So all the assumptions of Theorem 2.1 are satisfied and forV in C0([0, τ ];D2(RN,RN)) and f in H1(RN),

dJ(Ω; V ) =∫

Ω

12A′(0)∇y ·∇y − [div V (0)f +∇f · V (0)] y

dx. (2.43)

For V autonomous, (2.43) is continuous with respect to the space D1(RN,RN) andthe shape gradient is of order 1. We know by the structure Theorem 3.6 of Chapter 9that for Ω open and a C2-boundary Γ,

∃g(Γ) ∈ D1(Γ)′, dJ(Ω; V ) = ⟨g(Γ), V ⟩D1(Γ).

The characterization of g(Γ) will be given in the next section.We complete this section with the proof of Lemma 2.1.


So for t small enough the right-hand side of (2.4) is less than 3ε and this completesthe proof of the first part of (2.41).

(ii) The second part of (2.41) can be obtained by the change of variables∫

RN|f T−1

t − f |2 dx =∫

RN|f − f Tt|2J−1

t dX

and the fact that β−1 < J−1t < α−1. This completes the proof of the lemma.

2.5 Domain and Boundary Integral Expressions of the ShapeGradient

Expression (2.43) for the shape gradient is a volume (or domain) integral, and it iseasy to check that the map

V -→ dJ(Ω; V ) :−→V 0,1 → R (2.46)

is linear and continuous (cf. section 3.4 in Chapter 9). So by Corollary 1 to thestructure Theorem 3.6 in Chapter 9, we know that for a domain Ω with a C2-boundary Γ there exists a scalar distribution g(Γ) in D(Γ)′ such that

dJ(Ω; V ) = ⟨g(Γ), V (0) · n⟩D1(Γ). (2.47)

The next objective is to further characterize the boundary expression. Recallthat we have assumed that f ∈ H1(RN). So for a C2 boundary Γ the solutiony = y(Ω) of the problem

−y = f in Ω, y = 0 on Γ

belongs to H2(Ω). For velocity fields V in C0([0, τ ];D1(RN,RN)) satisfying as-sumption (V ) the transported solution yt in H2(Ω) is the solution of the system

−div (A(t)∇yt) = Jtf Tt in Ω, yt = 0 on Γ. (2.48)

Knowing that for all t in [0, τ ], yt ∈ H2(Ω)∩H10 (Ω), we can repeat the computation

of ∂tE(t,ϕ) for ϕ in H2(Ω)⋂

H10 (Ω) instead of H1

0 (Ω). With this extra smoothnesswe can use the formula

d

dt

∫

Ωt

F (t, x) dx

∣∣∣∣t=0

=∫

ΓF (0, x)V (0) · n dΓ +

∫

Ω

∂F

∂t(0, x) dx (2.49)

for a sufficiently smooth function F : [0, τ ]×RN → R. Prior to applying this formulato expression (2.15) in section 2.2, notice that for ϕ in H2(Ω),

ϕ =d

dtϕ T−1

t

∣∣t=0 = −∇ϕ · V (0) ∈ H1(Ω), (2.50)


but it generally does not belong to H10 (Ω). Then

∂tE(t,ϕ)∣∣t=0 =

∫

Γ

12|∇ϕ|2 − fϕ

V (0) · n dΓ +

∫

Ω∇ϕ ·∇ϕ− f ϕ dx.

(2.51)

Substitute ϕ = y in (2.51):

∂tE(t, y)∣∣t=0 =

∫

Γ

12|∇y|2 − fy

V (0) · n dΓ

−∫

Ω∇y ·∇(∇y · V (0))− f(∇y · V (0)) dx.

(2.52)

But y ∈ H2(Ω)⋂

H10 (Ω) and

∫

Ω∇y ·∇(∇y · V (0)) dx = −

∫

Ωy∇y · V (0) dx +

∫

Γ

∂y

∂n∇y · V (0) dΓ

and

∂tE(t, y)∣∣t=0 =

∫

Γ

[12|∇y|2 − fy

]V (0) · n− ∂y

∂n∇y · V (0)

dΓ. (2.53)

Buty = 0 on Γ ⇒ ∇y =

∂y

∂nn on Γ,

and finally∂tE(t, y)

∣∣t=0 = −

∫

Γ

12

∣∣∣∣∂y

∂n

∣∣∣∣2

V (0) · n dΓ. (2.54)

This is the boundary expression that is continuous for V (0) in the space D0(RN,RN).It has been obtained via a parametrization of the function space appearing in themin formulation. Thus

dJ(Ω; V ) = −∫

Ω

12

∣∣∣∣∂y

∂n

∣∣∣∣2

V (0) · n dΓ, (2.55)

as predicted in section 2.1.

Remark 2.3.An original application of the computations made for the generic example can befound in M. C. Delfour, G. Payre, and J.-P. Zolesio [3]. In that paper thederivative of the energy function with respect to the nodes in a P 1 finite elementapproximation is obtained from the volume expression of the shape semiderivativeof the continuous problem. This technique also applies to a broad class of boundaryvalue problems and to mixed hybrid finite element approximations. As observedby several authors the boundary integral expression is not suitable since the fi-nite element solution does not have the appropriate smoothness under which theboundary integral formula is obtained. This technique is used to obtain a triangu-larization which minimizes the approximation error of the solution. Other formulaeof the same type have been obtained for mixed finite element approximations in

An original application of the computations made for the generic example can be found in M. C. Delfour, G. Payre, and J.-P. Zole sio [3]. In that paper the derivative of the energy function with respect to the nodes in a P1 finite element approximation is obtained from the volume expression of the shape semiderivative of the continuous problem. This technique also applies to a broad class of boundary value problems and to mixed hybrid finite element approximations

As observed by several authors the boundary integral expression is not suitable since the finite element solution does not have the appropriate smoothness under which the boundary integral formula is obtained. This technique is used to obtain a triangularization which minimizes the approximation error of the solution


M. C. Delfour, Z. Mghzali, and J.-P. Zolesio [1] and several other boundaryvalue problems. The reader is also referred to M. C. Delfour, G. Payre, andJ.-P. Zolesio [1, 2, 4, 5] for application of those techniques to thermal problemssuch as diffusers and space radiators. The last paper combines the above techniqueswith the systematic handling of parametrized geometries.

3 Buckling of ColumnsAuchmuty’s dual principle was used for the functional associated with the optimaldesign of a column against buckling in section 5 of Chapter 5. In this section wefirst use this construction to compute the directional semiderivative with respect tothe cross-sectional area and then use it to give a necessary and sufficient conditionthat characterizes the maximizers. Indeed, recall the identity

µ(A) = − 12λ(A)

and notice that A is a maximizer of λ over A if and only if A is a maximizer of µover A.

We have seen in Theorem 5.3 of section 5 of Chapter 5 that the concave uppersemicontinuous functional

µ(A) def= infL(A, v) : v ∈ H2

0 (0, 1)

, (3.1)

L(A, v) def=12

∫ 1

0A |v′′|2 dx−

[∫ 1

0|v′|2 dx

]1/2

(3.2)

has maximizers over the weakly compact convex set A. Therefore, if µ(A) has adirectional semiderivative dµ(A;B), a maximizer is completely characterized by

∃A ∈ A, ∀B ∈ A, dµ(A;B −A) ≤ 0.

This directional semiderivative always exists for concave continuous functions, andTheorem 2.1 can be used to get an explicit expression of dµ(A;B).

Given A ∈ A, B ∈ L∞(0, 1), and t ≥ 0, let

Atdef= A + tB.

In view of the fact that A ≥ A0 > 0, there exists τ > 0 such that for all 0 ≤ t ≤ τ

∀t, 0 ≤ t ≤ τ, At ≥ A0/2.

Define

L(t, v) def=12

∫ 1

0At |v′′|2 dx−

[∫ 1

0|v′|2 dx

]1/2

⇒ µ(At) = inf

L(t, v) : v ∈ H20 (0, 1)

, X(t) def= E(At).


Theorem 3.1. Let B be an arbitrary function in L∞(0, 1) and A be an element of A.

(i) The directional semiderivative at A in the direction B is given by the followingexpression:

dµ(A;B) = infv∈E(A)

12

∫ 1

0B |v′′|2 dx, (3.3)

and since µ is concave and continuous, dHµ(A;B) also exists.

(ii) The maximizing elements of µ(A) or λ(A) in A are completely characterizedby the variational inequality

∃A ∈ A, ∀B ∈ A, infv∈E(A)

12

∫ 1

0(B −A) |v′′|2 dx ≤ 0 (3.4)

or equivalently


12

∫ 1

0B |v′′|2 dx ≤ 1

2λ(A)= −µ(A). (3.5)

If we replace E(A) by

E(A) def=

⎧⎪⎨

⎪⎩u ∈ H2

0 (0, 1) :

∫ 1

0|u′|2 dx = 1 and ∀v ∈ H2

0 (0, 1)∫ 1

0A u′′ v′′ dx = λ(A)

∫ 1

0u′ v′ dx

⎫⎪⎬

⎪⎭,

then the maximizing elements of µ(A) or λ(A) in A are completely character-ized by the variational inequality


∫ 1

0B |v′′|2 dx ≤ λ(A). (3.6)

Proof. (i) From Theorem 5.2 in Chapter 5, X(t) = ∅ and assumption (H1) issatisfied. The partial derivative of L(t, v) with respect to t is given by the expression

∂tL(t, v) =12

∫ 1

0B |v′′|2 dx, (3.7)

which is independent of t and exists for all B ∈ L∞(0, 1). Hence assumptions (H2)and (H4) are trivially satisfied. From (5.7) in Theorem 5.1 of Chapter 5

∃ut ∈ H20 (0, 1), ∀v ∈ H2

0 (0, 1),∫ 1

0At u′′

t v′′ dx = λ(At)∫ 1

0u′

t v′ dx (3.8)

⇒ A0

2

∫ 1

0|u′′

t |2 dx ≤∫ 1

0At |u′′

t |2 dx = λ(At)∫ 1

0|u′

t|2 dx =1

λ(At). (3.9)


But

0 <A0

2≤ At ≤ AB

def= A1 + τ∥B∥L∞

⇒ 12µ(A0) ≤ µ(At) ≤ µ(AB) and 2λ(A0) ≤ λ(At) ≤ λ(AB).

Therefore for any sequence tn 0, there exist a subsequence tnk, µ, and u ∈H2

0 (0, 1) such that

µkdef= µ(Atnk

)→ µ < 0, λkdef= λ(Atnk

) = − 12µk→ λ = − 1

2µ> 0,

uk u in H20 (0, 1)-weak ⇒ uk → u in H1

0 (0, 1).

Going to the limit in (3.8)

∀v ∈ H20 (0, 1),

∫ 1

0Atnk

u′′k v′′ dx = λ(Atnk

)∫ 1

0u′

k v′ dx, ∥u′k∥L2 =

1λk

,

we get the following variational equation for u ∈ H20 (0, 1):

∀v ∈ H20 (0, 1),

∫ 1

0A u′′ v′′ dx = λ

∫ 1

0u′ v′ dx,

∥u′∥L2 = limk→∞

∥u′k∥L2 = lim

k→∞

1λk

=1λ

> 0.

To show that u ∈ X(0) = E(A), let u0 be any element of E(A). By definition,

12

∫ 1

0Atnk

|u′′k |2 dx− ∥u′

k∥L2 ≤ 12

∫ 1

0Atnk

|u′′0 |2 dx− ∥u′

0∥L2 . (3.10)

But Atnk→ A in L∞(0, 1)-strong and

lim infk→∞

∫ 1

0Atnk

|u′′k |2 dx = lim inf

k→∞

∫ 1

0(Atnk

−A) |u′′k |2 dx +

∫ 1

0A |u′′

k |2 dx

≥ lim infk→∞

−∥Atnk−A∥L∞

∫ 1

0|u′′

k |2 dx +∫ 1

0A |u′′

k |2 dx

≥∫ 1

0A |u′′|2 dx

since ∥u′′k∥L2 is bounded. Since ∥u′

k∥L2 → ∥u′∥L2 , by going to the limit in (3.10),we get

L(A, u) =12

∫ 1

0A |u′′|2 dx− ∥u′∥L2

≤ lim infk→∞

12

∫ 1

0Atnk

|u′′k |2 dx− ∥u′

k∥L2

≤ lim infk→∞

12

∫ 1

0Atnk

|u′′0 |2 dx− ∥u′

0∥L2 =12

∫ 1

0A |u′′

0 |2 dx− ∥u′0∥L2

= µ(A) = infv∈V

L(A, v).

4. Eigenvalue Problems 535

Therefore, by definition of the minimum u ∈ E(A) = X(0) and assumption (H3) (i)is satisfied.

To check the second part of (H3) we first show that uk converges not only inH2

0 (0, 1)-weak but also in H20 (0, 1)-strong. Then assumption (H4) directly follows

from that property12∂L(t, uk) =

∫ 1

0B |u′′

k |2 dx→∫ 1

0B |u′′|2 dx = ∂L(t, u).

The strong convergence follows from the following chain of inequalities:

A0

2

∫ 1

0|u′′

k − u′′|2 dx ≤∫ 1

0

A0

2|u′′

k |2 dx +∫ 1

0

A0

2|u′′|2 dx−

∫ 1

0A0u

′′k u′′ dx

≤∫ 1

0Ak|u′′

k |2 dx +∫ 1

0A0|u′′|2 dx− 2

∫ 1

0A0u

′′k u′′ dx

= λk

∫ 1

0|u′

k|2 dx +∫ 1

0A0|u′′|2 dx− 2

∫ 1

0A0u

′′k u′′ dx

→ λ

∫ 1

0|u′|2 dx−

∫ 1

0A0|u′′|2 dx = 0

since λ = λ(A). This complete the proof of part (i).(ii) The first characterization now follows directly from the fact that µ is

concave and continuous and A is closed and convex. The second characterizationuses the following identity, which comes from (3.8) for the eigenvectors and theirnormalization (5.11) in Theorem 5.2 of Chapter 5 as elements of E(A): for allv ∈ E(A)

12

∫ 1

0A |v′′|2 dx = λ(A)

12

∫ 1

0|v′|2 dx =

12λ(A)

= −µ(A).

