scale convergence in homogenization

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SCALE CONVERGENCE INHOMOGENIZATIONMaria Luísa Mascarenhas a & Anca-Maria Toader aa Universidade de Lisboa , C.M.A.F., Faculdade de Ciências, Av. Prof.Gama Pinto 2, Lisboa , 1649-003 , PortugalPublished online: 31 Jan 2014.

To cite this article: Maria Luísa Mascarenhas & Anca-Maria Toader (2001) SCALE CONVERGENCE INHOMOGENIZATION, Numerical Functional Analysis and Optimization, 22:1-2, 127-158

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SCALE CONVERGENCE IN HOMOGENIZATION

Maria Luısa Mascarenhas and Anca-Maria Toader

C.M.A.F., Faculdade de Ciencias, Universidade de Lisboa,Av. Prof. Gama Pinto 2, 1649-003 Lisboa, Portugal

ABSTRACT

In order to treat non-periodic oscillations we extend the concept oftwo-scale convergence, using Young measures. We present examples andapplications.

Key Words: Young measures; Scale convergence; Quasiperiodicity;�-convergence.

1. INTRODUCTION

The effort to understand the asymptotic behavior of oscillating partial differ-ential equations has been at the origin of a great variety of mathematicaltools. Homogenization techniques are such an example: beginning with theasymptotic expansions (see [BLP], [SP] and [JKO]), essentially adapted to thestudy of periodic problems, these techniques also developed in order to treatmore general situations, through the concepts of G-convergence, introducedby S. Spagnolo (see [S]), of H-convergence, due to F. Murat and L. Tartar(see [Mu] and [T1]), and of �-convergence, due to the italian school ofDe Giorgi (for a rather complete bibliography see [DM]).

More recently, G. Nguetseng (in [N1] and [N2]) introduced the notionof two scale convergence, later developed by G. Allaire in [A], and W. E in [E].This new concept, not only justified the heuristic process of the asymptoticexpansions, but also turned out to be a powerful tool to treat the periodiccase. Many adaptations have been made since then, namely to stochastic

127

Copyright # 2001 by Marcel Dekker, Inc. www.dekker.com

NUMER. FUNCT. ANAL. AND OPTIMIZ., 22(1&2), 127–158 (2001)

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PDE (see [BMW]) and to homogenization problems involving different scalesof convergence (see [PS]).

In order to treat nonlinear problems, the Young measures, due toL.C. Young, were revisited by L. Tartar in [T2]. As remarked byM. Valadier in [V3], there exists a close relationship between these twonotions: the two scale limit represents in fact the barycenter of a Youngmeasure.

The same idea of M. Valadier was used by the authors to generalizethe two scale convergence, under the form of a new convergence, in orderto find the �-limit of a sequence of non periodic time dependent functionals(see [M] and [To]). In this paper we systematize this new concept under thename of scale convergence presenting new developments and applications.As in the periodic case, to describe the asymptotic behavior of an oscillatingproblem it is reasonable to consider, among all the possible oscillations,only those that synchronize with the given oscillating coefficients of the prob-lem. In fact the new concept generalizes the multiscale convergence intro-duced by G. Allaire, in [A], and by both G. Allaire and M. Briane, in [AB].It also simplifies the quasiperiodic setting treated by R. Alexander in [Ax].The scale convergence not only generalizes to non periodic oscillationsthe two-scale convergence, but also the proofs, being a natural consequenceof the fundamental theorem of Young measures, become simpler. It is ourconviction that the present notion is useful to understand essentially non-periodic problems.

In Section 2 we state some preliminary results concerning Youngmeasures. In Section 3 we introduce the scale convergence and related results.In Section 4 we present examples and applications and, in particular, thosethat motivated the definition of the scale convergence, namely we use itto derive the �-limit of a sequence of non-periodically oscillating functionals.In Section 5 we present a method inspired inMonte-Carlo, for the numericalapproach of a Young measure.

2. PRELIMINARIES

We present some definitions and properties about Young measures beused in the sequel. We follow, essentially, the notations and statements in [V2].

Let � be an open bounded subset of RN and let S be a metrizable,

locally compact, separable space. We represent by mN the N-dimensionalLebesgue measure, by Fð�Þ the family of all Lebesgue measurable subsetsof � and by BðSÞ the Borel �-field of S.

Definition 2.1. Consider a function f : �� S! R:1) The function f is said to be an integrand if f is Fð�Þ � BðSÞ-

measurable;

128 MASCARENHAS AND TOADER

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2) the function f is said to be a normal integrand if f is an integrandand f ðx; Þ is lower semicontinuous for mN - a.e. x 2 �;

3) the function f is said to be a Caratheodory integrand if f is anintegrand and f ðx; Þ is continuous for mN - a.e. x 2 �.

Definition 2.2. We call Young measure on �� S any positive measure �whose projection on � coincides with the Lebesgue measure (i.e.�ðA� SÞ ¼ mNðAÞ, for all A 2 Fð�ÞÞ. We denote by Yð�� SÞ the set ofall Young measures on �� S.

