chambolle cll submitted rev

8/2/2019 Chambolle CLL Submitted Rev

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A REMARK ON THE ANISOTROPIC OUTER MINKOWSKI CONTENT

ANTONIN CHAMBOLLE, STEFANO LISINI, AND LUCA LUSSARDI

Abstract. We study an anisotropic version of the outer Minkowski content of a closed

set in Rn. In particular, we show that it exists on the same class of sets for which the

classical outer Minkowski content coincides with the Hausdorff measure, and we give

its explicit form.

Keywords: outer Minkowski content, finite perimeter sets

2010 Mathematics Subject Classification: 28A75, 49Q15, 52A39

1. Introduction

As it is well known, the classical Minkowski content of a closed set S Rn is defined by

(1.1) M(S) := lim0+

|{x Rn : dist(x, S) }|2

whenever the limit in (1.1) exists and is finite; here | | denotes the Lebesgue measure in Rn.The quantity

Mmeasures the area of (n

1)-dimensional sets, and it is an alternative to

the more classical Hausdorff measure Hn1. With the role of surface measure, the Minkowskicontent turns out to be important in many problems arising from real applications: for instance

M is related to evolution problems for closed sets [1, 10, 13].Clearly, it poses as natural problems its existence and comparison with Hn1. Let us

mention some known results in this direction. In [9, p. 275] the author proves that M(S) existsand coincides with Hn1(S) whenever S is compact and (n 1)-rectifiable, i.e. S = f(K) forsome K Rn1 compact and f: Rn1 Rn Lipschitz. A generalization of this result iscontained in [4, p. 110]. Here, the authors consider countable Hn1-rectifiable compact sets

in Rn, i.e. sets which can be covered by a countable family of sets Sj , with j N, such that S0is Hn1-negligible and S

jis a (n

1)-dimensional surface in Rn of class C1, for any j > 0.

In this case, M(S) exists and coincides with Hn1(S) if a further density assumption on Sholds: more precisely there must exist > 0 and a probability measure on Rn satisfying

(B(x, r)) rn, for each r (0, 1) and for each x S, where B(x, r) is the open ballcentered in x of radius r. Counterexamples [4, p. 109] show that the countable rectifiability

is indeed not sufficient to ensure the existence of M.More recently [2], motivated by problems in stochastic geometry, a generalization of the

Minkowski content has been introduced, the so-called outer Minkowski content SM, which is1

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2 A. CHAMBOLLE, S. LISINI, AND L. LUSSARDI

defined by

(1.2) SM(E) := lim0+

|{x Rn : dist(x, E) } \ E|

, E Rn compact.

In [2] the authors investigate general conditions ensuring the existence of SM: in particular,they prove that if E is a set with finite perimeter and M(E) exists and coincides with theperimeter of E, then also SM(E) exists and coincides with the perimeter of E (in ).

Now, notice that the quantity which appears in the argument of the limit in (1.2) can be

rewritten as (provided the set E is nice enough)

1

(|E+ B(0, 1)| |E|).

We consider in this short note a variant of this content, which is an anisotropic outer

Minkowski content. The idea is to study the limit, as 0+, of

(1.3) 1

(|E+ C| |E|),

where C Rn is a closed convex body. It is standard that if E is convex, then |E+ C| is apolynomial in (of degree n) whose coefficient of the first degree term (see also Remark 3.6

below) is precisely the anisotropic perimeter

(1.4)

E

hC(E) dHn1 ,

where hC is the support function of C, defined by hC() = sup x Cx , and E the outernormal to E, see [11] for details. The convergence of (1.3) to (1.4) follows for convex sets E

and can be easily extended to (very) smooth sets.We show here two (expected) results: first, as a functional defined on sets, (1.3) -converges

to the natural limit (1.4) as 0.Second, we show in Theorem 3.4 that given any set for which the (classical) outer Minkowski

content equals the perimeter, then the limit of (1.3), as 0+, coincides with (1.4).The proof of Theorem 3.4 is quite technical, because we wanted to work under the only

assumption of the convergence of the classical content. We show that this convergence implies

that the boundary is flat enough in a relatively uniform way, so that the convergence of (1.3)

holds. It would be easy to adapt our proof to get, in Theorem 3.4, an if and only if, under

the additional assumption, though, that C is elliptic and regular in some appropriate sense,

we leave this to the reader.Eventually, we also deduce a -convergence result (see [5, 7] for details) for functionals of

the type

(1.5)1

ess supxC

u u(x) dxwhich coincides with (1.3) on characteristic functions of sets. The limit is (quite obviously)

given by hC(Du) (where the minus signs accounts for the fact that the outer normal was


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THE ANISOTROPIC OUTER MINKOWSKI CONTENT 3

appearing in (1.4), and not the inner normal which corresponds more naturally to the gradient

of the characteristic function E)

As a simple corollary, one, if wants, is able to approximate functionals of the typeE

(E) dHn1,

for a positively one-homogenous convex function : Rn [0, +) (and positive away from0). Indeed, it suffices to choose the convex body

C := {x Rn : x () }and apply our results.