4 Eigenvalue ProblemsThe first eigenvalue λ(Ω) of a linear boundary value problem defined on a boundedopen subset Ω of RN is a classical example of a shape functional which comes inthe form of an infimum. It is generally not differentiable since the eigenvalue canbe repeated, but Theorem 2.1 of section 2.3 can be used to compute its Euleriansemiderivative.

In this section we give the volume expression of the Eulerian semiderivativefor the Laplacian and the bi-Laplacian with a homogeneous Dirichlet boundarycondition for a general bounded open domain and its boundary expression when thedomain is sufficiently smooth. In the case of the Laplacian over a smooth domain,the first eigenvalue is simple and differentiable. As in section 3 it is technicallyadvantageous to work with G. Auchmuty [1]’s dual principle rather than withthe Rayleigh quotient. It is also technically advantageous to embed the problem ofthe computation of the Eulerian semiderivative for the domain Ω into a larger and


sufficiently smooth holdall D which will contain all the perturbations Ωt = Tt(V )(Ω)of Ω for t sufficiently small. The smoothness conditions that would normally occuron Ω will then occur on D. Once the expression for the semiderivative is obtained, Dcan be thrown away, leaving a final expression that does not require any smoothnessassumption on Ω.

4.1 Transport of Hk0 (Ω) by W k,∞-Transformations of RN

Recall Theorem 2.3 in Chapter 8 on the embedding of Hk0 (Ω) into Hk

0 (D). Asa consequence, the homogeneous Dirichlet boundary value problem in Ω is thefollowing: find y ∈ H1

0 (Ω) such that

∀ϕ ∈ H10 (Ω),

∫

Ω∇y ·∇ϕ dx =

∫

Ωf ϕ dx

is completely equivalent to the variational problem, find Y ∈ H10 (Ω; D) such that

∀Φ ∈ H10 (Ω; D),

∫

D∇Y ·∇Φ dx =

∫

Df Φ dx

and Y |Ω = y.Let k ≥ 1 and let T be a transformation of RN such that

T, T−1 ∈W k,∞(RN,RN).

Associate with T the map

ϕ -→ T (ϕ) def= ϕ T−1 : D(RN)→ Hk0 (RN).

By assumption on T , the norms ∥ϕ∥Hk and ∥T (ϕ)∥Hk = ∥ϕT−1∥Hk are equivalentand by density T extends to a linear bijection

ϕ -→ T (ϕ) def= ϕ T−1 : Hk0 (RN)→ Hk

0 (RN)

such that both T and T −1 are uniformly Lipschitzian. Given any bounded opensubset Ω of RN, the map

ϕ -→ TΩ(ϕ) def= T (ϕ)|Ω : Hk0 (Ω)→ Hk

0 (T (Ω))

is again a linear bijection such that both TΩ and T −1Ω are uniformly Lipschitzian.

Given a bounded open subset D of RN, further assume that T is such that

T (D) = D.

As a result T (RN \D) = RN \D and T (∂D) = ∂D and

TD : Hk0 (D)→ Hk

0 (D).


Let Ω be an open subset of RN such that Ω ⊂ D. Then T (Ω) ⊂ D, T−1(Ω) ⊂D, and H1

0 (Ω) and H10 (T (Ω)) can be identified with the subspaces H1

0 (Ω; D) andH1

0 (T (Ω);D) of H10 (D). Moreover,

∀ϕ ∈ D(Ω), e0(ϕ) T−1 = e0(ϕ T−1)

and the following diagram commutes:

Hk0 (Ω) e0→ Hk

0 (D)

↓ TΩ ↓ TD

Hk0 (T (Ω)) e0→ Hk

0 (D)

⇒ TD(Hk0 (Ω; D)) = Hk

0 (T (Ω);D).

4.2 Laplacian and Bi-LaplacianFor an arbitrary bounded open subset Ω of RN the bounded open holdall D can bearbitrarily chosen such that Ω ⊂ D. For instance, D can be a large enough openball. This will make it possible to throw on D any smoothness assumption thatwould normally occur on Ω and work with an arbitrary Ω. In this setup, D is usedas an intermediate step and can be thrown away once the semiderivative has beencomputed.

Given a velocity field

V ∈ C([0, τ ];W k,∞0 (D;RN)), (4.1)

the flow mapping Tt = Tt(V ) maps D onto itself and ∂D onto itself, and is equalto the identity on RN \D. So it transports H1

0 (D) onto itself.

Theorem 4.1. For k = 1, 2 and 0 ≤ t < τ

ϕ ∈ Hk0 (D) ⇐⇒ ϕ Tt(V ) ∈ Hk

0 (D),

ϕ ∈ Hk0 (Ω; D) ⇐⇒ ϕ Tt(V ) ∈ Hk

0 (Ω; D).

Hence for Ωt = Tt(Ω)

Hk0 (Ωt;D) =

ϕ T−1

t (V ) : ∀ϕ ∈ Hk0 (Ω; D)

. (4.2)

Proof. The second statement follows from the fact that Tt and its inverse T−1t are

both Lipschitz continuous and that, by the previous considerations or Lemma 6.2 inChapter 8, they transport sets of zero capacity onto sets of zero capacity: Tt(D\Ω) =D\Tt(Ω) = D\Ωt.

As Ω is compact and Ω ⊂ D, there exists τV > 0 such that for all t, 0 ≤ t < τV ,Ωt(V ) ⊂ D. In both cases the first eigenvalue is given by the Rayleigh quotient:

λ(Ωt(V )) = inf

akΩt

(ϕ,ϕ)∫Ωtϕ2 dx

: ∀ϕ ∈ Hk0 (Ωt), ϕ = 0

, k = 1, 2,


where

a1Ωt

(ϕ,ψ) def=∫

Ωt

∇ϕ ·∇ψ dx and a2Ωt

(ϕ,ψ) def=∫

Ωt

∆ϕ∆ψ dx.

In view of the previous constructions Hk0 (Ωt) can be replaced by Hk

0 (Ωt;D):

λ(Ωt(V )) = inf

akD(ϕ,ϕ)∫D ϕ2 dx

: ∀ϕ ∈ Hk0 (Ωt;D), ϕ = 0

, k = 1, 2. (4.3)

Note that if ϕ = 0 is a minimizer, then for any real number α = 0, αϕ is also aminimizer.

Theorem 4.2. Given k = 1, 2, let Ω be a bounded open subset of RN. Assume that

V ∈ C([0, τ ];W k,∞0 (D;RN))

for some bounded open domain D in RN such that Ω ⊂ D. There exists at least onenonzero solution ϕ ∈ Hk

0 (Ωt;D) to the minimization problem (4.3), λ(Ωt(V )) ≥λ(D) > 0, and

∀ϕ ∈ H10 (D),

∫

Dϕ2 dx ≤ λ(D)−1

∫

D|∇ϕ|2 dx,

∀ϕ ∈ H20 (D),

∫

Dϕ2 dx ≤ λ(D)−1

∫

D|∆ϕ|2 dx.

The solutions are completely characterized by the following variational equation:there exists ϕ ∈ Hk

0 (Ωt;D) such that

∀ψ ∈ Hk0 (Ωt;D), ak

D(ϕ,ψ) = λ(Ωt(V ))∫

Dϕψ dx (4.4)

or equivalently

∀ψ ∈ Hk0 (Ωt), ak

Ωt(ϕ,ψ) = λ(Ωt(V ))

∫

Ωt

ϕψ dx. (4.5)

Proof. Same technique as in the proof of Theorem 2.1 in Chapter 8.

The minimization problem can be rewritten on the unit sphere in L2(D) bynormalization:

λ(Ωt(V )) = inf∫

D|∇ϕ|2 dx : ∀ϕ ∈ H1

0 (Ωt;D), ∥ϕ∥L2(D) = 1

. (4.6)

The minimizing elements of (4.6) cannot be zero since the injection of H10 (D) into

L2(D) is compact (cf. Theorem 2.3 in Chapter 8).

Remark 4.1.One of the consequences of the use of the embedding of H1

0 (Ω) into H10 (D) is that

the characterization of the first eigenvalue on Ω requires only that Ω be a boundedopen subset of RN. It is not necessary to assume that Ω is Lipschitzian in order touse Rellich’s theorem, since D can be chosen sufficiently large (Ω ⊂ D) and smoothin order to contain all variations Ωt ⊂ D as t goes to zero. This technique will beexploited in the computation of the Eulerian semiderivative of λ(Ω).


G. Auchmuty [1]’s dual variational for this eigenvalue problem can be cho-sen as

µ(Ωt)def= inf

Lk(Ωt,ϕ) : ϕ ∈ Hk

0 (Ωt)

, (4.7)

Lk(Ωt,ϕ) def=12akΩt

(ϕ,ϕ)−[∫

Ωt

ϕ2 dx

]1/2

. (4.8)

By using the embedding of H10 (Ωt) into H1

0 (D), this problem can be rewritten as

µ(Ωt)def= inf

Lk(D,ϕ) : ϕ ∈ Hk

0 (Ωt;D)

, (4.9)

Lk(D,ϕ) def=12ak

D(ϕ,ϕ)−[∫

Dϕ2 dx

]1/2

. (4.10)


V ∈ C0([0, τ [ ; W k,∞0 (D;RN))

for some bounded open domain D in RN such that Ω ⊂ D. Then for 0 ≤ t < τ

µ(Ωt) = − 12λ(Ωt)

(4.11)


Ek(Ωt)def=

⎧⎪⎨

⎪⎩ϕ ∈ Hk

0 (Ωt;D) :ϕ is solution of (4.4) and∫

D|ϕ|2 dx

1/2

= 1/λ(Ωt)

⎫⎪⎬

⎪⎭. (4.12)

Proof. From the previous theorem the set E(Ωt) is not empty, and for any ϕ ∈ E(Ωt)

µ(Ωt) ≤ L(t,ϕ) = − 12λ(Ωt)

< 0.

Therefore the minimizers of (4.8) are different from the zero functions. For ϕ = 0,the functional L(t,ϕ) is differentiable and its directional derivative is given by

dL(t,ϕ;ψ) = akD(ϕ,ψ)− 1

∥ϕ∥L2(D)

∫

Dϕψ dx, (4.13)

and any minimizer of L(t,ϕ) is a stationary point of dL(t,ϕ;ψ); that is,

∀ψ ∈ Hk0 (Ωt;D), ak

D(ϕ,ψ)− 1∥ϕ∥L2(D)

∫

Dϕψ dx = 0. (4.14)

Therefore, ϕ is a solution of the eigenvalue problem with

λ′ =1

∥ϕ∥L2(D)⇒ − 1

2λ′ = µ(Ωt) ≤ −1

2λ(Ωt).

By minimality of λ(Ωt), we necessarily have λ′ = λ(Ωt) and this concludes the proofof the theorem.


We now turn to the computation of the Eulerian semiderivative dλ(Ω; V ) viadµ(Ω; V ) from the identity

λ(Ωt) = − 12µ(Ωt)

,

since whenever dµ(Ω; V ) exists

dλ(Ω; V ) =1

2µ(Ω)2dµ(Ω; V ) = 2λ(Ω)2 dµ(Ω; V ).

Assuming that V satisfies assumption (4.1), we use the function space parametri-zation of section 2.4 in conjunction with Theorem 2.1. From the characterization(4.2) of Hk

0 (Ωt;D), define the following new functional: for each ϕ ∈ H10 (D),

L(t,ϕ) def= L(D,ϕ T−1t (V ))

=12ak

D(ϕ T−1t (V ),ϕ T−1

t (V ))−∫

D|ϕ T−1

t (V )|2 dx

1/2

.

After a change of variables for k = 1 and ϕ ∈ H10 (D),

L(t,ϕ) =12

∫

DA(t)∇ϕ ·∇ϕ dx−

∫

DJt |ϕ|2 dx

1/2

,

A(t) = Jt DT−1t

∗DT−1t , Jt = det(DTt),

and for k = 2 and ϕ ∈ H20 (D)

L(t,ϕ) =12

∫

D|div (B(t)∇ϕ)|2 Jt dx−

∫

DJt |ϕ|2 dx

1/2

,

B(t) = DT−1t

∗DT−1t .

To apply Theorem 2.1, choose

X(t) def=ϕt def= ϕt Tt : ∀ϕt ∈ Ek(Ωt)

,

endowed with the weak topology of H10 (D). Assumption (H1) is clearly satisfied.

For all ϕ = 0, L(t,ϕ) is differentiable. For k = 1 and 0 = ϕ ∈ H10 (D),

∂tL(t,ϕ) =12

∫

DA′(t)∇ϕ ·∇ϕ dx− 1

∫D Jt |ϕ|2 dx

1/2

∫

D|ϕ|2 J ′

t dx,

and for k = 2 and 0 = ϕ ∈ H20 (D)

∂tL(t,ϕ) =∫

Ddiv (B′(t)∇ϕ) div (B(t)∇ϕ) Jt dx

+12

∫

D|div (B(t)∇ϕ)|2 J ′

t dx

− 1 ∫

D Jt |ϕ|2 dx1/2

∫

D|ϕ|2 J ′

t dx.