Since S is a metrizable, locally compact, separable space, any boundedBorel measure on S is a Radon measure and so we can use the disintegrationtheorem proved in [V1] (see also [V2]), stated as follows:

Theorem 2.3. Let � be a bounded positive measure on ð�� S;Fð�Þ � BðSÞÞwhose projection on ð�;Fð�ÞÞ we denote by �. Then there exists a family ofprobability measures on ðS;BðSÞÞ, ð�xÞx2�, depending measurably on x, suchthat the following disintegration formula holds

Z�xS

ðx; �Þ d�ðx; �Þ ¼

Z�

ZS

ðx; �Þ d�xð�Þ d�ðxÞ;

for all : �� S�R, positive �-measurable or �-integrable.We refer the family ð�xÞx2� as the disintegration of � with respect to its

projection �.Since a Young measure is a bounded positive measure on

ð�� S;F � BðSÞÞ, we can apply Theorem 2.3 when � 2 Yð�� SÞ and, inwhat follows, we will not distinguish a Young measure � from its disintegra-tion ð�xÞx2 with respect to the Lebesgue measure.

For each measurable function � : �! S we associate a Young measure��, with support in the graph of �, defined, for each A 2 Fð�Þ and B 2 BðSÞ,by

��ðA� BÞ :¼ mNðA \ ��1ðBÞÞ:

Then, for all C in Fð�Þ � BðSÞ, we have

��ðCÞ ¼

Z��S

Cðx; �Þ d��ðx; �Þ ¼

Z�

Cðx; �ðxÞÞ dx;

and, if : �� S! R is positive ��-measurable or ��-integrable,

h��; i ¼

Z��S

ðx; �Þ d��ðx; �Þ ¼

Z�

ðx; �ðxÞÞ dx:

SCALE CONVERGENCE IN HOMOGENIZATION 129

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On Yð�� SÞ we consider the narrow topology, i. e., the weakest topologythat makes continuous the maps

��

Z��S

ðx; �Þ d�ðx; �Þ;

for all bounded Caratheodory integrands . The following proposition holds:

Proposition 2.4. 1) The sequence of Young measures associated to a sequence offunctions uniformly bounded in L1

ð�;RdÞ admits a convergent subsequence in

Yð��RdÞ.

2) If S is a metrizable compact space, then every sequence of Youngmeasures associated to a sequence of measurable functions �n : �! Sadmits a convergent subsequence in Yð�� SÞ.

3) Let S be a metrizable compact space, �n : �! S a sequence ofmeasurable functions and ðvnÞ a uniformly bounded sequence in L

1ð�;Rd

Þ.Then the sequence of Young measures associated to the coupled sequenceð�n; vnÞ admits a convergent subsequence in Yð�� ðS �R

dÞÞ.

For the proof of the above results we refer to [V2] (Section 3: Theorem 11and definition of tight set).

Remark 2.5. 1) Given an uniformly bounded sequence ð�nÞ in L1ð�;Rd

Þ,using Proposition 2.4 1), we may assume that, up to a subsequence of ð�nÞ,the sequence of their associated Young measures, narrow converges to some� ¼ ð�xÞx2� 2 Yð��R

dÞ.

2) For a sequence �n : �! S we say that � is the Young measureassociated to the sequence ð�nÞ if the Young measures associated to ð�nÞnarrow converge to �, i.e., for all bounded Caratheodory integrand ,Z

�

ðx; �nðxÞÞ dx!

Z��S

ðx; �Þ d�ðx; �Þ:

The following result (see [V2] (Section 4) and [B]), allows us to considera larger class of admissible functions with respect to the narrow convergence,as soon as some uniform integrability condition is satisfied.

Theorem 2.6. Let S be a metrizable, locally compact, separable, complete space.Let �n : �! S be a sequence of measurable functions and let � be the Youngmeasure associated to ð�nÞ, in the sense of Remark 2.5 2). Then:

1) If : �� S! R is a normal integrand such that the sequence of thenegative parts ð ð; �nðÞÞ

�Þ is uniformly integrable in �, thenZ

��S

ðx; �Þd�ðx; �Þ � lim infn

Z�

ðx; �nðxÞÞ dx:


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2) If : �� S! R is a Caratheodory integrand such that ð ð; �nðÞÞ isuniformly integrable in �, thenZ

��S

ðx; �Þ d�ðx; �Þ ¼ limn

Z�

ðx; �nðxÞÞ dx: ð2:1Þ

3. THE SCALE-CONVERGENCE

In this section we present a generalization to the non-periodic caseof some notions and results introduced by G. Ngeutseng and developedby G. Allaire, W. E and M. Valadier for periodic oscillations, under thename of two-scale convergence. These ideas were first introduced in [M],in order to treat a non-periodic homogenization problem.

Let � be a metrizable compact space. Recalling the notations of theprevious section, we will use, in the sequel, S ¼ � or S ¼ ��R. Considera sequence of measurable functions �n : ��. In view of Proposition 2.4 2)and of Remark 2.5 2), let � be the Young measure associated to ð�nÞ, thenequality (2.1) holds.

We remark that the following definitions and results also hold in thevectorial case as well as in Lpð�;Rd

Þ, for 1 < p < þ1. For the sake ofsimplicity we only consider p ¼ 2 and d ¼ 1.

