progress in nonlinear differential equations978-1-4612-1562...differential equations, and a large...
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Progress in Nonlinear Differential Equations and Their Applications Volume 37
Editor Haim Brezis Universite Pierre et Marie Curie Paris and Rutgers University New Brunswick, N.J.
Editorial Board Antonio Ambrosetti, Scuola Nonnale Superiore, Pisa A. Bahri, Rutgers University, New Brunswick Felix Browder, Rutgers University, New Brunswick Luis Cafarelli, Institute for Advanced Study, Princeton Lawrence C. Evans, University of California, Berkeley Mariano Giaquinta, University of Pisa David Kinderlehrer, Carnegie-Mellon University, Pittsburgh Sergiu Klainennan, Princeton University Robert Kohn, New York University P. L. Lions, University of Paris IX Jean Mawhin, Universite Catholique de Louvain Louis Nirenberg, New York University Lambertus Peletier, University of Leiden Paul Rabinowitz, University of Wisconsin, Madison John Toland, University of Bath
Bemard Dacorogna
Paolo Marcellini
Implicit Partial Differential Equations
Springer Science+Business Media, LLC
Bemard Dacorogna Department of Mathematics Ecole Polytechnique Federale de Lausanne 1015 Lausanne, Switzerland
Library of Congress Cataloging-in-Publication Data Dacorogna, Bemard, 1953-
Pao10 Marcellini Dipartimento di Matematica "u. Dini" Universita di Firenze 50134 Firenze,ltaly
Implicit partial differential equations I Bemard Dacorogna, Paol0 Marcellini.
p. cm. - (Progress in nonlinear partial differential equations ; v. 37)
Inc\udes bibliographical references and index. ISBN 978-1-4612-7193-2 ISBN 978-1-4612-1562-2 (eBook) DOI 10.1007/978-1-4612-1562-2 1. Differential equations, Nonlinear. I. MarcelJini, Paolo.
II. Title. III. Series. QA377.D331999 515' .323-dc21 99-38323
CIP
AMS Subject Classifications: 34A60, 34B99, 35G30, 35R70, 49J45, 49K15, 19K20, 49L25, 73C50
Printed on acid-free paper. © 1999 Springer Science+Business Media New York
Originally published by Birkhiiuser Boston in 1999
Softcover reprint of the hardcover 1 st edition 1999
AU rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher Springer Science+Business Media, LLC. except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dis similar methodology now known or hereafter developed is forbidden. The use of general descriptive names, trade names, trademarks, etc., in this publication, even if the former are not especially identified, is not to be taken as a sign that such names, as understood by the Trade Marks and Merchandise Marks Act, may accordingly be used freely by anyone.
ISBN 978-1-4612-7193-2 SPIN 19901625
Typeset by the authors in It\TEX.
987 6 543 2 1
Contents
Preface ix
Acknowledgments xi
1 Introduction 1 1.1 The first order case ........ 1
1.1.1 Statement of the problem . 1 1.1.2 The scalar case ...... 2 1.1.3 Some examples in the vectorial case 4 1.1.4 Convexity conditions in the vectorial case 8 1.1.5 Some typical existence theorems in the vectorial case . 9
1.2 Second and higher order cases ............ 10 1.2.1 Dirichlet-Neumann boundary value problem 10 1.2.2 Fully nonlinear partial differential equations . 12 1.2.3 Singular values . 13 1.2.4 Some extensions 14
1.3 Different methods . . . . . 15 1.3.1 Viscosity solutions 15 1.3.2 Convex integration 17 1.3.3 The Baire category method . 18
1.4 Applications to the calculus of variations . 20 1.4.1 Some bibliographical notes . 21 1.4.2 The variational problem 22 1.4.3 The scalar case ....... 23
vi Contents
1.4.4 Application to optimal design in the vector-valued case. 24 1.5 Some unsolved problems . . . . . 26
1.5.1 Selection criterion .... 26 1.5.2 Measurable Hamiltonians 26 1.5.3 Lipschitz boundary data . 27 1.5.4 Approximation of Lipschitz functions by smooth functions 27 1.5.5 Extension of Lipschitz functions
1.5.6 1.5.7 1.5.8 1.5.9
and compatibility conditions . . . . . . . . Existence under quasiconvexity assumption Problems with constraints Potential wells .... Calculus of variations ..