Hence assumptions (H2) and (H4) are satisfied. In t = 0 the above expressionssimplify. For k = 1 and ϕ ∈ E1(Ω),

∂tL(0,ϕ) =12

∫

DA′(0)∇ϕ ·∇ϕ dx− λ(Ω)

∫

D|ϕ|2 div V (0) dx

= −∫

Dε(V (0))∇ϕ ·∇ϕ dx +

∫

D

[12|∇ϕ|2 − λ(Ω)|ϕ|2

]div V (0) dx,

A′(0) = div V (0)I −DV (0)− ∗DV (0), ε(U) def=12

[DU + ∗DU ] ,

and for k = 2 and ϕ ∈ E2(ω),

∂tL(0,ϕ) =2∫

Ddiv (ε(V (0))∇ϕ) ∆ϕ dx

+∫

D

[12|∆ϕ|2 − λ(Ω)|ϕ|2

]div V (0) dx,

B′(0) = −DV (0)− ∗DV (0) = −2ε(V (0)).

For volume-preserving velocities, div V (0) = 0, and the above expressions reduce tothe first integral term.

In order to apply Theorem 2.1 it remains to check assumption (H3). This isthe first example where the set of minimizers is not necessarily unique and for whichwe have a complete description of the Eulerian semiderivative.


V ∈ C0([0, τ [ ; W k,∞0 (D;RN))

for some bounded open domain D in RN such that Ω ⊂ D. Then for k = 1

12dλ(Ω; V ) = inf

ϕ∈E1(Ω)−∫

Ωε(V (0))∇ϕ ·∇ϕ dx

+∫

Ω

[12|∇ϕ|2 − λ(Ω)|ϕ|2

]div V (0) dx,

ε(U) def=12

[DU + ∗DU ] ,

E1(Ω) def=

ϕ ∈ H1

0 (Ω) :−∆ϕ = λ(Ω)ϕ in D(Ω)′

and∫

Ω|ϕ|2 dx = 1

;

for k = 212dλ(Ω; V ) = inf

ϕ∈E2(Ω)

∫

Ω2 div (ε(V (0))∇ϕ) ∆ϕ dx

+∫

Ω

[12|∆ϕ)|2 − λ(Ω)|ϕ|2

]div V (0) dx,

E2(Ωt)def=

⎧⎨

⎩ϕ ∈ H20 (Ω) :

∆(∆ϕ) = λ(Ω)ϕ in D(Ω)′

and∫

Ω|ϕ|2 dx = 1

⎫⎬

⎭ .


In both cases λ(Ω) has a Hadamard semiderivative in the sense of Definition 3.2 (iii)in Chapter 9, which is continuous with respect to V (0) ∈W k,∞

0 (D;RN):

dλ(Ω; V ) = dHλ(Ω; V (0)).

Proof. As Ω is compact and Ω ⊂ D, there exists τV > 0 such that for all t, 0 ≤ t <τV , Ωt(V ) ⊂ D. Observe that

µ(D) = infϕ∈H1

0 (D)L(ϕ) ≤ inf

ϕ∈H10 (Ωt;D)

L(ϕ) = µ(Ωt).

By continuity of t -→ L(t,ϕ), for any ϕt ∈ X(t) and ϕ0 ∈ X(0),

L(t,ϕt) ≤ L(t,ϕ0) ⇒ lim supt0

L(t,ϕt) ≤ L(0,ϕ0) = µ(Ω)

⇒ µ(D) ≤ lim supt0

µ(Ωt) ≤ µ(Ω).

Therefore there exists τ ′, 0 < τ ′ ≤ τV , such that for all t, 0 ≤ t ≤ τ ′,

µ(D) ≤ µ(Ωt) ≤12µ(Ω) < 0 ⇒ 0 < λ(D) ≤ λ(Ωt) ≤ 2λ(Ω).

Since A(t)→ I and Jt → 1 as t goes to zero, there exist c > 0 and 0 < τ ′′ ≤ τ suchthat for all 0 ≤ t ≤ τ ′′,

L(t,ϕ) =12

∫

DA(t)∇ϕ ·∇ϕ dx−

∫

DJt |ϕ|2 dx

1/2

≥c∥∇ϕ∥2L2(D) − ∥ϕ∥L2(D)

≥ c∥∇ϕ∥2L2(D) − c′∥∇ϕ∥L2(D)

for some c′ > 0 by Poincare’s inequality. Therefore for all ϕt ∈ X(t),

c ∥∇ϕt∥2L2(D) − c′∥∇ϕt∥L2(D) ≤ L(t,ϕt) ≤ 0 ⇒ ∥∇ϕt∥L2(D) ≤ c′/c.

For any sequence tn 0, there exist subsequences µ, and ϕ0 ∈ H10 (Ω; D) such that

µn = µ(Ωtn)→ µ and ϕn = ϕtn ϕ0 in H10 (Ω; D)-weak,

λn = λ(Ωtn)→ λ = − 12µ

> 0.

Therefore, since A(tn)→ I, Jtn → 1, and ϕn converges in H1-weak,

∀ψ ∈ H10 (Ω; D),

12

∫

DA(tn)∇ϕn ·∇ψ dx = λn

∫

Dϕn ψ dx

⇒ ∀ψ ∈ H10 (Ω; D),

12

∫

DA(0)∇ϕ0 ·∇ψ dx = λ

∫

Dϕ0 ψ dx,

λn =∫

DJtn |ϕn|2 dx→

∫

D|ϕ0|2 dx ⇒

∫

D|ϕ0|2 dx = λ > 0,


and λ is an eigenvalue for the problem on Ω. But we know that

µ(D) ≤ lim supn→∞

µ(Ωtn) = − 12λ

, µ(D) ≤ − 12λ≤ − 1

2λ(Ω),

λ(D) ≤ λ ≤ λ(Ω) ⇒ λ = λ(Ω),

since λ(Ω) is minimal and hence ϕ0 ∈ X(0). This proves condition (H3)(i). To provecondition (H3)(ii) and complete the proof we first prove that the weakly convergentsubsequence strongly converges in H1

0 (Ω; D). By compactness of the injection ofH1

0 (D) into L2(D) (cf. Theorem 2.3 in Chapter 8)

ϕn ϕ0 in H10 (D)-weak ⇒ ϕn → ϕ0 in L2(D)-strong, (4.15)

and for some α > 0 independent of n,

α

∫

D|∇(ϕn − ϕ0)|2 dx ≤

∫

DA(tn)∇(ϕn − ϕ0) ·∇(ϕn − ϕ0) dx

=∫

DA(tn)∇ϕn ·∇ϕn − 2A(tn)∇ϕ0 ·∇ϕn

+ A(tn)∇ϕ0 ·∇ϕ0 dx

=∫

DλnJtn ϕn ϕn − 2A(tn)∇ϕ0 ·∇ϕn

+ A(tn)∇ϕ0 ·∇ϕ0 dx

→∫

Dλϕ0 ϕ0 − 2∇ϕ0 ·∇ϕ0

+∇ϕ0 ·∇ϕ0 dx = 0.

By the same technique as in section 2.4 when n→∞ and t 0

∂tL(t,ϕn) =12

∫

DA′(t)∇ϕn ·∇ϕn dx

− 1 ∫

D Jt |ϕn|2 dx1/2

∫

D|ϕn|2 J ′

t dx

→ 12

∫

DA′(0)∇ϕ0 ·∇ϕ0 dx

− 1 ∫

D |ϕ0|2 dx1/2

∫

D|ϕ0|2 div V (0) dx = ∂tL(0,ϕ0).

The case k = 2 is analogous.

If Ω is assumed to be of class C2k, the eigenvector functions belong to Hk0 (Ω)∩

H2k(Ω) and the volume expressions of the previous theorem can be expressed asboundary integrals as in section 2.5. In that case we can use formulae (2.49) and(2.50) to compute the partial derivative of Lt = L(Ωt,ϕ T−1

t ). For k = 1,

Lt =12

∫

Ωt

|∇(ϕ T−1t )|2 dx−

∫

Ωt

|ϕ T−1t |2 dx

1/2

.


From formula (2.49)

L′ def= ∂tLt|t=0 =12

∫

Γ|∇ϕ|2 V (0) · n dΓ− 1

2

∫Γ |ϕ|2 V (0) · n dΓ ∫

Ω |ϕ|2 dx1/2

+∫

Ω∇ϕ ·∇ϕ dx−

∫Ω ϕ ϕ dx

∫Ω |ϕ|2 dx

1/2 .

For ϕ ∈ E1(Ω), ϕ = 0 on Ω, −∆ϕ = λ(Ω)ϕ in Ω, and λ(Ω)∥ϕ∥L2(Ω) = 1,

L′ =∫

Γ

12|∇ϕ|2 V (0) · n dΓ +

∫

Ω∇ϕ ·∇ϕ dx− λ(Ω)

∫

Ωϕ ϕ dx

=∫

Γ

12|∇ϕ|2 V (0) · n dΓ +

∫

Γ

∂ϕ

∂nϕ dΓ.

But ϕ = 0 on Γ implies that ∇ϕ = ∂ϕ/∂n n on Γ, and from identity (2.50)

ϕ = −∇ϕ · V (0) ⇒ ϕ = −∂ϕ∂n

V (0) · n on Γ

⇒ L′ = −∫

Γ

12

∣∣∣∣∂ϕ

∂n

∣∣∣∣2

V (0) · n dΓ.

For k = 2 the computation of the partial derivative of Lt = L(Ωt,ϕ T−1t ) is

similar, with obvious changes:

Lt =12

∫

Ωt

|∆(ϕ T−1t )|2 dx−

∫

Ωt

|ϕ T−1t |2 dx

1/2

.

From formula (2.49)


∫

Γ|∆ϕ|2 V (0) · n dΓ− 1

2

∫Γ |ϕ|2 V (0) · n dΓ ∫

Ω |ϕ|2 dx1/2

+∫

Ω∆ϕ∆ϕ dx−

∫Ω ϕ ϕ dx

∫Ω |ϕ|2 dx

1/2 .

For ϕ ∈ E2(Ω), ϕ = 0 on Ω, ∆(∆ϕ) = λ(Ω)ϕ in Ω, and λ(Ω)∥ϕ∥L2(Ω) = 1,

L′ =∫

Γ

12|∆ϕ|2 V (0) · n dΓ +

∫

Ω∆ϕ∆ϕ dx− λ(Ω)

∫

Ωϕ ϕ dx

=∫

Γ

12|∆ϕ|2 V (0) · n dΓ +

∫

Γ

∂∆ϕ∂n

ϕ−∆ϕ∂ϕ

∂ndΓ.

But ϕ = 0 and ∂ϕ/∂n = 0 on Γ imply that ∇ϕ = 0 on Γ and

D2ϕ = (D2ϕn) ∗n on Γ ⇒ ∆ϕ = (D2ϕn) · n on Γ.


From identity (2.50) we have on Γ

ϕ = −∇ϕ · V (0) ⇒ ϕ = 0 on Γ

⇒ ∇ϕ = −∇(∇ϕ · V (0)) = −D2ϕV (0)− ∗DV (0)∇ϕ = −D2ϕV (0)

⇒ ∂ϕ

∂n= −D2ϕV (0) · n = −D2ϕn · n V (0) · n = −∆ϕV (0) · n,

L′ = −∫

Γ

12

|∆ϕ|2 V (0) · n dΓ.

The quantity D2ϕn · n is equal to ∂2ϕ/∂n2. Let bΩ be the oriented function asso-ciated with the domain Ω of class C4. It is C4 in a neighborhood of Γ. Therefore,in that neighborhood

ψdef= ∇ϕ ·∇bΩ, ψ|Γ =

∂ϕ

∂n,

∂

∂n

(∂ϕ

∂n

)=∂ψ

∂n= ∇ψ ·∇bΩ|Γ ,

∇ψ = ∇(∇ϕ ·∇bΩ) = D2bΩ∇ϕ+ D2ϕ∇bΩ,

∇ψ ·∇bΩ = D2bΩ∇ϕ ·∇bΩ + D2ϕ∇bΩ ·∇bΩ = D2ϕ∇bΩ ·∇bΩ,

∂2ϕ

∂n2 =∂

∂n

(∂ϕ

∂n

)= D2ϕ∇bΩ ·∇bΩ

∣∣Γ = D2ϕ

∣∣Γ n · n = ∆ϕ|Γ.

We summarize the results in the next theorem.

Theorem 4.5. Given k = 1, 2, let Ω be a bounded open subset of RN of class C2k.Assume that

V ∈ C0([0, τ [ ; W k,∞0 (D;RN))

for some bounded open domain D in RN such that Ω ⊂ D. Then for k = 1

dλ(Ω; V ) = infϕ∈E1(Ω)

−∫

Γ

∣∣∣∣∂ϕ

∂n

∣∣∣∣2

V (0) · n dΓ,

E1(Ω) def=

ϕ ∈ H1

0 (Ω) ∩H2(Ω) :−∆ϕ = λ(Ω)ϕ in Ω

and∫

Ω|ϕ|2 dx = 1

;

for k = 2

dλ(Ω; V ) = infϕ∈E2(Ω)

−∫

Γ

∣∣∣∣∂2ϕ

∂n2

∣∣∣∣2

V (0) · n dΓ,

E2(Ωt)def=

⎧⎨

⎩ϕ ∈ H20 (Ω) ∩H4(Ω) :

∆(∆ϕ) = λ(Ω)ϕ in Ω

and∫

Ω|ϕ|2 dx = 1

⎫⎬

⎭ .

In both cases λ(Ω) has a Hadamard semiderivative in the sense of Definition 3.2 (iii)in Chapter 9 which is continuous with respect to V (0) ∈ C0(D;RN):

dλ(Ω; V ) = dHλ(Ω; V (0)).