The paper is organized as follows: in section 3 we define the setting and we state the

results, then in section 4 we prove the -convergence result for (1.5), and then the pointwise

convergence result for (1.3).

2. Notation and preliminaries

2.1. Notation. Let n 1 be integer. Given a measurable set A Rn, we will denote by |A|its Lebesgue measure. If k {0, . . . , n}, the k-dimensional Hausdorff measure of S Rn willbe denoted by Hk(S). We will use the notation x y for the standard scalar product in Rnbetween x and y, B(x, r) for the open ball of radius r centered in x. Finally, here convergence

in L1loc() means convergence in L1(B ) for any ball B.

We say that a sequence of sets Ej

R

n converges to E

R

n in L1loc() as j

+

, if Ej

converges to E in L1loc() as j +, where S denotes the characteristic function of theset S.

2.2. Geometric measure theory. In this paragraph we recall with some basic notions of

geometric measure theory we will need; for an exhaustive treatment of the subject we refer

the reader to [12].

Let n 1 be integer and let k N with k n. We say that S Rn is Hk-rectifiable ifS can be covered by a countable family of sets Sj, with j N, such that S0 is Hk-negligibleand Sj is a k-dimensional surface in R

n of class C1, for any j > 0.

Let E

R

n be a measurable set and

R

n be an open domain. We say that E has finite

perimeter in if the distributional derivative of E is a real Radon measure on ; we willdenote Per(E;) := |DE|(). The upper and lower n-dimensional densities of E at x arerespectively defined by

n(E, x) := lim supr0

|E B(x, r)|nrn

, n(E, x) := lim infr0

|E B(x, r)|nrn

,

where n is the volume of the (n 1)-dimensional unit disc (the diameter of the unit ball).If n(E, x) = n(E, x) their common value is denoted by n(E, x). For every t [0, 1] we


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define

(2.1) Et := {x Rn : n(E, x) = t}.

The essential boundary of E is the set E :=R

n

\ (E0

E1

). It turns out that ifE has finiteperimeter in , then Hn1(E\ E1/2) = 0, and Per(E; ) = Hn1(E ).

Moreover, one can define a subset of E1/2 as the set of points x where there exists a unit

vector E(x) such that:

E x

{y Rn : y E(x) 0}, as 0 in L1loc(Rn),

and which is referred to as the outer normal to E at x. The set where E(x) exists is called

the reduced boundary and is denoted by E. One can show that Hn1(E\ E) = 0,moreover, one has the decomposition DE = (E)Hn1 E.

3. Statement of the results

Let us assume that C is a closed convex body, that is, bounded and with 0 in its interior.

We denote its support function by hC() = supxCx , and its polar function is hC(x) :=suphC()1 x . It is well known, then, that both hC and hC are convex, one-homogeneousand Lipschitz functions, moreover C = {hC 1}.

By assumptions, there also exists a, b with 0 < a < b such that B(0, a) C B(0, b), inparticular, we have for all , x Rn

(3.1) a|| hC() b|| , 1b|x| hC(x)

1

a|x| .

Let Rn be an open domain. Given a Lebesgue measurable set E , we introducethe outer , C-Minkowski content,

(3.2) SM0,C(E;) :=1

(| (E+ C)| |E|) .

Actually, this definition is not very practical, since it can change drastically with Lebesgue-

negligible changes of the set E. For this reason, we introduce the functional, defined for a

measurable function u:

(3.3) F,C(u;) :=1

ess sup(xC)

u u(x) dxwhich takes values in [0, +]. Notice that one can check easily (using Fatous lemma) thatF,C is l.s.c. in L

1loc(). We then define

(3.4) SM,C(E;) := F,C(E; ) .It is also easy to check that the definition coincides with SM0,C on smooth sets, and in generalfor a measurable set E we have

SM,C(E; ) = min|EE|=0

SM0,C(E; ) = SM0,C(E1; ) = SM0,C( \ E0; )


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where E1 (resp., E0) is the set of points of Lebesgue density 1 (resp., 0) in (see (2.1)),

and denotes the symmetric difference. Eventually, one can check easily that F satisfies ageneralized coarea formula [14, 6]: for any function u L1loc(),

(3.5) F,C(u; ) =

SM,C({u > s}; ) ds .

Moreover, one can show that for any measurable sets, E and F,

SM,C(E F; ) + SM,C(E F; ) SM,C(E; ) + SM,C(F; )from which it follows that F is convex on L

1loc(), see [6] for details.