I First and Second Order PDE's
27 28 28 29 30
31
2 First Order Equations 33 2.1 Introduction.... 33 2.2 The convex case . . 34
2.2.1 The main theorem 34 2.2.2 An approximation lemma 36 2.2.3 The case independent of (x, u) . 40 2.2.4 Proof of the main theorem . 43
2.3 The nonconvex case . . . . . . . . . 47 2.3.1 The pyramidal construction 47 2.3.2 The general case . . 52
2.4 The compatibility condition . 56 2.5 An attainment result . 60
3 Second Order Equations 69 3.1 Introduction..................... 69 3.2 The convex case . . . . . . . . . . . . . . . . . . . 70
3.2.1 Statement of the result and some examples 70 3.2.2 The approximation lemma . . . . . . . . . 72 3.2.3 The case independent of lower order terms 73 3.2.4 Proof of the main theorem . 77
3.3 Some extensions .................. 81 3.3.1 Systems of convex functions . . . . . . . . 81 3.3.2 A problem with constraint on the determinant . 82 3.3.3 Application to optimal design ........ . 90
4 Comparison with Viscosity Solutions 95 4.1 Introduction....... 95 4.2 Definition and examples 97 4.3 Geometric restrictions. . . 100
Contents vii
4.4
4.3.1 Main results. . . . . . .. . ..... . 4.3.2 Proof of the main results. . . . . . . . Appendix . . . . . . . . . . . . . . ......... . 4.4.1 Subgradient and differentiability of convex functions . 4.4.2 Gauges and their polars. . . . . . . . . . . . . . . 4.4.3 Extension of Lipschitz functions . . . . . . . 4.4.4 A property of the sub and super differentials.
II Systems of Partial Differential Equations
5 Some Preliminary Results 5.1 Introduction ....................... . 5.2 Different notions of convexity ............. .
5.2.1 Definitions and basic properties (first order case) 5.2.2 Definitions and basic properties (higher order case) . . 5.2.3 Different envelopes. . ....
5.3 Weak lower semicontinuity .. 5.3.1 The first order case .. 5.3.2 The higher order case.
5.4 Different notions of convexity for sets 5.4.1 Definitions ............ . 5.4.2 The different convex hulls .... . 5.4.3 Further properties of rank one convex hulls 5.4.4 Extreme points . . . . . . . . . . . . . . ..
6 Existence Theorems for Systems 6.1 Introduction.............. 6.2 An abstract result . . . . . .
6.2.1 The relaxation property .. 6.2.2 Weakly extreme sets . . . . . . . .
6.3 The key approximation lemma . . . . . . . 6.4 Sufficient conditions for the relaxation property
6.4.1 One quasiconvex equation ...... . 6.4.2 The approximation property . . . . . . 6.4.3 Relaxation property for general sets
6.5 The main theorems . . . . . . . . . . . . . . . .
III Applications
7 The Singular Values Case 7.1 Introduction ................... . 7.2 Singular values and functions of singular values .
7.2.1 Singular values ............. .
· .100 · . 103 · .113 · . 113 · . 113 · . 115 · .117
119
121 · . 121 · . 121 · . 121 · . 124 · . 126 · . 127 · . 127 · . 129 · .130 · .130 · . 131 · . 135 · .138
141 · 141 .142 .142
· . 147 · . 148 · . 152 · . 152 · . 153
.154 · .157
167
169 · 169 · 171
· . 171
viii Contents
7.2.2 Functions depending on singular values 7.2.3 Rank one convexity in dimension two ..
.... 174
. ... 181 · .185 · . 186 · . 187
7.3 Convex and rank one convex hulls . . . . . . . 7.3.1 The case of equality ofthe Qi ••••••
7.3.2 The main theorem for general matrices 7.3.3 The diagonal case in dimension two .. 7.3.4 The symmetric case in dimension two ..
. ...... 193
8
9
7.4 Existence of solutions (the first order case). . . . 7.5 Existence of solutions (the second order case)
The Case of Potential Wells 8.1 Introduction . . . . . . .......... 8.2 The rank one convex hull 8.3 Existence of solutions . . .......... The Complex Eikonal Equation 9.1 Introduction . . . . . . . . . . . . . . · .... 9.2 The convex and rank one convex hulls · .... 9.3 Existence of solutions . . . . . . . . . · ....
IV Appendix
10 Appendix: Piecewise Approximations 10.1 Vitali covering theorems and applications
10.1.1 Vitali covering theorems . . . . . . . . . 10.1.2 Piecewise affine approximation .....
10.2 Piecewise polynomial approximation . . . . . . . 10.2.1 Approximation of functions of class eN . 10.2.2 Approximation of functions of class WN•oo .
References
Index
.195
.199
.200
205 .205 .206 .215
217 .217 .218 .222
223
225 .225 .225 .232
· .241 .242 .245
249
271
Preface
Nonlinear partial differential equations has become one of the main tools of modern mathematical analysis; in spite of seemingly contradictory terminology, the subject of nonlinear differential equations finds its origins in the theory of linear differential equations, and a large part of functional analysis derived its inspiration from the study of linear pdes.