4.3 Linear ElasticityThe constructions and results of the previous section readily extend to the vectorialcase of linear elasticity: find U ∈ H1

0 (Ω)3 such that

∀W ∈ H10 (Ω)3,

∫

ΩCε(U)·· ε(W ) dx =

∫

ΩF · W dx (4.16)

for some distributed loading F ∈ L2(Ω)3 and a constitutive law C which is a bilinearsymmetric transformation of

Sym def=τ ∈ L(R3;R3) : ∗τ = τ

(L(R3;R3) is the space of all linear transformations of R3 or 3× 3-matrices) underthe following assumption.

Assumption 4.1.The constitutive law is a linear bijective and symmetric transformation C : Sym→Sym for which there exists a constant α > 0 such that Cτ ·· τ ≥ α τ ·· τ for allτ ∈ Sym.

For instance, for the Lame constants µ > 0 and λ ≥ 0, the special constitutivelaw Cτ = 2µ τ + λ tr τ I satisfies Assumption 4.1 with α = 2µ.

The associated bilinear form is

aΩ(U, W ) def=∫

ΩCε(U)·· ε(W ) dx,

where the unknown is now a vector function. To make sense of (4.16) we shall useKorn’s inequality on a larger bounded open Lipschitzian domain D, Ω ⊂ D:

∃cD > 0, ∀W ∈ H10 (D)3,

∫

D|W |2 dx ≤ cD

∫

D|ε(W )|2 dx.

In view of the considerations of the previous section, for a bounded open domain Ωand the velocity fields V ∈ C0([0, τ [ ; W 1,∞

0 (D;R3)), for all 0 ≤ t < τ

∃cD > 0, ∀W ∈ H10 (Ωt(V ))3,

∫

Ωt(V )|W |2 dx ≤ cD

∫

Ωt(V )|ε(W )|2 dx.

So for any F ∈ L2(D)3, the variational equation (4.16) with Ωt in place of Ω has aunique solution Ut in H1

0 (Ωt(V ))3.With the same assumptions the first eigenvalue is given by the Rayleigh quo-

tient

λ(Ωt(V )) = inf

aΩt(U, U)∫Ωt

|U |2 dx: ∀U ∈ H1

0 (Ωt)3, U = 0

.

In view of the previous constructions H10 (Ωt) can be replaced by H1

0 (Ωt;D):

λ(Ωt(V )) = inf

aΩt(U, U)∫Ωt

|U |2 dx: ∀U ∈ H1

0 (Ωt;D)3, U = 0

. (4.17)


Theorem 4.6. Let Ω be a bounded open subset of R3. Assume that

V ∈ C0([0, τ [ ; W 1,∞0 (D;R3))

for some bounded open Lipschitzian domain D in R3 such that Ω ⊂ D. There existsat least one nonzero solution U ∈ H1

0 (Ωt;D)3 to the minimization problem (4.17),λ(Ωt(V )) ≥ λ(D) > 0, and

∀W ∈ H10 (D)3,

∫

D|W |2 dx ≤ λ(D)−1

∫

D∥ε(W )∥2 dx.

The solutions are completely characterized by the following variational equation:there exists U ∈ H1

0 (Ωt;D)3 such that

∀W ∈ H10 (Ωt;D)3, aD(U, W ) = λ(Ωt(V ))

∫

DU · W dx (4.18)

or equivalently

∀W ∈ H10 (Ωt)3, aΩt(U, W ) = λ(Ωt(V ))

∫

Ωt

U · W dx. (4.19)

G. Auchmuty [1]’s dual variational principle for this eigenvalue problem canbe chosen as

µ(Ωt)def= inf

L(Ωt, U) : U ∈ H1

0 (Ωt)3

, (4.20)

L(Ωt, U) def=12aΩt(U, U)−

[∫

Ωt

|U |2 dx

]1/2

. (4.21)

By using the embedding of H10 (Ωt) into H1

0 (D), this problem can be rewritten as

µ(Ωt)def= inf

L(D, U) : U ∈ H1

0 (Ωt;D)3

, (4.22)

L(D, U) def=12aD(U, U)−

[∫

D|U |2 dx

]1/2

. (4.23)


V ∈ C0([0, τ [ ; W 1,∞0 (D;R3))

for some bounded open Lipschitzian domain D in R3 such that Ω ⊂ D. Then for0 ≤ t < τ ,

µ(Ωt) = − 12λ(Ωt)

(4.24)


E(Ωt)def=

⎧⎪⎨

⎪⎩U ∈ H1

0 (Ωt;D)3 :U is solution of (4.18) and∫

D|U |2 dx

1/2

= 1/λ(Ωt)

⎫⎪⎬

⎪⎭. (4.25)


From the identity

λ(Ωt) = − 12µ(Ωt)

,

if dµ(Ω; V ) exists, then dλ(Ω; V ) exists and

dλ(Ω; V ) =1

2µ(Ω)2dµ(Ω; V ) = 2λ(Ω)2 dµ(Ω; V ).

We again use the function space parametrization of section 2.4 in conjunction withTheorem 2.1. From the characterization (4.2) of H1

0 (Ωt;D), define the new func-tional: for each U ∈ H1

0 (D)3

L(t, U) def= L(D, U T−1t (V ))

=12aD(U T−1

t (V ), U T−1t (V ))−

[ ∫

D|U T−1

t (V )|2 dx

]1/2

.

After a change of variables for U ∈ H10 (D)3,

L(t, U) =12

∫

DCε(U T−1

t ) Tt·· ε(U T−1t ) Tt Jt dx−

[ ∫

DJt |U |2 dx

]1/2

,

where Jt = det(DTt) and

D(U T−1t ) Tt = D(U)(DTt)−1,

2 ε(U T−1t ) Tt = D(U)(DTt)−1 + ∗(DTt)−1 ∗D(U).

Defining the transformation

4 C(t) τ ··σ def= Cτ(DTt)−1 + ∗(DTt)−1 ∗τ

··σ(DTt)−1 + ∗(DTt)−1 ∗σ

,

the previous expression can be written in the following more compact form:

L(t, U) =12

∫

DC(t)D(U)··D(U) Jt dx−

[ ∫

DJt |U |2 dx

]1/2

.

To apply Theorem 2.1, choose

X(t) def=

U t def= Ut Tt : ∀Ut ∈ E(Ωt)

endowed with the weak topology of H10 (D)3. Assumption (H1) is clearly satisfied.

For all U = 0, L(t, U) is differentiable. For 0 = U ∈ H10 (D)3,

∂tL(t, U) =12

∫

DC ′(t)D(U)··D(U) Jt + C(t)D(U)··D(U) J ′

t dx

− 1 ∫

D Jt |U |2 dx1/2

∫

D|U |2 J ′

t dx,


where

4 C ′(t) τ ··σ =C τT ′(t) + ∗T ′(t) ∗τ ··σ(DTt)−1 + ∗(DTt)−1 ∗σ

+ Cτ(DTt)−1 + ∗(DTt)−1 ∗τ

·· σT ′(t) + ∗T ′(t) ∗σ ,

T (t) def= (DTt)−1, T ′(t) = −(DTt)−1DV (t) Tt.

Hence assumptions (H2) and (H4) are satisfied. In t = 0 the above expressionssimplify. For U ∈ E(Ω)

∂tL(0, U) =12

∫

DC ′(0)D(U)··D(U) + C ε(U)·· ε(U) div V (0) dx

− 1 ∫

D |U |2 dx1/2

∫

D|U |2 div V (0) dx

and for all U and W

2 C ′(0)D(U)··D(W ) =− C D(U) DV (0) + ∗DV (0) ∗D(U) ·· ε(W )− C ε(U)·· D(W ) DV (0) + ∗DV (0) ∗D(W ) .

This is another example of the application of Theorem 2.1 to a case where theset of minimizers is not a singleton and for which we have a complete descriptionof the Eulerian semiderivative.


V ∈ C0([0, τ [ ; W 1,∞0 (D;R3))

for some bounded open Lipschitzian domain D in R3 such that Ω ⊂ D. Then

12dλ(Ω; V ) = inf

U∈E(Ω)−∫

ΩC ′(0)D(U)··D(U) dx

+∫

Ω

[12C ε(U)·· ε(U)− λ(Ω)|U |2

]div V (0) dx

= infU∈E(Ω)

−∫

ΩC ε(U)·· D(U) DV (0) + ∗DV (0) ∗D(U) dx

+∫

Ω

[12C ε(U)·· ε(U)− λ(Ω)|U |2

]div V (0) dx,

E(Ω) def=

⎧⎨

⎩U ∈ H10 (Ω)3 :

−−→div (ε(U)) = λ(Ω)U in D(Ω)′

and∫

Ω|U |2 dx = 1

⎫⎬

⎭ .

λ(Ω) has a Hadamard semiderivative in the sense of Definition 3.2 (iii) in Chapter 9which is continuous with respect to V (0) ∈W 1,∞

0 (D;R3):

dλ(Ω; V ) = dHλ(Ω; V (0)).


If Ω is assumed to be of class C2, the eigenvector functions belong to H10 (Ω)3∩

H2(Ω)3 and the volume expressions of the previous theorem can be expressed asboundary integrals as in section 2.5. In that case we can use formulae (2.49) and(2.50) to compute the partial derivative of Lt = L(Ωt, U T−1

t ):

Lt =12

∫

Ωt

C ε(U T−1t )·· ε(U T−1

t ) dx−∫

Ωt

|U T−1t |2 dx

1/2

.

From formula (2.49)


∫

ΓC ε(U)·· ε(U) V (0) · n dΓ− 1

2

∫Γ |U |2 V (0) · n dΓ ∫

Ω |U |2 dx1/2

+∫

ΩC ε(U)·· ε(U) dx−

∫Ω U · U dx

∫Ω |U |2 dx

1/2 .

For U ∈ E(Ω), U = 0 on Γ, λ(Ω)∥U∥L2(Ω) = 1, and −−→div (ε(U)) = λ(Ω)U in Ω,

L′ =12

∫

ΓC ε(U)·· ε(U) V (0) · n dΓ +

∫

ΩC ε(U)·· ε(U)− λ(Ω) U · U dx

=∫

Γ

12C ε(U)·· ε(U) V (0) · n + [C ε(U)]n · U dΓ.

But U = 0 on Γ implies that DΓ(U) = 0, D(U) = [D(U)]n ∗n on Γ, and fromidentity (2.50)

U = −D(U)V (0) ∈ H10 (Ω)3 ⇒ U = −[D(U)]n V (0) · n on Γ

⇒ L′ = −∫

Γ[C ε(U)]n · [D(U)]n V (0) · n dΓ.

This expression can be rewritten only in terms of ε(U) as follows:

2 ε(U) = D(U)n ∗n + n ∗(D(U)n), 2 ε(U)n = D(U)n + (D(U)n) · n n,

2 ε(U)n · n = 2D(U)n · n,

[C ε(U)]n · D(U)n = 2 [C ε(U)]n · ε(U)n− [C ε(U)]n · n (ε(U)n) · n.

We summarize the results in the next theorem.

Theorem 4.9. Let Ω be a bounded open subset of R3 of class C2. Assume that

V ∈ C0([0, τ [ ; W 1,∞0 (D;R3))

for some bounded open Lipschitzian domain D in R3 such that Ω ⊂ D. Then

dλ(Ω; V ) = infU∈E(Ω)

−∫

Γ[C ε(U)]n · [D(U)]n V (0) · n dΓ,


−∫

Γ2 [C ε(U)]n · ε(U)n− [C ε(U)]n · n (ε(U)n) · n V (0) · n dΓ,

E(Ω) def=

⎧⎨

⎩U ∈ H10 (Ω)3 ∩H2(Ω)3 :

−−→div (ε(U)) = λ(Ω) U in Ω

and∫

Ω|U |2 dx = 1

⎫⎬

⎭ .

5. Saddle Point Formulation and Function Space Parametrization 551

For the special constitutive law Cτ = 2µτ + λtr τ I,


−∫

Γ

4 µ|[ε(U)]n|2 + (λ− 2µ) |tr ε(U)|2

V (0) · n dΓ.

λ(Ω) has a Hadamard semiderivative in the sense of Definition 3.2 (iii) inChapter 9 which is continuous with respect to V (0) ∈ C0(D;R3):

dλ(Ω; V ) = dHλ(Ω; V (0)).

5 Saddle Point Formulation and Function SpaceParametrization

5.1 An Illustrative ExampleLet Ω be a bounded open domain in RN with a smooth boundary Γ. Let y = y(Ω)be the solution of the Neumann problem

−y + y = f in Ω,∂y

∂n= 0 on Γ, (5.1)

where f is a fixed function in H1(RN). Associate with y(Ω) the objective function

J(Ω) =12

∫

Ω|y(Ω)− yd|2 dx, (5.2)

where yd is a fixed function in H1(RN).The solution of (5.1) coincides with the minimizing element of the following

variational problem:

infE(Ω,ϕ) : ϕ ∈ H1(Ω)

, (5.3)

E(Ω,ϕ) def=12

∫

Ω

|∇ϕ|2 + ϕ2 − 2 fϕ

dx. (5.4)

The minimizing element y of (5.4) is the solution in H1(Ω) of Euler’s equation

dE(Ω, y;ϕ) = 0, ∀ϕ ∈ H1(Ω), (5.5)

dE(Ω, y;ϕ) =∫

Ω[∇y ·∇ϕ+ yϕ− fϕ] dx, (5.6)

which is the variational equation for y.The objective function J(Ω) is a shape functional, and the solution of (5.1)

will be called the state. It is convenient to introduce the objective function

F (Ω,ϕ) def=12

∫

Ω|ϕ− yd|2 dx, (5.7)

which clearly expresses the dependence on Ω and ϕ. To sum up, we consider theobjective function

J(Ω) = F(Ω, y(Ω)

), (5.8)


where y = y(Ω) is the solution of

y ∈ H1(Ω), ∀ϕ ∈ H1(Ω), dE(Ω, y;ϕ) = 0. (5.9)

We wish to find an expression for the shape derivative dJ(Ω; V ).