Before to state our results we say that a family of functionals G defined on the Lebesgue

measurable subsets ofRn -converges to G in L1loc() as 0 if for any Lebesgue measurable set E, and for any family of Lebesgue measurable sets

E

E in L1loc() as

0, we have

G(E) liminf0

G(E) ,

for any Lebesgue measurable set E, there exist a family of Lebesgue measurable setsE such that E E in L1loc() as 0 and

lim sup0

G(E) G(E).

In the same way, we say that a family of functionals F defined on L1loc() -converges to

F in L1loc() as 0 if for any u L1loc(), and for any family of elements of L1loc() such that u u in

L1loc() as 0, we have

F(u) lim inf0

F(u) ,

for any u L1loc(), there exist a family of elements of L1loc() u such that u uin L1loc() as 0 and

limsup0

F(u) F(u).

We will show the following result:

Theorem 3.1. As 0, SM,C and SM0,C converge to

(3.6) PerhC(E) :=

EhC(E(x)) dHn1(x) if E has finite perimeter,

+ elsein L1loc(), where here, we let E(x) be the outer

1 normal to E at x. Moreover, if {E}>0are sets with locally finite measure and sup>0 SM,C(E; ) < , then, up to subsequences,E converge to some set E in L

1loc().

1Observe that with this classical but not so natural choice, we have PerhC (E) =hC(DE).


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In particular, we deduce from [6]:

Corollary 3.2. As 0, F,C converges to

T VC(u;) :=

hC(Du) if u BV() ,+ else

in L1loc(), where hC(Du) stands for hC dDu

d|Du|

d|Du| [8]. Moreover, if {u}>0 are

functions in L1loc() with sup F,C(u; ) < , then {u}>0 is precompact in L1loc().

For any measurable set E we can also consider

M,C(E; ) : = (SM,C(E; ) + SM,C( \ E;))/2.From Theorem 3.1 the following Corollary follows easily:

Corollary 3.3. As 0, M,C converges to (PerhC (E) + PerhC ( \ E))/2 in L1loc().

Concerning the pointwise convergence of SM0,C, we also have the following interestingresult, from which the -lim sup inequality in Theorem 3.1 follows in a straightforward way.

Theorem 3.4. Assume that the set E is a finite-perimeter set such that

(3.7) lim0

SM0,B(0,1)(E; ) = Per(E; ) .Then,

(3.8) lim0SM

0,C(E;) = PerhC(E; ) .

Remark 3.5. The sets which satisfy (3.7) are studied in [2]. A sufficient condition is that the

Minkowski content of the reduced boundary coincides with its (n 1)dimensional measure,that is,

lim0

|{x : dist(x, E) }|2

= Per(E; ) .

The proof is quite elementary (see [2, Thm 13]): in that case, we can introduce, for x ,the signed distance function

dE(x) := dist(x, E) dist(x, \ E)

and we have that (thanks to the co-area formula)

1

2|{x : dist(x, E) }| 1

2|{x : |dE(x)| }|

=1

2

Per({dE < s}; ) ds = 12

11

Per({dE < s}; ) ds .

In particular, since

Per(E; ) lim inf0

Per({dE < s}; )


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for all s, by Fatous lemma, we find that also

(3.9) lim0

1

2|{x : |dE(x)| }| = Per(E; ),

while (for the same reasons)

(3.10)Per(E; ) liminf

0

1

|{x : 0 < dE(x) }|,

Per(E; ) liminf0

1

|{x : 0 < dE(x) }|.

It follows from (3.9) and (3.10) that the inequality in (3.10) must in fact be an equality, and

the lim inf a lim. In particular, we deduce (3.7).

Remark 3.6. In case E is a convex body, then it is well known that (see [11])

|E+ C

|=

|E

|+ PerhC (E; ) + O(

2) .

Also, if E is compact and rectifiable (that is, included in the image of a Lipschitz map

from Rn1 to Rn), and Hn1(E\ E) = 0, then the Minkowski content coincides with theperimeter, see [9, Thm 3.2.39 p. 275]. It is easy to build examples, though, where this is not

true and still, (3.8) holds, see again [2].

As before, for any measurable set E we letM0,C(E; ) := (SM0,C(E; ) + SM0,C( \ E;))/2.

Then the following pointwise convergence result holds.

Theorem 3.7. Assume that the set E is a finite-perimeter set such that

lim0

M0,B(0,1)(E; ) = Per(E; ) .Then,

lim0

M0,C(E; ) = (PerhC (E; ) + PerhC ( \ E;))/2 .In particular, we get

lim0

M0,C(E; ) =

E

hC(E(x)) + hC(E(x))2

dHn1(x).