Let L2�ð��Þ represent the space of all �-measurable functions on

�� with �-integrable square, and L2ð�;Cð�ÞÞ the L2-functions, with

respect to the Lebesgue measure, defined in � and with values in theBanach space of continuous functions on �.

Definition 3.1. We say that a sequence ð�nÞ in L2ð�Þ, ð�nÞ-converges to

� 2 L2�ð��Þ ifZ

�

�nðxÞ�ðx; �nðxÞÞ dx!

Z��

�ðx; �Þ� ðx; �Þ d�ðx; �Þ; ð3:1Þ

for all � 2 L2ð�;Cð�ÞÞ. We will say that � is the ð�nÞ-limit of the sequence

ð�nÞ.The above definition is justified, in view of the following compactness

result.

Theorem 3.2. From each bounded sequence ð�nÞ in L2ð�Þ we may extract a

subsequence ð�nkÞ, ð�nkÞ-converging to some � in L2�ð��Þ. In particular ð�nkÞ

is weakly convergent and its weak limit �0 satisfies mN a.e. in x 2 �,

�0ðxÞ ¼

Z�

�ðx; �Þ d�xð�Þ: ð3:2Þ


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Proof. Since by Proposition 2.4 3), with S ¼ � and d ¼ 1, the sequence ofYoung measures associated to ð�n; �nÞ is relatively compact inYð�� ð��RÞÞ, in view of Remark 2.5 2), we consider, up to a subsequence,� as the Young measure associated to ð�n; vnÞ.

For � in L2ð�;Cð�ÞÞ, the sequence ð�nðxÞ� ðx; �nðxÞÞÞ is uniformly

integrable. In fact, since ð�nÞ is bounded in L2ð�Þ by some constant c and

jvnðxÞ� ðx; �nðxÞÞj � j�nðxÞj sup�2�

j� ðx; �Þj;

we obtain, for any measurable subset E of �,

ZE

j�nðxÞ�ðx; �nðxÞÞj dx � c

ZE

sup�2�

j�ðx; �Þj2 dx:

Since x� sup�2� j�ðx; �Þj is in L1ð�Þ, we conclude that ð�nðxÞ

� ðx; �nðxÞÞÞ is uniformly integrable.Applying Theorem 2.6 2) with S ¼ ��R and ðx; �; Þ ¼ � ðx; �Þ we

obtain

Z�

�nðxÞ� ðx; �nðxÞÞ dx!

Z��R

� ðx; �Þ d� ðx; �; Þ:

By disintegrating � with respect to its projection � on ��,� ¼ ð�ðx;�ÞÞðx;�Þ2�x�, we obtain

Z��R

� ðx; �Þ d�ðx; �; Þ ¼

Z�

Z�

� ðx; �Þ

ZR

d�ðx;�Þð Þ d�xð�Þ

� �dx

and, defining � ðx; �Þ :¼RR d�ðx;�Þð Þ, we get

Z�

�nðxÞ�ðx; �nðxÞÞ dx!

Z��

� ðx; �Þ�ðx; �Þ d�ðx; �Þ: ð3:3Þ

The function � is �-measurable by construction. It remains to show that� 2 L2

�ð��Þ. Applying Jensen’s inequality to each probability measure�ðx;�Þ,

k�k2L2�ð��Þ

¼

Z�

Z�

ZR

d�ðx;�Þð Þ

��2 d�xð�Þ dx

�

Z�

Z�

ZR

2 d�ðx;�Þð Þ d�xð�Þ dx


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and, by Theorem 2.6 1), with S ¼ ��R and ðx; �; Þ ¼ j j2, we have

Z��R��

2 d�ðx; �; Þ � lim infn!þ1

Z�

�2nðxÞ dx < þ1;

therefrom � 2 L2�ð��Þ.

By taking in (3.1) test functions depending only on x we obtain that theweak limit �0 of ð�nkÞ satisfies formula (3.2). œ

Proposition 3.3. Let w 2 L2ð�;Cð�ÞÞ and define wnðxÞ :¼ wðx; �nðxÞÞ. Then,

for all Caratheodory integrands � : �� ð��RÞ�R such that there exist apositive constant C and a function pðxÞ 2 L1

ð�Þ satisfying

j�ðx; �; Þj � CðpðxÞ þ j j2Þ; ð3:4Þ

for all ð�; Þ 2 ��R and mN a.e. x 2 �, we have

Z�

�ðx; �nðxÞ;wnðxÞÞ dx!

Z��

�ðx; �;wðx; �ÞÞ d�ðx; �Þ: ð3:5Þ

In particular ðwnÞ ð�nÞ-converges to w.*

Proof. Using (3.4) and since w 2 L2ð�;Cð�ÞÞ, we have

j�ðx; �nðxÞ;wnðxÞÞj � C�pðxÞ þ jwðx; �nðxÞÞj

2�

� C�pðxÞ þ sup

�2�jwðx; �Þj2

�;

with the second member of the last inequality in L1ð�Þ, which implies

that the sequence ðx; �nðxÞ;wnðxÞÞ is uniformly integrable. Theorem 2.6 2),with S ¼ � and ðx; �Þ ¼ �ðx; �;wðx; �ÞÞ, yields convergence (3.5).To complete the proof we take ’ 2 L2

ð�;Cð�ÞÞ. We have, for allð�; Þ 2 ��R and mN a.e. x 2 �,

j ’ðx; �Þj � j j2 þ sup�2�

j’ðx; �Þj2:


*The function w is called by M. Valadier mother function with respect to the sequence ðwnÞ, inthe periodic setting developed in [V3].