5.2 Saddle Point Formulation

The basic approach is the one of control theory. Equation (5.1) (or in its variationalform (5.9)) is considered as a state constraint in the minimization problem. Weconstruct a Lagrangian functional by introducing a Lagrange multiplier function orthe so-called adjoint state ψ:

G(Ω,ϕ,ψ) = F (Ω,ϕ) + dE(Ω,ϕ;ψ). (5.10)

Then the objective function is given by

J(Ω) = minϕ∈H1(Ω)

supψ∈H1(Ω)

G(Ω,ϕ,ψ) (5.11)

since

supψ∈H1(Ω)

G(Ω,ϕ,ψ) =

F (Ω, y(Ω)), if ϕ = y(Ω),+∞, if ϕ = y(Ω).

(5.12)

In our example the Lagrangian G is convex and continuous with respect to thevariable ϕ and concave and continuous with respect to the variable ψ. Moreover,the space H1(Ω) is convex and closed. So the functional G has a saddle point ifand only if the saddle point equations have a solution (y, p) (cf. I. Ekeland andR. Temam [1]):

p ∈ H1(Ω), dG(Ω, y, p; 0,ψ) = 0, ∀ψ ∈ H1(Ω), (5.13)

y ∈ H1(Ω), dG(Ω, y, p;ϕ, 0) = 0, ∀ϕ ∈ H1(Ω). (5.14)

They are completely equivalent to

y ∈ H1(Ω), dE(Ω, y;ψ) = 0, ∀ψ ∈ H1(Ω), (5.15)

p ∈ H1(Ω), dF (Ω, y;ϕ) + d2E(Ω, y; p;ϕ) = 0, ∀ϕ ∈ H1(Ω), (5.16)

or

−y + y = f in Ω,∂y

∂n= 0 on Γ, (5.17)

−p + p + y − yd = 0 in Ω,∂p

∂n= 0 on Γ. (5.18)

System (5.15)–(5.16) has a unique solution in H1(Ω)×H1(Ω) which coincides withthe unique saddle point of G(Ω,ϕ,ψ) in H1(Ω)×H1(Ω).


5.3 Function Space ParametrizationWe have shown that the objective function J(Ω) can be expressed as a min maxof a functional G with a unique saddle point (y, p) which is completely character-ized by the variational equations (5.15)–(5.16). The same result holds when Ω istransformed into a domain Ωt = Tt(Ω) under the action of the velocity field V fort ≥ 0:

J(Ωt) = minϕ∈H1(Ωt)

supψ∈H1(Ωt)

G(Ωt,ϕ,ψ), (5.19)

where the saddle point (yt, pt) is completely characterized by

yt ∈ H1(Ωt), dE(Ωt, yt;ψ) = 0, ∀ψ ∈ H1(Ωt), (5.20)

pt ∈ H1(Ωt), dF (Ωt, yt;ϕ) + d2E(Ωt, yt; pt,ϕ) = 0, ∀ϕ ∈ H1(Ωt). (5.21)

We are looking for a theorem that will give an expression for the derivative ofa min sup with respect to a parameter t ≥ 0. However, in (5.19) the space H1(Ωt)depends on the parameter t. To get around this difficulty and obtain a min supexpression for J(Ωt) over spaces that are independent of t ≥ 0, we introduce thefollowing parametrization:

H1(Ωt) =ϕ T−1

t : ϕ ∈ H1(Ω)

(5.22)

since Tt and T−1t are diffeomorphisms. This parametrization does not affect the

value of the saddle point J(Ωt) but will change the parametrization of the func-tional G:

J(Ωt) = infϕ∈H1(Ω)

supψ∈H1(Ω)

G(Ωt,ϕ T−1t ,ψ T−1

t ). (5.23)

This parametrization is apparently unique to shape analysis. It amounts to intro-ducing the new Lagrangian functional for all ϕ and ψ in H1(Ω):

G(t,ϕ,ψ) = G(Tt(Ω),ϕ T−1

t ,ψ T−1t

). (5.24)

Our next objective is to find an expression for the limit

dg(0) = limg(t)− g(0)

t, (5.25)

where

g(t) = J(Ωt) = infϕ∈H1(Ω)

supψ∈H1(Ω)

G(t,ϕ,ψ). (5.26)

This will be done in the next section.Before closing, it is useful to look at the expression for G and the resulting

saddle point (yt, pt) in H1(Ω)×H1(Ω). By definition, G is given by the expression

G(t,ϕ,ψ) =12

∫

Ωt

|ϕ T−1t − yd|2 dx

+∫

Ωt

[∇(ϕ T−1

t ) ·∇(ψ T−1t )

+(ϕ T−1t )− f (ψ T−1

t )]

dx,

(5.27)

that the objective function J(Ω) can be expressed as a minmax of a functional G with a unique saddle point (y,p) which is completely characterized by the variational equations (5.15)–(5.16).


and its saddle point is the solution of the variational equations

yt ∈ H1(Ω) and ∀ψ in H1(Ω),∫

Ωt

[∇(yt T−1

t ) ·∇(ψ T−1t )

+ (yt T−1t )(ψ T−1

t )− f(ψ T−1t )

]dx = 0,

(5.28)

pt ∈ H1(Ω) and ∀φ in H1(Ω),∫

Ωt

[(yt T−1t − yd)(ϕ T−1

t ) +∇(yt T−1t ) ·∇(ϕ T−1

t )

+ (yt T−1t )(ϕ T−1

t )] dx = 0.

(5.29)

It is readily seen that (yt T−1t , pt T−1

t ) coincide with the saddle point (yt, pt) inH1(Ωt)×H1(Ωt):

yt = yt T−1t , pt = pt T−1

t ,

or equivalently

yt = yt Tt, pt = pt Tt. (5.30)

The solutions (yt, pt) can easily be interpreted as the solutions (yt, pt) on Ωt trans-ported back to the fixed domain Ω by the transformation Tt.

In view of this observation, we can rewrite expressions (5.27) to (5.29) onthe fixed domain Ω by using the coordinate transformation Tt. Expression (5.27)becomes

G(t,ϕ,ψ) =12

∫

Ω|ϕ− yd Tt|2Jtdx

+∫

ΩA(t)∇ϕ ·∇ψ + Jt [ϕψ − (f Tt)ψ] dx,

(5.31)

where for t > 0 small,

DTt = Jacobian matrix of Tt, (5.32)Jt = detDTt (since det DTt = |det DTt| for t ≥ 0 small), (5.33)

A(t) = Jt[DTt]−1 ∗[DTt]−1. (5.34)

Similarly the variational equations (5.28)–(5.29) reduce to

yt ∈ H1(Ω) and ∀ψ ∈ H1(Ω),∫

Ω

A(t)∇yt ·∇ψ + Jt

[ytψ − (f Tt),ψ

]dx = 0,

(5.35)

pt ∈ H1(Ω) and ∀ϕ ∈ H1(Ω),∫

Ω

A(t)∇pt ·∇ϕ+ Jt

[ptϕ+ (yt − yd Tt)ϕ

]dx = 0.

(5.36)


5.4 Differentiability of a Saddle Point with Respect to aParameter

Consider a functional

G : [0, τ ]×X × Y → R (5.37)

for some τ > 0 and sets X and Y . For each t in [0, τ ] define

g(t) = infx∈X

supy∈Y

G(t, x, y) (5.38)

and the sets

X(t) =

xt ∈ X : supy∈Y

G(t, xt, y) = g(t)

, (5.39)

Y (t, x) =

yt ∈ Y : G(t, x, yt) = supy∈Y

G(t, x, y)

. (5.40)

Similarly defineh(t) = sup

y∈Yinf

x∈XG(t, x, y) (5.41)

and the sets

Y (t) =

yt ∈ Y : infx∈X

G(t, x, yt) = h(t)

, (5.42)

X(t, y) =

xt ∈ X : G(t, xt, y) = infx∈X

G(t, x, y)

. (5.43)

In general we always have the inequality

h(t) ≤ g(t). (5.44)

To complete the set of notations, we introduce the set of saddle points

S(t) = (x, y) ∈ X × Y : g(t) = G(t, x, y) = h(t) , (5.45)

which may be empty.Our objective is to find realistic conditions under which the limit

dg(0) = limt0

g(t)− g(0)t

(5.46)

exists. A case of special interest is when G has a saddle point for all t in [0, τ ]. It canbe viewed as an extension of Theorem 2.1 in section 2.3 on the differentiability of amin with respect to a parameter. It is used when the functional to be minimized is afunction of the state, which is itself a function of the domain through the boundaryvalue problem. In that case the saddle point equations coincide with the “stateequation” and the “adjoint state equation” as illustrated in the previous section.The main advantage of this approach is to avoid the problem of the existence and


characterization of the derivative of the state xt with respect to t. In a controlproblem this would be the directional derivative of the state with respect to thecontrol variable. In particular, it is not necessary to invoke any implicit functiontheorem with possibly restrictive differentiability conditions. It will be sufficient tocheck two continuity conditions for the set-valued maps X(·) and Y (·). To completethis discussion we recall the following.

Lemma 5.1. Fix t in [0, τ ]. Then

∀(xt, yt) ∈ X(t)× Y (t), h(t) ≤ G(t, xt, yt) ≤ g(t), (5.47)

and if h(t) = g(t),X(t)× Y (t) = S(t). (5.48)

Proof. (i) If X(t) × Y (t) = ∅, there is nothing to prove. If there exist xt ∈ X(t)and yt ∈ Y (t), then by definition

h(t) = infx∈X

G(t, x, yt) ≤ G(t, xt, yt) ≤ supy∈Y

G(t, xt, yt) = g(t). (5.49)

(ii) If h(t) = g(t), then in view of (5.49), X(t) × Y (t) ⊂ S(t). Conversely, ifthere exists (xt, yt) ∈ S(t), then h(t) = G(t, xt, yt) = g(t) and by the definition ofX(t) and Y (t), (xt, yt) ∈ X(t)× Y (t).

It is important to keep in mind that identity (5.48) is always true when

h(t) = supy∈Y

infx∈X

G(t, x, y) = infx∈X

supy∈Y

G(t, x, y) = g(t)

but that S(t) may be empty.

Theorem 5.1 (R. Correa and A. Seeger [1]). Let the sets X and Y , the realnumber τ > 0, and the functional

G : [0, τ ]×X × Y → R

be given. Assume that the following assumptions hold:

(H1) S(t) = ∅, 0 ≤ t ≤ τ ;

(H2) for all (x, y) in [∪X(t) : 0 ≤ t ≤ τ × Y (0)] ∪ [X(0) × ∪Y (t) : 0 ≤ t ≤ τ]the partial derivative ∂tG(t, x, y) exists everywhere in [0, τ ];

(H3) there exists a topology TX on X such that for any sequence tn : 0 < tn ≤ τ,tn → t0 = 0, there exist x0 ∈ X(0) and a subsequence tnk of tn, and foreach k ≥ 1, there exists xnk ∈ X(tnk) such that

(i) xnk → x0 in the TX-topology, and(ii) for all y in Y (0),

lim inft0

k→∞

∂tG(t, xnk , y) ≥ ∂tG(0, x0, y); (5.50)

but that S(t) may be empty. Theorem 5.1 (R. Correa and A. Seeger [1]). Let the sets X and Y , the realnumber τ > 0, and the functional G:[0,τ]×X×Y →Rbe given. Assume that the following assumptions hold: (H1) S(t)=∅, 0≤t≤τ;(H2) for all (x,y) in [∪X(t) : 0 ≤ t ≤ τ×Y(0)]∪[X(0)×∪Y(t) : 0 ≤ t ≤ τ] the partial derivative ∂tG(t,x,y) exists everywhere in [0,τ]; (H3) there exists a topology TX on X such that for any sequence tn : 0 < tn ≤ τ,tn → t0 = 0, there exist x0 ∈ X(0) and a subsequence tnk of tn, and foreach k ≥ 1, there exists xnk ∈ X(tnk ) such that (i) xnk → x0 in the TX -topology, and(ii) for all y in Y (0), lim inf ∂tG(t, xnk , y) ≥ ∂tG(0, x0, y);t0 k→∞ (5.50)


(H4) there exists a topology TY on Y such that for any sequence tn : 0 < tn ≤ τ,tn → t0 = 0, there exist y0 ∈ Y (0) and a subsequence tnk of tn, and foreach k ≥ 1, there exists ynk ∈ Y (tnk) such that

(i) ynk → y0 in the TY -topology, and(ii) for all x in X(0),

lim supt0

k→∞

∂tG(t, x, ynk) ≤ ∂tG(0, x, y0). (5.51)

Then there exists (x0, y0) ∈ X(0)× Y (0) such that

dg(0) = infx∈X(0)

supy∈Y (0)

∂tG(0, x, y) = ∂tG(0, x0, y0)= sup

y∈Y (0)inf

x∈X(0)∂tG(0, x, y). (5.52)

Thus (x0, y0) is a saddle point of ∂tG(0, x, y) on X(0)× Y (0).

Proof. (i) We first establish upper and lower bounds to the differential quotient

∆(t)t

, ∆(t) def= g(t)− g(0).

Choose arbitrary x0 in X(0), xt in X(t), y0 in Y (0), and yt in Y (t). Then bydefinition

G(t, xt, y0) ≤ G(t, xt, yt) ≤ G(t, x0, yt),−G(0, xt, y0) ≤ −G(0, x0, y0) ≤ −G(0, x0, yt).

Add the above two chains of inequalities to obtain

G(t, xt, y0)−G(0, xt, y0) ≤ ∆(t) ≤ G(t, x0, yt)−G(0, x0, yt).