4. Proof of the results

4.1. Proof of Theorem 3.1. First, the compactness statement is proved as follows. Noticethat for any E measurable we can rewrite F,C(E; ) as

(4.1) F,C(E;) :=1

ess sup(xC)

|E E(x)| dx.

By assumption the family {E}>0 satisfies F,C(E; ) c for some c > 0. Let > 0 andlet

= {x B(0, 1/) : d(x, ) > };


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of course we have . Since B(0, a) C, there exists r > 0 such that for any Rnwith || 1 and any < 0, for 0 sufficiently small, it holds, using the form (4.1) for F,C,

|

E(x + r)

E(x)

||| dx F,C(

E; ) cand thus

sup 0 and

function u a functionu =

kZ

sk{sk+1u>sk},

where sk (k, (k + 1)) is a level appropriately chosen so thatSM,C({u > sk}; ) (1 + sup

>0F,C(u;))/.

Then, the previous compactness result (and a diagonal argument) shows that there exists a

a positive infinitesimal sequence k such that u1/nk converges to some u

1/n in L1loc(), for all

n 1. Since u1/mk u1/nk 2/ min{m, n} and u1/m u1/n 2/ min{m, n} for allm,n,k, we easily deduce that (up to a subsequence), there exists u such that uk

u in

L1loc().

In order to prove the convergence, we must show that for any E,

if E E in L1loc(), then(4.2) PerhC (E; ) lim inf0 SM,C(E; ) ,

and that there exists E E with(4.3) limsup

0SM,C(E; ) PerhC(E; ).

As it is standard that one can approximate any set E with finite perimeter by meansof (almost) smooth sets such that Per(Ek; ) Per(E; ) (for instance, minimizers ofPer(F; ) + k|EE|) then (4.3) will follow, using a diagonal argument and Remark 3.5,from Theorem 3.4 (which we will prove later on).

Hence, we focus on the proof of (4.2). Let us introduce the anisotropic (essential) distance

function to a set E:

distC(x, E) := ess infyE

hC(x y) .


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(Equivalently, this is the hC-distance to the set E1 of points where the Lebesgue density of E

is 1, or to the complement of E0.) Then, distC(x, E) < if and only if there exists a set of

positive measure in E of points y with hC(xy) < , or, in other words, such that xy C,which is equivalent to say that x E1 + C. In particular, it follows that

(E1 + C\ E1) = {x \ E1 : distC(x, E) < }.On the other hand, if one lets d(x) := distC(x, E), it is standard that d is Lipschitz and that

hC(d) = 1 a.e. in {d > 0}, and 0 a.e. in {d = 0} E1. The proof follows the same lines asin [3]. First, for any x, y , if > 0, one can find a set with positive measure in E of pointy with d(y) hC(y y) d(y) + . Then, for these points,(4.4) d(x) d(y) hC(x y) hC(y y) + hC(x y) + and sending to zero and using (3.1), it follows that d is Lipschitz. Moreover, d = 0 a.e.

in {d = 0}. Now, from (4.4) we also see that d(x + tz) d(x) thC(z) for all z; therefore, if

d is differentiable at x it follows that d(x) z 1 for all z C, hence hC(d(x)) 1.We show the reverse inequality for points x where d(x) > 0: for such a point, there exists

y E1 with d(x) = hC(x y). For each x (y, x] (which means that x = y and x lies onthe line segment with extreme points y and x), one has d(x) = hC(x

y) > 0, otherwisethere would exist y with hC(x

y) < hC(x y), but then, it would follow thathC(x y) hC(x x) + hC(x y) < hC(x x) + hC(x y) = hC(x y)

since x (y, x], a contradiction. It follows that for z = x y, t (0, 1),d(x

tz) = hC(x

tz

y) = (1

t)d(x),

and if in addition x is a point of differentiability, it follows that

d(x) z = d(x) = hC(z).But since hC(d(x)) 1 and z/hC(z) C, it follows that hC(d(x)) = 1. If moreover hCis differentiable as well in x y, we find in addition that d(x) = hC(x y). If hC isdifferentiable in d(x), we find that y = x d(x)hC(d(x)) and in particular, in that case,the projection y must be unique. For a general convex set C this might not be the case, even

at points of differentiability.

Now, let {E}>0 be a family of sets, with |(EE) | 0 as 0, and assume thatlim inf0 SM,C(E; ) < +. We consider a subsequence Ek := Ek such that this lim infis in fact a limit. We have

((E1k + kC) \ E1k) {x : 0 < distC(x, Ek) < k}(the difference being the possible set of points x E1k with distC(x, Ek) = 0). It follows,letting dk(x) := max{distC(x, Ek)/k, 1},

SMk,C(Ek; ) 1

k

{0


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In particular, (dk)k1 have equibounded total variation: we may assume that a subsequence

(not relabelled) converges to some limit d, with values in [0, 1], in L1loc(). (And, in fact, we

may even assume that the convergence is pointwise, out of a negligible set.)