In recent years, several mathematicians have investigated nonlinear equations, particularly those of the second order, both linear and nonlinear and either in divergence or nondivergence form. Quasilinear and fully nonlinear differential equations are relevant classes of such equations and have been widely examined in the mathematical literature.
In this work we present a new family of differential equations called "implicit partial differential equations", described in detail in the introduction (c.f. Chapter 1). It is a class of nonlinear equations that does not include the family of fully nonlinear elliptic pdes. We present a new functional analytic method based on the Baire category theorem for handling the existence of almost everywhere solutions of these implicit equations. The results have been obtained for the most part in recent years and have important applications to the calculus of variations, nonlinear elasticity, problems of phase transitions and optimal design; some results have not been published elsewhere.
The book is essentially self-contained, and includes some background material on viscosity solutions, different notions of convexity involved in the vectorial calculus of variations, singular values, Vitali type covering theorems, and the approximation of Sobolev functions by piecewise affine functions. Also, a comparison is made with other methods - notably the method of viscosity solutions and
x Preface
briefly that of convex integration. Many mathematical examples stemming from applications to the material sciences are thoroughly discussed.
The book is divided into four parts. In Part 1 we consider the scalar case for first (Chapter 2) and second (Chapter 3) order equations. We also compare (Chapter 4) our approach for obtaining existence results with the celebrated viscosity method. While most of our existence results obtained in this part of the book are consequences of vectorial results considered in the second part, we have avoided (except for very briefly in Section 3.3) vectorial machinery in order to make the material more readable.
In Part 2 we first (Chapter 5) recall basic results on generalized notions of convexity, such as quasiconvexity, as well as on some important lower semicontinuity theorems of the calculus of variations. Central existence results of Part 2 are in Chapter 6, where Nth order vectorial problems are discussed.
In Part 3 we study in great detail applications of vectorial existence results to important problems originating, for example, from geometry or from the material sciences. These applications concern singular values, potential wells and the complex eikonal equation.
Finally, in Part 4 we gather some nonclassical Vitali type covering theorems, as well as several fine results on the approximation of Sobolev functions by piecewise affine or polynomial functions. These last results may be relevant in other contexts, such as numerical analysis.
Acknowledgments
It has been possible to complete this book with the encouragement and help of many colleagues and friends. We refer in particular to Hai'm Brezis, Editor of the PNLDE Series who expressed his appreciation of our work upon several occasions.
We thank Andrea Dall' Aglio and Nicola Fusco, with whom we discussed some measure theory properties of sets and covering results, in particular, the proof of Theorem 10.3.
We wish to recall here that recently we dedicated an article, on the same subject of this book, respectively to the memory of Ennio De Giorgi and to Stefan Hildebrandt on his 60th birthday.
We benefitted from discussions and the encouragement of several other colleagues and friends; in particular Luigi Ambrosio, John Ball, Lucio Boccardo, Giuseppe Buttazzo, Italo Capuzzo Dolcetta, Pierre Cardaliaguet, Arrigo Cellina, Gui-Qiang Chen, Gianni Dal Maso, Francesco Saverio De Blasi, Emmanuele Di Benedetto, Craig Evans, Irene Fonseca, Wilfrid Gangbo, Nicolas Georgy, Enrico Giusti, Pierre Louis Lions, Anna Migliorini, Giuseppe Modica, Luca Mugnai, Stefan Milller, Fran~ois Murat, Giulio Pianigiani, Laura Poggiolini, Carlo Sbordone, Vladimir Sverak, Rabah Tahraoui, Giorgio Talenti, Chiara Tanteri, Luc Tartar. We thank all of them.
We received some help from Giuseppe Modica and Camil-Demetru Petrescu to format the latex file of this book; their help has been very useful. We also thank Ann Kostant, Executive Editor of Mathematics, for the excellent work developed by the staff of Birkhiiuser.
This research has been supported by the Troisieme Cycle Romand de Mathematiques, Fonds National Suisse, under the contract 21-50472.97, by the Ital-
xii Acknowledgments
ian Consiglio Nazionale delle Ricerche, Contracts 96.00176.01, 97.00906.01 and Ministero dell'Universita e della Ricerca Scientifica e Tecnologica. We thank our Institutions, i.e., the Department de Mathematiques at the Ecole Poly technique Federale de Lausanne, and the Dipartimento di Matematica "U.Dini" of the Universita di Firenze.
Bernard Dacorogna and Paolo Marcellini
Firenze and Lausanne, March 1999
Implicit Partial Differential Equations