By assumption (H2), there exist θt, 0 < θt < 1, and αt, 0 < αt < 1, such that

G(t, xt, y0)−G(0, xt, y0) = t∂tG(θtt, xt, y0),G(t, x0, yt)−G(0, x0, yt) = t∂tG(αtt, x0, yt),

and by dividing by t > 0,

∂tG(θtt, xt, y0) ≤∆(t)

t≤ ∂tG(αtt, x0, yt). (5.53)

(ii) Define

dg(0) = lim inft0

∆(t)t

, dg(0) = lim supt0

∆(t)t

.

There exists a sequence tn : 0 < tn ≤ τ, tn → 0, such that

limn→∞

∆(tn)tn

= dg(0).

Thus (x0, y0) is a saddle point of ∂tG(0, x, y) on X(0) × Y (0).

dg(0) = inf sup ∂tG(0, x, y) = ∂tG(0, x0, y0) (5.51) (5.52) x∈X(0) y∈Y (0) = sup inf ∂tG(0, x, y). y∈Y (0) x∈X(0)

(H4) there exists a topology TY on Y such that for any sequence tn : 0 < tn ≤ τ,tn → t0 = 0, there exist y0 ∈ Y (0) and a subsequence tnk of tn, and foreach k ≥ 1, there exists ynk ∈ Y (tnk ) such that (i) ynk → y0 in the TY -topology, and(ii) for all x in X(0), lim sup ∂tG(t, x, ynk ) ≤ ∂tG(0, x, y0).t0 k→∞Then there exists (x0, y0) ∈ X(0) × Y (0) such that dg(0) = inf sup ∂tG(0, x, y) = ∂tG(0, x0, y0) (5.51) (5.52) x∈X(0) y∈Y (0) = sup inf ∂tG(0, x, y).y∈Y (0) x∈X(0) Thus (x0, y0) is a saddle point of ∂tG(0, x, y) on X(0) × Y (0).


By assumption (H3), there exist x0 ∈ X(0) and a subsequence tnk of tn foreach k ≥ 1, there exists xnk ∈ X(tnk) such that xnk → x0 in TX , and

∀y ∈ Y (0), lim inft0

k→∞

∂tG(t, xnk , y) ≥ ∂tG(0, x0, y).

Thus, from the first part of the estimate (5.53) for any y ∈ Y (0) and t = tnk ,

∂tG(θtnktnk , xnk , y) ≤ ∆(tnk)

tnk

and∂tG(0, x0, y) ≤ lim inf

k→∞∂tG(θtnk

tnk , xnk , y) ≤ limk→∞

∆(tnk)tnk

= dg(0).

Therefore,∃x0 ∈ X(0),∀y ∈ Y (0), ∂tG(0, x0, y) ≤ dg(0)

andinf

x∈X(0)sup

y∈Y (0)∂tG(0, x, y) ≤ sup

y∈Y (0)∂tG(0, x0, y) ≤ dg(0). (5.54)

By a dual argument and assumption (H4) we also obtain

∃y0 ∈ Y (0),∀x ∈ X(0), ∂tG(0, x, y0) ≥ dg(0),

dg(0) ≤ infx∈X(0)

∂tG(0, x, y0) ≤ supy∈Y (0)

infx∈X(0)

∂tG(0, x, y), (5.55)

and necessarily

inf x ∈ X(0) supy∈Y (0)

∂tG(0, x, y)

= dg(0) = dg(0) = supy∈Y (0)

infx∈X(0)

∂tG(0, x, y).

In particular, from (5.54) and (5.55),

supy∈Y (0)

∂tG(0, x0, y) = dg(0) = infx∈X(0)

inf ∂tG(0, x, y0)

and (x0, y0) is a saddle point of ∂tG(0, ·, ·).

Remark 5.1.In the applications this formulation of the theorem presents some definite technicaladvantages over its original version.

(i) From identity (5.52), ∂tG(0, ·, ·) has a saddle point with respect to X(0)×Y (0).

(ii) Another important feature is the use of subsequences in assumptions (H3)and (H4). This makes it possible to work with weak topologies in reflexiveBanach spaces and use the eventual boundedness of the sets of saddle points.


(iii) Finally, assumption (H2) and conditions (5.50) and (5.51) in (H3) and (H4)need only be checked on the family of saddle points at t = 0. For instance, thefirst part of assumptions (H3) and (H4) could be satisfied in H1(Ω)×H1(Ω).Yet, if the saddle points are smoother, say in H2(Ω) × H2(Ω), this extrasmoothness can be used to satisfy (H2) and (5.50) and (5.51) in (H3) and (H4).

5.5 Application of the TheoremOur example has a unique saddle point (yt, pt) for t ≥ 0 small, and we can usethe corollary to Theorem 5.1. The set-valued maps X and Y reduce to ordinaryfunctions

t -→ X(t) = yt, t -→ Y (t) = pt, (5.56)

and it is sufficient to show their continuity at t = 0 in H1(Ω). So we now checkassumptions (H1) to (H4).

Assume that V belongs to V1, that is, D1(RN,RN), and that f and y belongto H1(RN). Choose τ > 0 small enough such that

Jt = detDTt = |det DTt| = |Jt|, 0 ≤ t ≤ τ, (5.57)

and that there exist constants 0 < α < β such that

∀ξ ∈ RN, α|ξ|2 ≤ A(t) ξ · ξ ≤ β|ξ|2 and α ≤ Jt ≤ β. (5.58)

Since the bilinear forms associated with (5.35) and (3.34) are coercive, there existsa unique pair (yt, pt) solution of the system (5.35)–(5.36). Hence

∀t ∈ [0, τ ], X(t) = yt = ∅, Y (t) = pt = ∅. (5.59)

So assumption (H1) is satisfied. To check (H2) we use expression (5.31) and computefor ϕ and ψ in H1(Ω):

∂tG(t,ϕ,ψ)

=∫

Ω

12(ϕ− yd Tt)2div Vt − (ϕ− yd Tt)∇yd · VtJt

dx

+∫

ΩA′(t)∇ϕ ·∇ψ + div Vt(ϕψ − f Ttψ)− Jt∇f · Vtψ dx,

(5.60)

where

Vt(X) def= V (Tt(X)), A′(t) = (div Vt)I − ∗DVt −DVt, (5.61)

I is the identity matrix on RN, and DVt is the Jacobian matrix of Vt. By the choiceof V in D1(RN,RN), t -→ Vt and t -→ DVt are continuous on [0, τ ]. Moreover, f andyd belong to H1(RN). As a result expression (5.60) is well-defined and ∂tG(t,ϕ,ψ)exists everywhere in [0, τ ] for all ϕ and ψ in H1(Ω). This can be proved in manyways. For instance, we establish (5.60) for f and yd in D(RN). Then we showthat the affine map (f, yd) -→ ∂tG(·,ϕ,ψ) is continuous from H1(RN)×H1(RN) to


C1([0, τ ]). So it extends by uniform continuity to all (f, yd) in H1(RN)×H1(RN)and density of D(RN) in H1(RN). Assumption (H2) is satisfied.

To check assumptions (H3)(i) and (H4)(i), we first show that for any sequencetn ⊂ [0, τ ], tn → 0, there exists a subsequence of ytn, still denoted by ytn,such that

ytn y0 = y in H1(Ω)-weak,

ptn p0 = p in H1(Ω)-weak,

where (y, p) is the solution of system (5.15)–(5.16) or (5.17)–(5.18). By the choiceof τ satisfying condition (5.58), there exists a constant c > 0 such that

α∥yt∥H1(Ω) ≤ βc∥f∥L2(RN),

α∥pt∥H1(Ω) ≤ βc∥yt − yd∥L2(Ω).

So the pair yt, pt is bounded in H1(Ω) × H1(Ω) and there exist a subsequenceytn , ptn and a pair (z, q) in H1(Ω)×H1(Ω) such that

ytn z in H1(Ω)-weak and ptn q in H1(Ω)-weak.

The pair (z, q) can be characterized by going to the limit in the variational equations(5.35)–(5.36):

∫

Ω

A(tn)∇ytn ·∇ψ + Jtn

[ytnψ − (f Ttn)ψ

]dx = 0,

∫

Ω

A(tn)∇ptn ·∇ϕ+ Jtn

[ptnϕ+ (ytn − yd Ttn)ϕ

]dx = 0.

So we proceed as in section 2.4 and use Lemma 2.1 to obtain

∀ψ,

∫

Ω∇z ·∇ψ + zψ − fψ dx = 0,

∀ϕ,

∫

Ω∇q ·∇ϕ+ qϕ+ (z − yd)ϕ dx = 0.

By uniqueness (z, q) = (y, p). We now proceed as in section 2.4 and prove that

ytn → y in H1(Ω)-strong, ptn → p in H1(Ω)-strong

by the same argument. So assumptions (H3)(i) and (H4)(i) are satisfied for thestrong topology of H1(Ω). Finally, assumptions (H3)(ii) and (H4)(ii) are readilysatisfied in view of the strong continuity of (t,ϕ) -→ ∂tE(t,ϕ,ψ) and (t,ψ) -→∂tE(t,ϕ,ψ). In fact, it would have been sufficient to check assumption (H3) withH1(Ω)-strong and (H4) with H1(Ω)-weak.

So all assumptions of Theorem 5.1 are satisfied and

dJ(Ω; V ) =∫

Ω

12(y − yd)2div V − (y − yd)∇yd · V

dx

+∫

ΩA′(0)∇y ·∇p + div V (0), (yp− fp)−∇f · V (0)p dx, (5.62)


where (y, p) is the solution of (5.17)–(5.18) or, in variational form,

y ∈ H1(Ω), ∀ψ ∈ H1(Ω),∫

Ω∇y ·∇ψ + yψ − fψ dx = 0, (5.63)

p ∈ H1(Ω), ∀ϕ ∈ H1(Ω),∫

Ω∇p ·∇ϕ+ pϕ+ (y − yd)ϕ dx = 0. (5.64)

5.6 Domain and Boundary Expressions for the Shape GradientExpression (5.62) for the shape gradient is a volume or domain integral. For yd andf in H1(RN) it is readily seen that the map

V -→ dJ(Ω; V ) : D1(RN,RN)→ R (5.65)

is linear and continuous. So by Corollary 1 to the structure Theorem 3.6 in sec-tion 3.4 of Chapter 9, we know that for a domain Ω with a C2-boundary Γ, thereexists a scalar distribution g(Γ) in D1(Γ)′ such that

dJ(Ω; V ) = ⟨g(Γ), V · n⟩. (5.66)

We now further characterize this boundary expression. In view of the assumptionson f , yd, and Ω, the pair (y, p) is the solution in H2(Ω)×H2(Ω) of the system

−y + y = f in Ω,∂y

∂n= 0 on Γ, (5.67)

−p + p + (y − yd) = 0 in Ω,∂p

∂n= 0 on Γ. (5.68)

Similarly, for V in D1(RN,RN) the system

−div[A(t)∇yt

]+ Jty

t = Jtf Tt in Ω,∂yt

∂n= 0 on Γ, (5.69)

−div[A(t)∇pt

]+ Jtp

t + (yt − yd Tt)Jt = 0 in Ω,∂pt

∂n= 0 on Γ (5.70)

has a unique solution in H2(Ω)×H2(Ω) instead of H1(Ω)×H1(Ω). With this extrasmoothness we can use the formula

d

dt

∫

Ωt

F (t, x) dx

∣∣∣∣t=0

=∫

ΓF (0, x)V (0) · n dΓ +

∫

Ω

∂F

∂t(0, x) dx (5.71)

for a sufficiently smooth function F : [0, τ ]×RN → R. We easily obtain

∂tG(0,ϕ,ψ) =∫

Γ

12(ϕ− yd)2 +∇ϕ ·∇ψ + ϕψ − fψ

dΓ

+∫

Ω(ϕ− yd)ϕ+∇ψ ·∇ϕ+ ψϕ dx

+∫

Ω

∇ϕ ·∇ψ + ϕψ − f ψ

dx, (5.72)


whereϕ =

d

dtϕ T−1

t

∣∣∣∣t=0

= −∇ϕ · V (0) (5.73)

andψ =

d

dtψ T−1

t

∣∣∣∣t=0

= −∇ψ · V (0). (5.74)

Now substitute for (ϕ,ψ) the solution (y, p) of (5.63)–(5.64):

∂tG(0, y, p)

=∫

Γ

12(y − yd)2 +∇y ·∇p + yp− fp

V · n dΓ

+∫

Ω(y − yd)(−∇y · V ) +∇p ·∇(−∇y · V ) + p(−∇p · V ) dx

+∫

Ω∇y ·∇(−∇p · V ) + y(−∇p · V )− f(−∇p · V ) dx. (5.75)

We recognize that the second term is (5.64) with ϕ = −∇y · V and that the thirdterm is (3.63) with ψ = −∇p · V . So they are both zero, and finally

dJ(Ω; V ) =∫

Γ

12(y − yd)2 +∇y ·∇p + yp− fp

V · n dΓ. (5.76)

It must be emphasized that this last expression has been obtained under the as-sumption that both y and p belong to H2(Ω). We shall see later that shape gradientcan also be obtained by our technique for the finite element approximations of yand p. However, for piecewise linear elements, formula (5.76) fails since the finiteelement solutions yh and ph belong to H1(Ωh) but not to H2(Ωh). However, thedomain formula (5.62) will remain true. The crucial point is that for the continuousproblem, a smooth boundary plus f and yd in H1(RN) put the solution (y, p) inH2(Ω) × H2(Ω). However, the smoothness of the finite element solution (yh, ph)cannot be improved.