By assumption, |{0 < dk < 1}| ck, in particular we deduce that d {0, 1} a.e. in .Observe also that (Xc denoting here the complement of X in , that is \ X), if B is a ballin Rn,

|B {dk < k}E| = |B {dk < k} Ec| + |B {dk < k}c E|= |B {dk < k} E1k Ec| + |B {dk < k} (E1k)c Ec|+ |B {dk < k}c E1k E| + |B {dk < k}c (E1k)c E|

|B (EkE)| + |{dk < k} \ E1k|.

As |B (EkE)| 0 (by assumption), while |{dk < k} \ E1k | ck, we deduce that{dk < k} E in L1loc(). It follows that {d = 1} = \ E, hence dk 1 E as k .

Thanks to Reshetniaks lower semicontinuity Theorem, we deduce that

hC(DE) lim infk

hC(dk(x)) dx limk

SMk,C(Ek; ) .

Since hC(DE) = PerhC(E), (4.2) follows.

As already mentioned, the proof of (4.3) will follow from Theorem (3.4), which is given in

the next Section.

4.2. Proof of Theorem 3.4. Now, we consider a set E such that (3.7) holds. Wewill identify E with the set of points where its Lebesgue density is 1, moreover, a necessary

condition for (3.7) is that E =

>0 E + B(0, ) coincides with E up to a negligible set, in

other words, |E\ E| = 0.A first remark is that, clearly, using (3.1),

aSMa,B(0,1)(E; ) SM,C(E; ) bSMb,B(0,1)(E; )

hence any limit of SM,C(E; ) is in between aPer(E; ) and bPer(E; ). In particular, wecan introduce the measures

:= 1

(E+C E)

which are equibounded. Then, up to a subsequence, we have k

as measures in , with

aHn1 E bHn1 E. We introduce the Besicovitch derivative g(x) [a, b] ofthe measure w.r. Hn1 E, defined by

g(x) = lim0

(B(x, ))

Hn1(E B(x, ))


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and which is defined Hn1a.e. on E. Moreover, since E is rectifiable, it is also givenby

(4.5) g(x) = lim0

(B(x, ))

nn1.

(We recall n is the diameter of the unit disc.)

Theorem 3.4 will follow if we can show that g(x) = hC(E(x)) for Hn1a.e. x E. We

fix from now on a point x E where (4.5) exists. Observe that from (4.2), it follows thatg(x) hC(E(x)) Hn1-a.e. in E, so that we need to show that g(x) hC(E(x)). Werecall that

lim0

Hn1(E B(x, ))nn1

= 1

and

lim0

B(0,1)Ex {y : yE(x)0} dy = 0

hold. We denote = E(x) and without loss of generality we will assume that it is the

direction of the last coordinate xn. We will use the notation x = (x, xn) Rn1 R to

distinguish between the component x and xn (along ) of a point x Rn.We also introduce the measures

:=1

E+B(0,b) E ,

the main assumption of Theorem 3.4 ensures that these measures converge weakly- to =bHn1 E as 0. Now we use a classical procedure: since for a.e. > 0,

k(B(x, ))

(B(x, )) and k(B(x, ))

(B(x, )),

we can build a sequence (k)kN with k = k/k 0 such that

limk

k(B(x, k))

nn1k

= g(x)

and

limk

k(B(x, k))

nn1k

= b.

In fact, the rest of the proof would be relatively easy if we could ensure that k k ask , using then a blow-up argument. However, this is not clear in general, and we needto consider the situation where k = o(k), hence

k 0. As is usual, we do a blow-up using

the change of variables x = x + ky. We let Ek = (E x)/k, and we observe thatlim

k

1

nk

B(0,1)

(Ek+kC Ek) dy = g(x),(4.6)

limk

1

nk

B(0,1)

(Ek+kB(0,b) Ek) dy = b,(4.7)

limk

B(0,1)

Ek {y : y0} dy = 0.(4.8)


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Moreover, for any > 0 (small), one can check easily that if we replace in (4.7) and (4.8)

B(0, 1) with B(0, 1 ), or even with C(0, 1 ) := {(y, yn) B(0, 1) : |y| 1 }, (4.8)still holds and the right-hand side in (4.7) is replaced with b(1 )n1. Indeed, it followsfrom (4.2) (with C = B(0, b)) and (4.8) that for any open set A B(0, 1),

bHn1(A {y = 0}) lim infk

1

k

A

(Ek+kB(0,b) Ek) dy .