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So �ðx; �; Þ ¼ ’ðx; �Þ satisfies hypothesis (3.4), with C ¼ 1 and pðxÞ ¼sup�2� jwðx; �Þj

2, which yields

Z�

wnðxÞ’ðx; �nðxÞÞ dx!

Z��

wðx; �Þ’ðx; �Þ d�ðx; �Þ: œ

The following proposition proves that all the elements in L2�ð��Þ

are obtained as ð�nÞ-limits. The proof uses the separability of L2ð�;Cð�ÞÞ,

a density argument and a standard diagonalization process. The proof is veryclose to the one presented in [A], Lemma 1.13.

Proposition 3.4. For every � 2 L2�ð��Þ there exists a subsequence ð�nkÞ of

ð�nÞ and a sequence ð�kÞ in L2ð�Þ such that ð�kÞ ð�nkÞ-converges to �.

Proof. Since L2ð�;Cð�ÞÞ is dense in L2

�ð��Þ, consider ðwkÞ in L2ð�;Cð�ÞÞ

such that

kwk � �kL2�ð��Þ

< 1=k: ð3:6Þ

Let ð’jÞ be a countable dense family in L2ð�;Cð�ÞÞ and normalize it. For

each k and for all j we have, by Proposition 3.3, as n goes to þ1,

Z�

wkðx; �nðxÞÞ’jðx; �nðxÞÞ dx!

Z��

wkðx; �Þ’jðx; �Þ d�ðx; �Þ

and

Z�

jwkðx; �nðxÞÞj2 dx!

Z��

jwkðx; �Þj2 d�ðx; �Þ:

Then, for each � > 0 there exists n0ðk; j; �Þ such that, for n � n0ðk; j; �Þ andfor all i � j,

Z�

wkðx; �nðxÞÞ’iðx; �nðxÞÞ dx�

Z��

wkðx; �Þ’iðx; �Þ d�ðx; �Þ

�� < � ð3:7Þ

and

Z�

jwkðx; �nðxÞÞj2 dx�

Z��

jwkðx; �Þj2 d�ðx; �Þ

�� < �: ð3:8Þ


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Set � ¼ 1=k, j ¼ k, and nk ¼ n0ðk; k; 1=kÞ. Then, for all i � k we get from(3.6), (3.7) and (3.8)

Z�

wkðx; �nkðxÞÞ’iðx; �nkðxÞÞ dx�

Z��

�ðx; �Þ’iðx; �Þ d�ðx; �Þ

�� < 2

k

and

Z�

jwkðx; �nkðxÞÞj2 dx�

Z��

j�ðx; �Þj2 d�ðx; �Þ

�� < 1

kþ2c

k; ð3:9Þ

where c is a constant satisfying kwkkL2�ð��Þ

� c, independently of k,and where we have used the fact that k’ikL2

�ð��Þ� k’ikL2ð�;Cð�ÞÞ ¼ 1.

Defining �kðxÞ :¼ wkðx; �nkðxÞÞ then, for all j, as k goes to þ1, we obtain

Z�

�kðxÞ’jðx; �nkðxÞÞ dx!

Z��

�ðx; �Þ’jðx; �Þ d�ðx; �Þ: ð3:10Þ

Convergence (3.10) still holds for the non normalized initial dense family,by multiplying it by the norm of its elements. Since by (3.9) k�kkL2ð�Þ

is bounded independently of k, by a density argument we get convergence(3.10) for all ’ 2 L2

ð�;Cð�ÞÞ. œ

The following proposition generalizes a similar result obtained in [V3].It measures, in terms of the ð�nÞ-convergence, how far a weakly convergentsequence in L2

ð�;RÞ is from strong convergence.

Proposition 3.5. Consider a bounded sequence ð�nÞ in L2ð�Þ, let � be its ð�nÞ-

limit. For any w0 2 L2ð�Þ, in particular for the weak limit �0 of ð�nÞ

ðseeð3:2ÞÞ;we have

1) If � 2 L2ð�;Cð�ÞÞ and one of the two following limits exists, then

limn!þ1

k�n � w0k2L2ð�Þ ¼ lim

n!þ1k�n � �ð; �nðÞÞk

2L2ð�Þ þ k�� w0k

2L2�ð��Þ

:

2) If � 2 L2�ð��Þ and if the limit of k�n � w0kL2ð�Þ, as n goes to þ1,

exists, then, for every sequence ðwkÞ in L2ð�;Cð�ÞÞ strongly convergent to � in

L2�ð��Þ, we have

limn!þ1

k�n � w0k2L2ð�Þ ¼ lim

k!þ1limn!þ1

k�n � wkð; �nðÞÞk2L2ð�Þ þ k�� w0k

2L2�ð��Þ

:


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Proof. Writing �nðxÞ � w0ðxÞ ¼ �nðxÞ � �ðx; �nðxÞÞ þ �ðx; �nðxÞÞ � w0ðxÞ weobtain