6 Multipliers and Function Space Embedding6.1 The Nonhomogeneous Dirichlet ProblemLet Ω be a bounded open domain in RN with a sufficiently smooth boundary Γ.Let y = y(Ω) be the solution of the nonhomogeneous Dirichlet problem

−∆y = f in Ω, y = g on Γ, (6.1)

where f and g are fixed functions in H1/2+ε(RN) and H2+ε(RN), respectively,for some arbitrary fixed ε > 0. Associate with the solution of (6.1) the objectivefunction

J(Ω) =12

∫

Ω|y(Ω)− yd|2 dx (6.2)

for some fixed function yd in H1/2+ε(RN) and some arbitrary fixed ε > 0. We wantto compute the derivative of J(Ω) with respect to Ω subject to the state equation

We shall see later that shape gradient can also be obtained by our technique for the finite element approximations of y and p. However, for piecewise linear elements, formula (5.76) fails since the finite element solutions yh and ph belong to H1(Ωh) but not to H2(Ωh). However, the domain formula (5.62) will remain true. The crucial point is that for the continuous problem, a smooth boundary plus f and yd in H1(RN) put the solution (y,p) in H2(Ω) × H2(Ω). However, the smoothness of the finite element solution (yh, ph) cannot be improved.

6. Multipliers and Function Space Embedding 563

system (6.1). Our objective is to transform this problem into finding the saddlepoint of a volume Lagrangian functional. This technique can be applied to otherboundary value problems with Dirichlet conditions.

6.2 A Saddle Point Formulation of the State EquationWhen g = 0 problem (6.1) is equivalent to a variational problem on H1

0 (Ω). Wheng = 0 the extra constraint φ = g makes the Sobolev space dependent on g. To getaround this difficulty, we introduce a Lagrange multiplier and the new functional

L(φ,ψ, µ) =∫

Ω(∆φ+ f)ψ dx +

∫

Γ(φ− g)µ dΓ (6.3)

for all ψ ∈ H2(Ω) and µ ∈ H1/2(Γ). This is a convex-concave functional with aunique saddle point (φ, ψ, µ) which is completely characterized by the equations

∆φ+ f = 0 in Ω, φ− g = 0 in Γ, (6.4)

∀φ ∈ H2(Ω),∫

Ω∆φ ψ dx +

∫

Γφµ dΓ = 0. (6.5)

The last equation characterizes ψ and µ:

∆ψ = 0 in Ω, ψ = 0 on Γ, (6.6)

µ =∂ψ

∂non Γ (6.7)

(cf., for instance, I. Ekeland and R. Temam [1, Prop. 1.6]). Of course, thisimplies that the saddle point is unique and given by

(φ, ψ, µ) = (y, 0, 0). (6.8)

The purpose of the above computation was to find out the form of the multi-plier µ,

µ =∂ψ

∂non Γ, (6.9)

in order to rewrite the previous functional as a function of two variables instead ofthree:

L(Ω,φ,ψ) =∫

Ω(∆φ+ f)ψ dx +

∫

Γ(φ− g)

∂ψ

∂ndΓ (6.10)

for (φ,ψ) in H2(Ω)×H2(Ω). It is also advantageous for shape problems to get ridof boundary integrals whenever possible. So noting that

∫

Γ(φ− g)

∂ψ

∂ndΓ =

∫

Ωdiv [(φ− g)ψ] dx, (6.11)

we finally use the functional

L(Ω,φ,ψ) =∫

Ω(∆φ+ f)ψ + (φ− g)∆ψ + (φ− g) · ψ dx (6.12)


on H2(Ω) × H2(Ω). It is readily seen that it has a unique saddle point (φ, ψ) inH2(Ω)×H2(Ω) which is completely characterized by the saddle point equations

∆φ+ f = 0 in Ω, φ = g on Γ, (6.13)

∆ψ = 0 in Ω, ψ = 0 on Γ. (6.14)

6.3 Saddle Point Expression of the Objective FunctionNow repeat the above constructions taking into account the objective function.First introduce the objective function

F (Ω,φ) =12

∫

Ω|φ− yd|2 dx (6.15)

and the new Lagrangian functional

G(Ω,φ,ψ) = F (Ω,φ) + L(Ω,φ,ψ).

Then it is easy to verify that

J(Ω) = minφ∈H2(Ω)

maxψ∈H2(Ω)

G(Ω,φ,ψ). (6.16)

The Lagrangian G(Ω,φ,ψ) is given by the expression

G(Ω,φ,ψ) =12

∫

Ω|φ− yd|2 dx

+∫

Ω(∆φ+ f)ψ + (φ− g)∆ψ + (φ− g) · ψ dx

(6.17)

on H2(Ω)×H2(Ω). It is readily seen that it has a unique saddle point(φ, ψ) whichis completely characterized by the following saddle point equations:

∆φ+ f = 0 in Ω, φ = g on Γ, (6.18)

∀φ ∈ H2(Ω),∫

Ω(φ− yd)φ+ ∆φψ + φ∆ψ + φ · ψ dx = 0. (6.19)

But the last equation is equivalent to

∀φ ∈ H2(Ω),∫

Ω[(φ− yd) + ∆ψ]φ dx +

∫

Γ

∂φ

∂nψ dΓ = 0 (6.20)

or

∆ψ + (φ− yd) = 0 in Ω, ψ = 0 on Γ (6.21)

by using the theorem on the surjectivity of the trace. In what follows, we shall usethe notation (y, p) for the saddle point (φ, ψ). As a result, we have

J(Ω) = minφ∈H2(Ω)

maxψ∈H2(Ω)

G(Ω,φ,ψ). (6.22)


We shall now use the above Lagrangian formulation combined with the ve-locity method to compute the shape gradient of J(Ω). Given a velocity field V inD1(RN,RN) and the parametrized domains Ωt = Tt(Ω),

J(Ωt) = minφ∈H2(Ω)

maxψ∈H2(Ω)

G(Ωt,φ,ψ). (6.23)

There are two methods for getting rid of the time dependence in the underlyingfunction spaces:

• the function space parametrization and

• the function space embedding.

In the first case, we parametrize the functions in H2(Ωt) by elements of H2(Ω)through the transformation

φ -→ φ T−1t = H2(Ω)→ H2(Ωt), (6.24)

where “” denotes the composition of the two maps, and we introduce the parame-trized Lagrangian,

G(t,φ,ψ) = G(Tt(Ω),φ T−1t ,ψ T−1

t ), (6.25)

on H2(Ω)×H2(Ω). In the function space embedding method, we introduce a largeenough domain, D, which contains all the transformations Ωt : 0 ≤ t ≤ t of Ωfor some small t > 0.

In this section, we use the function space embedding method with D = RN

andJ(Ωt) = min

Φ∈H2(RN)max

Ψ∈H2(RN)G(Ωt, Φ, Ψ). (6.26)

As can be expected, the price to pay for the use of this method is the fact that theset of saddle points

S(t) = X(t)× Y (t) ⊂ H2(RN)×H2(RN) (6.27)

is not a singleton anymore since

X(t) = Φ ∈ H2(RN) : Φ|Ωt = yt, (6.28)

Y (t) = Ψ ∈ H2(RN) : Ψ|Ωt = pt, (6.29)

where (yt, pt) is the unique solution in H2(Ωt)×H2(Ωt) to the previous saddle pointequations on Ωt

∆yt + f = 0 in Ωt, yt = g on Γt, (6.30)∆pt + (yt − yd) = 0 in Ωt, pt = 0 on Γt. (6.31)

We can now apply the theorem of R. Correa and A. Seeger [1], which saysthat under appropriate assumptions (to be checked in the next section)

dJ(Ω; V ) = minΦ∈X(0)

maxΨ∈Y (0)

∂tG(Ωt, Φ, Ψ)∣∣t=0. (6.32)


Since we have already characterized X(0) and Y (0), we need only compute thepartial derivative of

G(Ωt, Φ, Ψ) =∫

Ωt

12|Φ− yd|2 + (∆Φ + f)Ψ + (Φ− g)∆Ψ

+ (Φ− g) · Ψ

dx.(6.33)

If we assume that Ωt is sufficiently smooth, then

f, yd ∈ H1/2+ε(RN) and g ∈ H2+ε(RN) ⇒ p ∈ H5/2+ε(Ω), (6.34)

and we can choose to consider our saddle points S(t) in H5/2+ε(RN)×H5/2+ε(RN)rather than H2(RN) ×H2(RN). If the functions Φ and Ψ belong to H5/2+ε(RN),then

∂tG(Ωt, Φ, Ψ) =∫

Γt

12|Φ− yd|2 + (∆Φ + f)Ψ + (Φ− g)∆Ψ

+ (Φ− g) · Ψ

V · nt dΓt.(6.35)

This expression is an integral over the boundary Γt which will not dependon Φ and Ψ outside of Ωt. As a result, the min and the max can be dropped inexpression (6.32), which reduces to

dJ(Ω; V ) =∫

Γ

12(y − yd)2 + (∆y + f)p + (y − g)∆p

+ (y − g) · p

V · n dΓ.

(6.36)

However, p = 0 and y − g = 0 imply

∇p =∂p

∂nn and ∇(y − g) =

∂

∂n(y − g)n on Γ (6.37)

and, finally,

dJ(Ω; V ) =∫

Γ

12|g − yd|2 +

∂

∂n(y − g)

∂p

∂n

V · n dΓ, (6.38)

where

−∆y = f in Ω, y = g on Γ,

∆p + (y − yd) = 0 in Ω, p = 0 on Γ.

6.4 Verification of the Assumptions of Theorem 5.1As we have seen the computations of the shape gradient are both quick and easy.We now turn to the step by step verification of the assumptions of Theorem 5.1.Many of the constructions given below are “canonical” and can be repeated fordifferent problems in different contexts.


Let yd and f ∈ H1(RN) and g ∈ H5/2(RN) so that

X = Y = H3(RN). (6.39)

The saddle points S(t) = X(t)× Y (t) are given by

X(t) = Φ ∈ X : Φ|Ωt = yt, (6.40)Y (t) = Ψ ∈ Y : Ψ|Ωt = pt. (6.41)

The sets X(t) and Y (t) are not empty since it is always possible to constructa continuous linear extension

Πm : Hm(Ω)→ Hm(RN) (6.42)

for each m ≥ 1. For instance with m = 1 and a boundary Γ which is W 1,∞,see S. Agmon, A. Douglis, and L. Nirenberg [1, 2], and for m > 1, see V. M.Babic [1] and J. Necas [1]. Using this Πm, we define the following extension:

Πmt : Hm(Ωt)→ Hm(RN), (6.43)

Πmt (φ) = [Πm(φ Tt)] T−1

t . (6.44)

In what follows, m is fixed and equal to 3, so we shall drop the superscript mand define the extensions

Yt = Πtyt, Pt = Πtpt (6.45)

of yt and pt, respectively. Hence,

Yt ∈ X(t) and Pt ∈ Y (t)⇒ S(t) = ∅. (6.46)

So condition (H1) is satisfied. Condition (H2) follows from the assumptions onf , yd, and g. To check conditions (H3) and (H4), we need two general theorems,which can be used in various contexts and problems.

Theorem 6.1. For V ∈ D1(RN,RN) and Φ ∈ L2(RN),

limt0

Φ Tt = Φ and limt0

Φ T−1t = Φ in L2(RN). (6.47)

Proof. (i) The space D(RN) of continuous functions with compact support in RN

is dense in L2(RN). So given ε > 0, there exists Φε in D(RN) such that

∥Φ− Φε∥2L2 <ε2

maxJ−1

t : 0 ≤ t ≤ τ.

Hence,

∥Φ Tt − Φ∥ ≤ ∥Φε Tt − Φε∥+ ∥Φ Tt − Φε Tt∥+ ∥Φ− Φε∥. (6.48)

But

∀t ∈ [0, τ ],∫

RN|Φ Tt − Φε Tt|2 dx =

∫

RN|Φ− Φε|2J−1

t dx ≤ ε2.


So the last two terms in (6.48) are less than 2ε. It remains to evaluate the first termfor a fixed function Φε with compact support K in RN. Recall that, since Φε = 0on the boundary ∂K of K, Tt(K) = K for all t in [0, τ ] (use M. Nagumo [1]’stheorem twice as in the proof of Theorem 5.1 (i) in Chapter 4). Moreover, by thecompactness of K, Φε is uniformly continuous on RN and

∃δ > 0,∀x, y ∈ RN, |x− y| < δ =⇒ |Φε(y)− Φε(x)| <ε

m(K)1/2 .

However, Tt is also uniformly continuous on K and

∃η > 0,∀t, 0 ≤ t < η,∀x ∈ K, |Ttx− x| < δ.

By construction,supp (Φε Tt) = Tt(supp Φε) ⊂ K,

andΦε = 0 and Φε Tt = 0 outside of K.

Finally,∫

RN|Φε(Ttx)− Φε(x)|2 dx =

∫

K|Φε(Ttx)− Φε(x)|2 dx ≤ ε2,

and this implies that

∀ε > 0, ∃η > 0, ∀0 ≤ t ≤ η, ∥Φ Tt − Φ∥L2(RN) ≤ 3ε.

(ii) For the second part of (6.47), we make a change of variables and use theresult of part (i)

∫

RN|Φ T−1

t − Φ|2 dx =∫

RN|Φ− Φ Tt|2Jt dx ≤ ε2.

This completes the proof.

Corollary 1. Under the assumptions of Theorem 6.1 for m ≥ 1, V in Dm(RN,RN),and Φ ∈ Hm(RN),

limt0

Φ Tt = Φ and limt0

Φ T−1t = Φ in Hm(RN). (6.49)

Theorem 6.2. Under the assumptions of Corollary 1 to Theorem 6.1,

yt → y0 in Hm(Ω)-strong (resp., -weak) (6.50)

implies thatYt → Y0 in Hm(RN)-strong (resp., -weak).