Together with (4.7), we deduce that as soon as Hn1(A {y = 0}) = 0,

(4.9) bHn1(A {y = 0}) = limk

1

k

A

(Ek+kB(0,b) Ek) dy .

We fix a (small) value of > 0. Then, we choose a value > 10b and we consider the points

z Zn1 such that the hypersquares (k(z + (0, 1)n1)) {0} are contained in B(0, 1 ).There is a finite number Nk of such squares and we enumerate the corresponding points

{zk1 , . . . , z

kNk}. For i = 1, . . . , N k, we let

Cki = [(k(z

ki + (0, 1)

n1)) R] B(0, 1) , Cik = (k(zki + (0, 1)n1)) {0} .We then let

(4.10) aki =

Cki

Ek {yn0} dy 2(k)n1and k =

Nki=1 a

ki : from (4.8) we know that k 0 as k . We then consider

Zk = {i = 1, . . . , N k : aki

k(k)

n1},Zk = {1, . . . , N k} \ Zk. It follows that(4.11) k

k(k)

n1#Zk (k)n1#Zk

k

which gives a control on the bad surface, of the cylinders Cki where the integral aki is large.

For each i = 1, . . . , N k, we have

(4.12)1

k

Cki

Ek+kB(0,b) Ek dy = 1

k

Ci

k

1y2

1y2

Ek+kB(0,b) Ek dyndy

by Cik : |({y} R) (B(0, 1) Ek)| > 0, |({y} R) (B(0, 1) \ Ek)| kb,

since clearly, each time a point (y, yn) Ek, then (y, yn + s) Ek + kB(0, b) for |s| kb.

We denote by D

k

i the set in the right-hand side of (4.12). For y

Dk

i ,1y2

1y2

Ek {yn0} dy kb

2

as soon as k

/(2b). It follows that |Cik \ Dki | 2aki /

, hence if i Zk, so thataki

k(

k)

n1 =

k|Cik|/n1, we get that (4.12) bounds

(4.13) b|Dki | b|Cik|

1 2

kn1

= b|Cik|

1 K

k

.


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To sum up, (4.12) and (4.13) show that for any i Zk,1

k Cki

Ek+kB(0,b)

Ekdy b|Cik|

1 K

k.

In particular, it follows that (using (4.11))

(4.14) lim infk

iZk

1

k

Cki

Ek+kB(0,b) Ek dy

b limk

Nki=1

Cik

(k)n1#Zk

1 K

k

= nb(1 )n1

and together with (4.9) (with A = C(0, 1 )) we deduce that

k :=iZk

1k

Cki

Ek+kB(0,b) Ek dy b|Cik|1 Kk k 0.So now we introduce

Zk =

i Zk : 1

k

Cki

Ek+kB(0,b) Ek dy b|Cik|1 Kk (k)n1k

,

and its complement Zk = Zk \ Zk. Then as before, one sees that

(4.15) k k(

k)

n1#Zk (k)n1#Zk k .

We see at this point that (4.14) still holds if Zk is replaced with Zk, and Zk with Z

k Zk.

Together with (4.9) (with again A = C(0, 1 )) it follows that

limsupk

1

k

C(0,1)\

iZk

Cki

Ek+kB(0,b) Ek dy = 0 ,

and as a consequence

(4.16) lim supk

1

k

C(0,1)\

iZk

Cki

Ek+kC Ek dy = 0 .

We now need to estimate the quantity (1/k)Cki |Ek+kCEk | dy for i Zk, hence when

(4.17)

b|Cik|

1 K

k

1

k

Cki

Ek+kB(0,b) Ek dy b|Cik|

1 K

k +

k

bn1

and

(4.18) |Cik \ Dki | 2|Cik|

kn1

.


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We choose k, i Zk so that (4.17) and (4.18) hold, and consider the change of variabley = zki +

k y. We let F = (Ek zki )/k, Q = (Cki zki )/k (0, )n1 (

/k,

/k),

D = (Dki zki )/k. We find that

(4.19)

Q

F+B(0,b) F

dy bn1

1 K

k +

k

bn1

,

(4.20)

(0,)n1

k

,

k

F {yn0} dy n1

kk

,

while (4.18) yields

(4.21)

|(0, )n1

\D

| 2k

.

Let us set Fb = F + B(0, b) = {y Rn : dist(y, F) b}, and

Fs = {y Fb : dist(y ,Fb) b s}.

We observe that F Fs for any s [0, b]. It is well known that if 0 < s < b, the boundariesFs are C

1,1, with curvatures between 1/(b s) and 1/s. We want to show that for k largeenough, these boundaries are essentially flat inside Q. Let now

D =

y (0, )n1 : |({y} R) Q F| = 0

,

D =

y (0, )n1 : ({y} R) Q Fb ,D =

y (0, )n1 : |({y} R) Q (Fb \ F)| 2b

.