Z�

jvnðxÞ � w0ðxÞj2 dx

¼

Z�

j�nðxÞ � �ðx; �nðxÞÞj2 dxþ

Z�

j�ðx; �nðxÞÞ � w0ðxÞj2 dx

þ 2

Z�

��nðxÞ � �ðx; �nðxÞÞ

��ðx; �nðxÞÞ � w0ðxÞ

�dx:

Using the definition of ð�nÞ-convergence, the fact that � 2 L2ð�;Cð�ÞÞ

and Proposition 3.3, the last integral in the right hand side of the previousequality converges to zero. From Proposition 3.3, the second integral con-verges to k�� w0k

2L2�ð��Þ

. Letting n going to þ1, we conclude the proofof statement 1).

The proof of statement 2) follows from a similar argument appliedto �nðxÞ � w0ðxÞ ¼ �nðxÞ � wkðx; �nðxÞÞ þ wkðx; �nðxÞÞ � w0ðxÞ and finallypassing to the limit as k tends to þ1. œ

As for the two scale convergence, the following result holds, showingthat the ð�nÞ-convergence also contains more information than the weakconvergence.

Corollary 3.6. Let ð�nÞ be a bounded sequence in L2ð�Þ and let � 2 L2

�ð��Þbe its ð�nÞ-limit. Assume that

limn!1

k�nkL2ð�Þ ¼ k�kL2�ð��Þ

: ð3:11Þ

Then, for every ðunÞ which ð�nÞ-converges to u 2 L2�ð��Þ, we have

unðxÞ�nðxÞ*

Z�

uðx; �Þ�ðx; �Þ d�xð�Þ; ð3:12Þ

weakly in L1ð�Þ. Moreover, if � 2 L2

ð�;Cð�ÞÞ, then

limn!þ1

k�nðxÞ � �ðx; �nðxÞÞkL2ð�Þ ¼ 0 ð3:13Þ

Proof. If � 2 L2ð�;Cð�ÞÞ convergence (3.12) is trivial, in view of Definition

3.1. If � does not belong to L2ð�;Cð�ÞÞ let ðwkÞ be a sequence converging


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to � in L2ð�;Cð�ÞÞ. Choose ’ 2 L1ð�Þ and consider the following inequality

Z�

unðxÞ�nðxÞ’ðxÞ dx�

Z��

uðx; �Þ�ðx; �Þ’ðxÞ d�ðx; �Þ

��

� k’kL1ð�Þ

Z�

junðxÞj2 dx

� Z�

jvnðxÞ � wkðx; �nðxÞÞj2 dx

�

þ k’kL1ð�Þ

Z�

unðxÞwkðx; �nðxÞÞ dx

��

Z��

uðx; �Þwkðx; �Þ d�ðx; �Þ

��þ k’kL1ð�Þ

Z��

uðx; �Þwkðx; �Þ d�ðx; �Þ

��

Z��

uðx; �Þvðx; �Þ d�ðx; �Þ

��;obtained by subtracting and adding the integrals

Z�

unðxÞwkðx; �nðxÞÞ dx e

Z��

uðx; �Þwkðx; �Þ d�ðx; �Þ;

and by using Schwartz’s inequality to estimate the first term of the decom-position.

Since ðunÞ is bounded in L2ð�Þ by a constant C and in view of the

definition of the ð�nÞ-convergence, letting n!þ1 we conclude, from theabove inequality, that

limn!þ1

Z�

unðxÞ�nðxÞ’ðxÞ dx�

Z��

uðx; �Þvðx; �Þ’ðxÞ d�ðx; �Þ

��

� Ck’kL1ð�Þ limn!þ1

kvn � wkð; �nÞkL2ð�Þ

þ k’kL1ð�ÞkukL2�ð��Þ

k�� wkkL2�ð��Þ

:

ð3:14Þ

By Proposition 3.5 2) with w0 ¼ 0 and (3.11) we have that

limk!þ1

limn!þ1

k�n � wkð; �nÞkL2�ð��Þ

¼ 0 ð3:15Þ

and then, when k!þ1, we obtain, from (3.14), convergence (3.12).If � 2 L2

ð�;Cð�ÞÞ, (3.15) can be written with wk ¼ �, and we get conver-gence (3.13) œ

4. EXAMPLES AND APPLICATIONS

In this section we present some examples of scale convergence andwe use this concept to determine the �-limit of non-periodic oscillatingfunctionals.


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4.1. Two-Scale Convergence and the Multi-Scale Convergence

In the above framework the two-scale and multi-scale convergenciesare particular cases of ð�nÞ-convergence. We recall the definition of two-scale convergence. Let Y :¼ ½0; 1�N .