Proof. The strong case is obvious. We prove the weak case for m = 0. By definition,

Yt = (Πyt) T−1t ,

and for all Φ in L2(RN), we consider∫

RNYtΦ dx =

∫

RN(Πyt) T−1

t Φ dx =∫

RNΠytΦ TtJt dx.

We have shown in Theorem 6.1 that

Φ Tt → Φ in L2(RN)-strong.

In addition, Jt → 1, and by linearity and continuity of Π,

Πyt → Πy in L2(RN)-weak.

Hence,

∀Φ ∈ L2(RN),∫

RNYtΦ dx→

∫

RNΠyΦ dx =→

∫

RNY0Φ dx.

This proves the weak convergence.

To satisfy condition (H3), we transform (yt, pt) on Ωt to (yt, pt) = (yt Tt, pt Tt) on Ω. The pair (yt, pt) is the transported pair of solutions from Ωt to Ω. It isthe unique solution in H1(Ω)×H1(Ω) of the system

−div [A(t)yt] = Jtf Tt in Ω, yt = g Tt on Γ, (6.51)

−div [A(t)pt] = Jt(yt − yd Tt) in Ω, pt = 0 on Γ, (6.52)

where

A(t) = Jt[DTt]−1∗[DTt]−1, Jt = |det DTt|, (6.53)

DTt is the Jacobian matrix of Tt, and ∗[DTt]−1 is the transpose of [DTt]−1.For sufficiently smooth domains Ω and vector fields V , the pair yt, pt is

bounded in H1(Ω) × H1(Ω) as t goes to zero. Since H1(Ω) is a Hilbert space,we can extract weakly convergent subsequences to some (y, p) in H1(Ω) ×H1(Ω).However, by linearity of the equation with respect to (yt, pt) and continuity of thecoefficients with respect to t, the limit point (y, p) will coincide with (y0, p0), sincethe system has a unique solution at t = 0. Then we go back to the equation foryt and y and show that the convergence is strong in H1(Ω). Finally, by using theregularity of the data and the classical regularity theorems, we show that (yt,pt)→(y,p) in H3(Ω)×H3(Ω).

For the verification of condition (H4), we go back to expression (6.35), whichcan be rewritten as a volume integral:

∂tG(Ωt, Φ, Ψ) =∫

Ωt

div[

12(Φ− yd)2 + (∆Φ + f)Ψ

+ (Φ− g)∆Ψ + (Φ− g) · Ψ]V

dx

(6.54)


for (Φ, Ψ) ∈ H3(RN)×H3(RN). Now introduce the map

(Φ, Ψ) -→F (Φ, Ψ)

=[12(Φ− yd)2 + (∆Φ + f)Ψ + (Φ− g)∆Ψ + (Φ− g) · Ψ

]V

: H3(RN)×H3(RN)→ (H1(RN))N .

It is bilinear and continuous. Finally, the map

(t, F ) -→∫

Γt

F nt dΓ =∫

Ωt

F dx =∫

Ω(div F ) TtJ

−1t dx (6.55)

from [0, τ ]×H1(RN) to R is continuous. Then

(t, Φ, Ψ) -→ ∂tG(Ωt, Φ, Ψ) =∫

Γt

F (Ψ, Ψ) · nt dΓt (6.56)

is continuous and condition (H4) is satisfied. This completes the verification of thefour conditions of Theorem 5.1.

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Index of Notation

Ah, 280As,h, 280

B(RN,RN), 125, 136B(Ω), 64B0(Ω), 64bA, 337BCb(D), 354, 381BCc

d(D), 325BCd(D), 299, 325BCb, 354BCd, 299Bk(RN,RN), 125Bk(Ω), 64Bk(Ω,Rm), 65Bk(RN,RN), 143, 150BPS(D), 266BV(D), 245BVloc(U), 245BX(D), 247, 325, 381

Ck (A), 280Ck b(A), 344capD, 430capx,r, 441C(Ω), 63Cb(D), 337Cc

d(D), 276C∞

c (Ω), 63Cd, 268Cd(D), 323Cc

d(D), 324Cc

d(E;D), 324Cc

d,loc(E;D), 324Cd(D), 268CD(x), 200

χΩ, 209C∞(Ω), 64C∞

0 (Ω), 64Ck(RN,RN), 125Ck

0 (RN,RN), 125C0(Ω), 63C0

b (D), 340Ck

0 (Ω), 64Ck

0 (Ω,Rm), 65Ck0 (RN), 202

Ck,1(RN,RN), 125Ck,1, 205Ck(Ω), 64Ck(Ω,Rm), 65Ck,λ(Ω), 65, 66Ck(Ω), 63Ck(Ω,Rm), 65Ck

c (Ω), 63Ck(RN), 205Ck(RN,RN), 143, 150Ck

0 (RN,RN), 143, 150Ck,1(RN,RN), 130Ck+1,1(RN), 130Ck(RN,RN), 143, 150C(λ,ω), 114Cb(D), 379Cb(E;D), 379Cb,loc(E;D), 379co X(D), 215crit (dA), 317C∞

0 (RN,RN), 125, 136

d2J(Ω; V ;W ), 506dA(x), 268d∗

A, 322

615

616 Index of Notation

∂bA, 347D(Ω), 63∂∗Ω, 220∂αf , 60∂∗Ω, 250df(x; v), 459dG , 125dHf(x; v), 459Diffk(RN,RN), 153dJΩ(f ; θ), 478Dk(Ω), 63Dℓ

Γ, 510

F0b (D), 340

F(B(RN,RN)), 125Fb(D), 340F(Bk(RN,RN)), 125F(Ck(RN,RN)), 125F(Ck,1(RN,RN)), 125F(C∞

0 (RN,RN)), 125F(D), 270F/G, 140Fk

0 , 124, 204F(Ck

0 (RN,RN))/G(Ω), 202Fk+1(RN), 130F(Θ), 124, 161

G(D), 276G(Ω0), 125G(Ω0), 138GΘ, 161

H, additive curvature, 74H, mean curvature, 74H, 81, 109H1

• (Ω; D), 225, 419H1

• (χ;D), 225H1

⋄ (Ω; D), 225, 260, 419H1

⋄ (χ;D), 225Hb(S), 343Hd, 72Hom(RN,RN), 153H(S), 273Hc(S), 278Hs, 73

I, interior, 220Ib(S), 343I(S), 273Ic(S), 278

Jb(S), 343J(S), 273Jc(S), 278

K, Gauss curvature, 74κi principal curvature, 74

L, 201Lc,r(O, D), 450LD, 201L(D, r,ω,λ), 254LD(x), 200, 472Lip L(D,RN), 472Lk

A, 479

M1(D), 245

N, 56#(A), 59# ∂A, 343

O, exterior, 220Oc,r(D), 442, 450Oc, 446Ω0, 220Ω1, 220Ω•, 220Ωf , 477Ωt(V ), 472Oω,λ,ν , 450

pA, 279pA(x), 284p∂A(x), 347ΠA, 279Π∂A, 344p∂A, 347

R, 56ρc

H(Ω2, Ω1), 276ρ([A]b, [B]b), 340ρ, 212

Index of Notation 617

Sing (∇bA), 344Sing (∇dA), 279Sk (A), 280

TD(X), 198TF F(Θ), 148, 151Θ, 124T f

t , 478

Uh(A), 280Us,h(A), 280

−→V k, 475V k

K , 475−→V m,k, 475V m,k

K , 475Vm,k

K , 201V fθ , 478

W k,c(RN,RN), 130W k,∞(RN,RN), 130

X(D), 214X(D, r,ω,λ), 255Xµ(D), 212X(RN), 214

Index

area function, 486Auchmuty

dual principle, 241, 242, 539

Bernoulli condition, 259Bouligand contingent cone, 194boundary

function, 484integral, 70, 94measure, 96measure theoretic, 220

bounded curvature, 299

Caccioppoli set, 252canonical density, 71, 95capacitary constraints, 429capacity

(r, c)-density condition, 441,443, 450, 451

strong(r, c)-density condition,441

thick, 441Wiener criterion, 441

Caratheodory set, 434Cartesian graph, 176chain rule, 465characteristic function, 209

strong convergence, 214weak convergence, 217

Clarke tangent cone, 200co-area formula, 251column

buckling, 4, 240compactivorous property, 276compliance, 11, 228composite material, 215

cone propertylocal uniform, 115uniform, 115

connectedcomponents, 59

connected space, 58continuity

Holder, 65Lipschitz, 65

convergence in measure, 248convex function

semiderivative, 467covariant

derivatives, 498partial derivatives, 499

cracks, 280b-cracks, 344

critical point, 317curvature

additive, 500bounded, 299, 354Gauss, 74locally bounded, 299mean, 74, 500principal, 74

cusp property, 115uniform, 115

densityWiener, 441

derivative, 458Frechet, 462Gateaux, 460Hadamard, 461

strong, 462weak, 462

619

620 Index

diffeomorphism, 153Ck,ℓ, 69

differentiableFrechet, 478Gateaux, 477

dilated setclosed, 280h-boundary, 280open, 280

Dirichlet boundary value problemclassical, 420overrelaxed, 420relaxed, 420

distancegeodesic, 58

distance function, 268oriented, 337W 1,p-topology, 292, 296, 350χΩ, 295χA, 294, 297

domainCk, 68C∞, 173connected, 58equi-Holderian, 88equi-Lipschitzian, 88Holderian, 87(k, ℓ)-Holderian, Ck,ℓ, 68integral, 504k-Lipschitzian, Ck,1, 68Lipschitzian, 87locally

Ck,ℓ, 87, 88Holderian, 87, 113Lipschitzian, 87, 113

path-connected, 58Sobolev, 374W ε,p(D), 252Wm,p, 374

dominating function, 81, 108, 113space, 81, 109

eigenvaluefirst, 535λA(Ω), 413

problem, 5, 240

embeddingHk

0 (Ω) into Hk0 (D), 62

embedding theoremconvex domains, 65path-connected domains, 66

Euler’s buckling load, 5, 240exterior

measure theoretic, 220

finite perimeterlocally, 247set, 247

flat cone property, 450free boundary, 259, 260

problem, 263, 521function

bounded variation, 245concave, 281continuous

(0,λ)-Holder, 65(k,λ)-Holder, 65Lipschitz, 65uniformly, 64

convex, 281locally

concave, 281convex, 281semiconcave, 281semiconvex, 281

semiconcave, 281semiconvex, 281support, 322uniformly Lipschitzian, 465

function spaceembedding, 565parametrization, 522, 553,

565fundamental form

firstaα, 74

second, 500bαβ , 74, 500D2b, 492, 500

third, 492cαβ , 74

Index 621

geodesic distance, 58Lipschitz domains, 67

geodesics, 166graph

locallyCk,ℓ, 89Lipschitzian, 92

group structure on P(D), 57, 58

Hausdorff measure, 72, 73H1(Ω)-norm function, 487holdall, 170, 472hole-free

set, 59homeomorphism, 153homogenization, 215, 219

inner product, 56double, 56

interiormeasure theoretic, 220

Laplace–Beltrami operator, 496level sets, 178, 251

Ck,ℓ, 75Lie bracket, 502linear elasticity, 546linear extension

operator, 93Lipschitz manifold, 317

metricCourant, 123, 202, 205Hausdorff, 270

complementary, 276pseudometric, 130right-invariant, 126semimetric, 129

microstructure, 215minimal surface problem, 12,

244

neighborhoodh-tubular, 280

normal derivative, 488nowhere dense, 286

objective functionalMumford–Shah, 32

orthogonal projection operator, 492

perimeter, 254perturbation of the identity, 148, 151,

183polar coordinates, 177Preface, xixprojection, 492

on A, 284on ∂A, 344, 347onto a set, 279

pseudometric, 130

Rayleighquotient, 5, 240, 241, 537

reduced boundary, 250regularization

Tikhonov, 32relaxed problem, 230representative

measure theoretic, 220nice, 210, 220open and closed, 221

saddle pointstructure, 232

semiderivative, 458Eulerian, 473

second-order, 506Gateaux, 459, 477Hadamard, 459, 469, 474perturbation

identity, 172velocity

method, 172, 473semimetric, 129set

boundary, 56Caccioppoli, 244Ck, 68closure, 56complement, 56convex, 223diameter, 72

622 Index

equi-Holderian, 88equi-Lipschitzian, 88finite perimeter, 244hole-free, 59interior, 56k-Lipschitzian, Ck,1, 68(k, ℓ)-Holderian, Ck,ℓ, 68level, 75Lipschitzian, 87locally

Ck,ℓ, 87–89convex, 381Holderian, 87, 113Lipschitzian, 87, 113semiconvex, 381strictly convex, 381

semiconvex, 381stable, 420

shapedifferentiable, 479

twice, 510function, 170

continuity, 202functional, 170, 202, 472

definition, 472gradient, 479semiderivative, 172variational principle, 521

skeleton, 344stable set, 420staircase, 292structure theorem, 479submanifold, 71surface

measure, 94, 96

tangent space, 148, 151tangent space to F(Θ), 148, 151tangential

calculus, 491divergence, 495gradient, 493Hessian matrix, 496Jacobian matrix, 495linear strain tensor, 495vectorial divergence, 496

topologyHausdorff, 270

complementary, 276, 277uniform metric, 340

transformFenchel, 322

transformations, 180, 193transmission problem, 11, 228, 235,

423transpose, 56transpose of a matrix, 6tubular norms, 355

uniform cone property, 254universe, 170, 472

velocity method, 171, 180viability condition, 195volume

function, 473, 481functional, 486integral, 482, 485

Wienerdensity, 441

Wiener criterion, 441

shapes and geometries: metrics, analysis, differential...

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