The definition of D ensures that if y (0, )n1 \ D, |({y} R) Q (Fb \ F)| b.From (4.19) and (4.21), we have that

bn1 +

k b|((0, )n1 \ D) \ D| + 2b|D| bn1 + b|D| 2

k

,

hence

(4.22) |D| 1b

k + 2

k

.

Now, we easily deduce from (4.20) that both |D| and |D \ D| are bounded by a constant(multiple of n1/

) times

k. It follows that there exists a constant K

(still depending

on , ) such that

(4.23) |D D| K

k +

k

.


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Now, each time y (0, )n1 \ (D D), in ({y} R) Q one can find points which arein F (hence in Fs for all s [0, b]), points which are not in Fb, and as a consequence there arealso points in Fs, for any s [0, b]. It follows that

(4.24) Hn1(Fs Q) n1 K

k +

k

,

and as a consequence, also (from the coarea formula applied to the distance function to Fb),

(4.25) |(Fb \ F0) Q| =b0H

n1(Fs Q) ds bn1 bK

k +

k

.

Inequalities (4.19) and (4.25) yield that (here the constant K may vary from line to line,

keeping the same kind of dependency in the parameters)

(4.26) |(F0 \ F) Q| bK k + k .

Let us choose > 0, small, and observe that (using (4.24) and (4.19))

2

Hn1(Fs Q) ds

|(Fb \ F0) Q| (b )

n1 K

k +

k

n1 + K

k +

k

so that there exists s [, 2] with

(4.27) Hn1(Fs Q) n1 + K

k +

k

.

We recall that Fs has a (1/)-Lipschitz normal, by construction. Eventually, we deduce thatFs must be almost flat if k is large enough: indeed, fix > 0 and assume there is a point

y Fs Q with, |Fs(y) | 1 . We let = Fs(y). Then, for t small, we considerthe ball B(y,t) (which we assume is in Q, and we let r = t). The regularity of Fs yields

that, in that ball, it consists (at least) of a C1,1 graph which passes in between two spherical

caps of radius , which are tangent in y and normal at that point to . We call S the subset

of B(y, r) bounded by these two caps (see Figure 1). A simple calculation shows that the

trace of these spherical caps on the sphere B (y, r) is given by the intersection of this sphere

with the hyperplanes {(y y) = t2 r} (hence, S B(y, r) {|(y y) | < t2 r}). Inparticular, the surface Hn1(Fs

B(y, r)) can be estimated from below with the surface of

the corresponding discs, that is, nrn1

1 t22/4n1.

Let us now estimate from below the surface Hn1(Fs Q \ B(y, r)). Since we know thatgiven any y (0, )n1 \ (D D), Fs ({y} R) Q = , it is enough to estimatefrom above the projection ofS onto (0, )n1, which we denote by (S). This, in turn, is

bounded by the projection of

B(y, r)

y : |(y y) | < t2

r

=

y + r(s+ ) : |s| < t

2, ||

1 s2 , = 0

.


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16/18


0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 01 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 10 0 0 0 0 0 00 0 0 0 0 0 01 1 1 1 1 1 11 1 1 1 1 1 1

The surface in the ball is at least this diameter

The surface out of the ball projects at

B(y,t)

Fs

y

S

least on this shaded region

Figure 1. If the normal at y to Fs is away from , then its surface must

exceed n1 by some quantity which is estimated.

Now, this projection is a subset of the vertical projection of the diameter of B(y, r) perpen-

dicular to , that is, = {y + r : || 1 , = 0}, plus the disk (B(0,rt/2)). It follows(see the expansion of the volume of Minkowski sums of convex sets in [11], cf Remark 3.6)

that

|(S)| |()| + Per(())rt

2 + o(rt) .

Here, Per(()) is the (n 2)-dimensional perimeter of () in (0, )n1, and a simplescaling argument shows that o(rt) is of the form rn1o(t), where the latter o depends only

on the geometry of the vertical projection of the unit ball, that is, on and, in fact,would be largest for = . Now, since Per(()) Hn2() = (n1)nrn2, we obtainthat

|(S)| ( )nrn1 + 2(n 1)nrn2 rt2

nrn1(1 (1 (n 1)t))

if t is small enough. It follows

(4.28)

Hn1(FsQ) n1K(

k +

k)+nr

n1

1 t

22

4

n1

1 + (1 (n 1)t)

.

For t = 0, the quantity between the right-hand side parentheses is > 0, and it decreases

with t. It follows that one can find t > 0 (depending only on n and ) such that (4.28) reads

Hn1(Fs Q) n1 K(

k +

k) + n(t)

n1

2.