Definition 4.1.1. A sequence ð�nÞ in L2ð�Þ is said to two-scale converge to

a limit �0 belonging to L2ð�� YÞ if, for any function in L2

ð�;C#ðYÞÞ,we have

limn!þ1

Z�

�nðxÞ ðx; nxÞ dx ¼

Z�

ZY

�0ðx; yÞ ðx; yÞ dx dy;

where L2ð�;C#ðYÞÞ denotes the space of L2-functions with values in the

Banach space of continuous Y-periodic functions.Defining T :¼ R

N=ZN as the N-dimensional torus, we may identifyC#ðYÞ with CðTÞ and define the measure � such that for all ’ 2 CðTÞ,h�; ’i :¼

RY ’ð yÞ dy. Considering, in Definition 3.1, � equal to T and

�nðxÞ :¼ ½nx�, where, for a 2 RN , ½a� represents its fractional part, then the

Young measure associated to ð�nÞ is � ¼ mN � �. In fact, by the Riemann-Lebesgue’s Theorem (c.f. [BLP], [JKO] or [SP]), for each ’ ¼ 12,1 2 L

2ð�Þ and 2 2 C#ðYÞ (which we identify with CðTÞ) the following

convergence holds

Z�

1ðxÞ2ð½nx�Þ dx!

Z�

1ðxÞ

ZY

2ðyÞ dy dx ¼

Z�

ZT

’ðx; �Þ d�ð�Þ dx:

So, the disintegration ð�xÞx2� at each point x is independent of x and coin-cides with � and, consequently, � ¼ mN � �.

The multi-scale convergence or ðmþ 1Þ-scale convergence (see [A] and[AB]) may also be recovered from the ð�nÞ-convergence by taking � :¼ Tm

and �n : ��, �nðxÞ :¼ ð½nx�; . . . ; ½nmx�Þ. More generally, consider

�nðxÞ :¼ ð½a1nx�; . . . ; ½a

mn x�Þ;

where ðainÞn, for i ¼ 1; . . . ;m, are m sequences of real numberssuch that limn!þ1 a

in ¼ þ1 and satisfying limn!þ1 a

in=a

iþ1n ¼ 0 for all

i ¼ 1; . . . ;m� 1. As a consequence of the Riemann-Lebesgue’s Theorem(see [AB]), the following convergence holds, for all ’ 2 L2

ð�;CðTmÞÞ,

Z�

’ðx; ½a1nx�; . . . ; ½amn x�Þ dx!

Z��Tm

’ðx; �1; . . . ; �mÞ d�1; . . . ; d�m dx;


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and we conclude that the Young measure associated to the sequenceð�nÞ is � ¼ mN � �� .

4.2. Quasiperiodic Oscillations

Also very important in the applications is the quasiperiodic case:considering the Y-periodic net in R

N (Y ¼ ½0; 1�N), instead of reproducingthe same structure in each cell, as in the periodic case, we allow it to varysmoothly from cell to cell. The torsion and the Neumann problems ina quasiperiodic perforated domain were first treated in [MP]. Later, inorder to optimize the microstructure of the perforated domain, the sameresults were improved in [CMT].

Another approach to quasiperiodicity, in some sense equivalentto the previous one, consists in considering a regular deformation ofthe Y-periodic net in R

N . The following example was treated in [Ax]where the notion of � � 2 convergence was introduced. Although theresults are the same, we think that our setting presented below is muchmore clear.

Let h be a C2 diffeomorphism in RN , so that h�1 will represent a defor-

mation of the Y-periodic net. Let y0 denote the geometric center ofY ¼ ½0; 1�N .

Let B represent an open ball in RN , centered at the origin and with

a radius r, depending on the Lipschitz constant of h�1, such that, forall k 2 Z

N ,

h�1ðy0 þ kÞ þ B �� h�1ðY þ kÞ:0

Fix an open bounded subset � of RN . For each n define

Hn;k :¼ h�1 y0 þ k

n

� þ1

nB and Hn :¼

[k2Z

N

Hn;k:

Consider, now, the sequence of domains �n :¼ � nHn. It seems naturalthat the asymptotic behavior of the sequence �n, as n!þ1, depends onlyon the gradient of the deformation h. In order to prove it, consider thefollowing open subset O of �� Y , that we identify with �� T ,

O :¼[x2�

½fxg � ðy0 þ rhðxÞBÞ�: ð4:2:1Þ

Define �n : �! T as �nðxÞ :¼ ½nhðxÞ�. The following proposition helpsto understand the role played by the ð�nÞ-convergence in the asymptoticbehavior of the domains �n.


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Proposition 4.2.1. Let ’ 2 L2ð�;CðTÞÞ:

1) If supp ’ � ð�� YÞ n O, then there exists n0 such that n > n0 impliesthat ’ðx; ½nhðxÞ�Þ ¼ 0 for all x 2 Hn.

2) If supp ’ � O, then there exists n0 such that n > n0 implies that’ðx; ½nhðxÞ�Þ ¼ 0 for all x 2 �n.

Proof. 1) Suppose supp ’ � ð�� YÞ n O. If x 2 h�1ðy0þkn Þ þ1n B, then, for

some z 2 B, we have

’ðx; ½nhðxÞ�Þ ¼ ’ h�1y0 þ k

n

� þz

n

�; y0 þ rh h�1

y0 þ k

n

� � zþ n oð1=nÞ

:

Since h 2 C2ðRN;RNÞ, jnoð1=nÞj < c=n, where c is a constant independent

of k. Since

h�1y0 þ k

n

� ; y0 þ rh h�1

y0 þ k

n

� � z

� 2 O;

there exists n0, independent of k, such that n > n0 implies that

h�1y0 þ k

n

� þz

n; y0 þ rh h�1

y0 þ k

n

� � zþ n oð1=nÞ

�

is in a neighborhood of O that does not intersect the support of ’. Then’ðx; ½nhðxÞ�Þ ¼ 0.