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Together with (4.27), it follows that if k is large enough (depending on K, , ), we get a

contradiction, and therefore Fs(y) 1 for any y Fs Q at distance at least tfrom Q. (The normal cannot be reverted, nor can y be close to the boundary Q B (0, 1),because of (4.20).)

We deduce that there exists a value such that

Fs Q {y Q : y + 2}

for k large enough, with moreover F Q Fs Q {y Q : y + 2}. In particular,it follows that (F Q) + C {y () + + 2}. A consequence is that

((F + C) \ F) Q ((Fs \ F) Q) ((Fb \ F) Q) \ ((b, b)n1 R)) {y Q : y + 2 + ()} .

Now, using again (4.21) we can show that

|((Fb \ F) Q) \ ((b, b)n1 R))| 2nb2n2 + 2b

k

,

while, using (4.26) and (4.27),

|(Fs \ F) Q| 2n1 + (b + 2)K

k +

k

.

We deduce that

|((F + C) \ F) Q| n1(2 + ()) + 2n1 + 2nb2n2 + Rk

where Rk is a rest which goes to zero with k and k. Returning to the original sets C

ki , we

find that if k, i Zk and k is large enough,

(4.29)1

k

Cki

|Ek+kC Ek | dy |Cik|

() + 2 + 2 +2nb2

+

1

n1Rk

.

Together with (4.16), (4.29) yields that

lim supk0

1k

C(0,1)

Ek+kC Ek dy n(1 )n1() + 2 + 2 + 2nb2

.

Sending first , then to zero and eventually to +, and using (4.6) and (4.9), we deduce

g(x) = limk0

1

k

B(0,1)

Ek+kC Ek dy bn(1 (1 )n1) + n(1 )n1() ,

and letting then 0 yields the desired inequality.


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18/18


References

[1] J. P. Aubin: Mutational equations in metric spaces. Set-Valued Anal.1 (1993) 3-46.

[2] L. Ambrosio, A. Colesanti and E. Villa: Outer Minkowski content for some classes of closed sets and

applications to stochastic geometry. Math. Ann.342 (4) (2008), 727-748.[3] L. Ambrosio and N. Dancer: Calculus of Variations and Partial Differential Equations: Topics on Geo-

metrical Evolution Problems and Degree Theory. (G. Buttazzo, A. Marino, M. K. V. Murthy Eds) Springer

Verlag, Berlin-Heidelberg, 2000.

[4] L. Ambrosio, N. Fusco and D. Pallara: Functions of Bounded Variation and Free Discontinuity Problems.

Oxford Science Publications, 2000.

[5] A. Braides: -convergence for beginners. Oxford University Press, 2002.

[6] A. Chambolle, A. Giacomini and L. Lussardi: Continuous limits of discrete perimeters. M2AN Math.

Model. Numer. Anal. 44 (2) (2010), 207-230.

[7] G. Dal Maso: Introduction to -convergence. Birkhuser, 1993.

[8] F. Demengel and R. Temam, Convex functions of a measure and applications, Indiana Univ. Math. J.

33:673-709, 1984.

[9] H. Federer: Geometric measure theory. Springer-Verlag New York Inc., New York, 1969.

[10] T. Lorenz: Set-valued maps for image segmentation. Comput. Vis.Sci.4 (2001) 41-57.

[11] R. Schneider: Convex Bodies: The Brunn-Minkowski Theory. Encyclopedia of Mathematics and its Ap-

plications 44. Cambridge University Press, 1993.

[12] L. Simon: Lectures on Geometric Measure Theory. Proc. Center Math. Anal. Australian National

Univ. (1983).

[13] J. Sokolowski and J. P. Zolesio: Introduction to shape optimization. Shape sensitivity analysis. Springer

Series in Computaional Mathematics, 16. Springer-Verlag, Berlin, 1992.

[14] A. Visintin: Nonconvex functionals related to multiphase systems. SIAM J. Math. Anal. (5) 21 (1990),

1281-1304.

(A. Chambolle) CMAP, cole Polytechnique, CNRS, F-91128, Palaiseau, FranceE-mail address, A. Chambolle: [email protected]

(S. Lisini) Dipartimento di Matematica F. Casorati, Universit degli Studi di Pavia, via Fer-

rata 1, I-27100 Pavia, Italy

E-mail address, S. Lisini: [email protected]

URL: http://www-dimat.unipv.it/lisini/

(L. Lussardi) Dipartimento di Matematica e Fisica N. Tartaglia, Universit Cattolica del

Sacro Cuore, via dei Musei 41, I-25121 Brescia, Italy

E-mail address, L. Lussardi: [email protected]

URL: http://www.dmf.unicatt.it/~lussardi/


22Mar2012

chambolle cll submitted rev

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