2) Suppose supp ’ � O. Let Q :¼� � 1=2; 1=2½N . For some k 2 ZN ,

x 2 h�1ðy0þkn þ 1nQÞ, then, for some y 2 Q, we have

x ¼ h�1y0 þ k

n

� þrh�1

y0 þ k

n

� y

nþ oð1=nÞ and ½nhðxÞ� ¼ y0 þ y: ð4:2:2Þ

If x 62 Hn;k, then nðx� h�1ðy0þkn ÞÞ 62 B and, by (4.2.2),

rh�1y0 þ k

n

� yþ n oð1=nÞ 62 B:

Consequently

yþ rh h�1y0 þ k

n

� � n oð1=nÞ 62 rh h�1

y0 þ k

n

� � B;


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which yields

h�1y0 þ k

n

� ; y0 þ yþ rh h�1

y0 þ k

n

� � n oð1=nÞ

� 2 ð�� YÞ n O:

Since, in view of the regularity of h, jrhðh�1ðy0þkn ÞÞn oð1=nÞj � c=n, with cindependent of k, we conclude from (4.2.2) that there exists n0, independentof k, such that if n > n0 we have ðx; ½n hðxÞ�Þ 62 supp ’ and then’ðx; ½n hðxÞ�Þ ¼ 0: œ

Corollary 4.2.2. The sequence of the characteristic functions n of the domains�n, ð�nÞ-converges to the characteristic function of ð�� YÞ n O.

Proof. By Theorem 3.2 there exists a subsequence nk of n and a function 2 L2

ð�� TÞ such that, for all ’ 2 L2ð�;CðTÞÞ,

Z�

nkðxÞ’ðx; ½nk hðxÞ�Þ dx!

Z��Y

ðx; yÞ’ðx; yÞ dx dy: ð4:2:3Þ

Using, in (4.2.3), ’, alternatively with support in O and with supportin ð�� YÞ n O, and since, in view of the regularity of h, the Lebesgue2N-dimensional measure of @O (see (4.2.1)) is zero, we conclude thatalmost everywhere in �� Y , coincides with the characteristic function of ð�� YÞ n O. Since the limit function is uniquely determined, conver-gence (4.3.3) holds for the entire sequence n. œ

Proposition 4.2.1 and Corollary 4.2.2 allow us to treat a large classof problems in the oscillating domains �n. Consider, as an example, a verysimple one: the following homogeneous Neumann problem:

�4un þ un ¼ f ; in �n;run �n ¼ 0; on @�n n @�;un ¼ 0; on @�;

8<: ð4:2:4Þ

where f 2 L2ð�Þ and �n represents the exterior normal to �n (we admit that �

is regular enough).We prove the following result (c.f. [MP], [CMT], [Br], [Ax]), where

the influence of the microstructure O (see (4.2.1)), as it is called in [CMT],is explicit. The proof is very close to the one presented in [MP], for the morecomplicated case where the term un is missing in (4.2.4), since there we cannotavoid uniform extensions, and also to the one presented in [A] for theperiodic case.


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Proposition 4.2.3. The zero extensions eunun of the solutions of problem (4.2.4)converge weakly in L2

ð�Þ to the solution u0 of the following homogenizedproblem

�div ðAru0Þ þ �u0 ¼ �f ; in �;un ¼ 0; on @�;

�ð4:2:6Þ

where A ¼ ½aij�; aijðxÞ ¼ �ðxÞ�ij �RYn½y0þrhðxÞB�

ry�xi ry�

xj dy; the functions �

xi ;

i ¼ 1; . . . ;N; are the solutions of the following microscopic problems

�4y �xi ¼ 0; in Y n ½y0 þ rhðxÞB�;

ry�xi � ¼ ��i; on @½y0 þ rhðxÞB�;R

Yn½y0þrhðxÞB��xi dy ¼ 0;

8<: ð4:2:7Þ

with �xi and ry�xi Y-periodic, �ðxÞ :¼

RY ðx; yÞ dy and �ij represents the

Kronecker tensor.

Proof. From the variational formulation of (4.2.4), i.e.,Z�n

runðxÞrðxÞ þ unðxÞ ðxÞ dx ¼

Z�n

f ðxÞ dx; ð4:2:8Þ

for all 2 H1ð�nÞ, such that ¼ 0 on the exterior boundary @�, we obtain

the uniform bounds of the sequences ðkunkL2ð�nÞÞ and ðkrunkL2ð�nÞ

Þ. Thus, byTheorem 3.2, there exist two functions u and in L2

ð�� YÞ such that, upto a subsequence, the zero extensions of un and run, respectively representedby eunun and grunrun, ð�nÞ-converge to u and , respectively. Using Corollary 4.2.2and Corollary 3.6 we conclude that

uðx; yÞ ¼ uðx; yÞ ðx; yÞ and ðx; yÞ ¼ ðx; yÞ ðx; yÞ:

To obtain the link between u and we consider ’ 2 ½Dðð�� YÞ n OÞ�N ,so that, by Proposition 4.2.1, for n large enough the support of ’ðx; ½nhðxÞ�Þis in �n. Integrating by parts

R�nrun’ðx; ½nhðxÞ�Þdx, multiplying be 1=n and

passing to the limit, as n goes to þ1 , we obtain

limn!þ1

Z�n

un ðdivy’Þðx; ½nhðxÞ�Þ rhðxÞ dx ¼ 0:

In view of the scale convergence we conclude thatZð��YÞnO

uðx; yÞ ðdivy’Þðx; yÞ rhðxÞ dx ¼ 0;


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for all ’ 2 ½Dðð�� YÞ n OÞ�N and, consequently,

uðx; yÞ ¼ u0ðxÞðx; yÞ;

for certain u0, independent of y.If, on the other hand, we consider ’ 2 ½Dðð�� YÞ n OÞ�N such that

divy’ ¼ 0, also by integrating by partsR�nrun’ðx; ½nhðxÞ�Þ dx, and passing

to the limit, we conclude thatZð��YÞnO

½ ðx; yÞ � ru0ðxÞ� ’ðx; yÞ dxdy ¼ 0;

for all ’ 2 ½Dðð�� YÞ n OÞ�N such that divy’ ¼ 0. Setting ’ðx; yÞ ¼ ’0ðxÞ’1ðx; yÞ; ’1 2 ½Dðð�� YÞ n OÞ�

N and divy’ ¼ 0, we get for almost x 2 �,ZYn½y0þrhðxÞB�

½ ðx; yÞ � ru0ðxÞ� ’1ðx; yÞ dy ¼ 0; ð4:2:9Þ

for all ’1 such that divy’1 ¼ 0 and ’1 � ¼ 0 on @½y0 þ rhðxÞB�. From theorthogonality condition (4.2.9) we obtain that, for almost x 2 �, there existsu1ðx; Þ 2 H

1ðR

NÞ, Y-periodic, satisfying for almost y 2 Yn ½y0 þrh ðxÞB�;

ðx; yÞ ¼ ru0 þ ryu1ðx; yÞ: ð4:2:10Þ

Then, by a slicing argument, (4.2.10) holds in almost x 2 ð�� YÞ n O:Finally, to obtain the homogenized equation, we consider in the varia-

tional formulation (4.2.8) test functions ðxÞ :¼ ’ðx; ½nhðxÞ�Þ; where’ðx; yÞ ¼ ’0ðxÞ þ

1n ’1ðx; yÞ½rhðxÞ�

�1: Then, passing to the limit as n tendsto þ1 and using the scale convergencies of eunun, grunrun and of n, we obtainZ

ð��YÞnO

ðru0 þryu1Þ ðr’0 þ ry’1Þ þ

Z�

u0 � ’0 ¼

Z�

f � ’0:

Varying ’0 and ’1 we get by the usual procedure (see [A] or [MP]), the limitproblems (4.2.6) and (4.2.7). œ

Remark 4.2.4. Unfortunately when there are concentration effects, forinstance, when the Lebesgue measure of the supports of �n tend to zero,we do not obtain much information from the Young measures associatedto the sequence ð�nÞ, since they all reduce to the Dirac mass.

4.3. The !-Limit of Non-periodic Oscillating Functionals

The scale convergence method appeared to be an appropriate toolto determine the �-limit of energy functionals involving non-periodic


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oscillations. In fact the first ideas on scale convergence where developed in[M] and [To], in order to study nonlocal effects in a nonlinear first orderproblem with oscillating coefficients. Here we establish the more generalresult stated in Theorem 4.3.3.

Since we are going to deal only with equicoercive sequences of func-tionals, defined on a reflexive and separable Banach space X , endowed withthe weak topology, we may consider the following simplified definition of�-convergence (see [DM], Proposition 8.16):

Definition 4.3.1. Let X be a reflexive and separable Banach space. LetFn : X ! R be an equicoercive sequence of functionals. We say thatF 0 : X ! R is the �-lower limit of the sequence ðFnÞ, with respect to theweak topology, if:

1) for every � 2 X and for every sequence ð�nÞ converging weakly to � inX one has lim infn!1 Fnð�nÞ � F

0ð�Þ;

2) for every � 2 X there exists a sequence ð ��nÞ converging weakly to �such that lim infn!1 Fnð ��nÞ ¼ F

0ð�Þ.

We say that F 00 : X ! R is the �-upper limit of the sequence ðFnÞ, withrespect to the weak topology, if:

3) for every � 2 X and for every sequence ð�nÞ converging weakly to � inX one has lim supn!1 Fnð�nÞ � F

00ð�Þ;

4) for every � 2 X there exists a sequence ð ~��nÞ converging weakly to �such that lim supn!1 Fnð ~��nÞ ¼ F

00ð�Þ.

We say that the F : X ! R is the �-limit of the sequence ðFnÞ, withrespect to the weak topology, if F 0 ¼ F 00 ¼ F .

Let � be a metrizable compact space and �n : � �� a sequence ofmeasurable functions. Suppose that � is the Young measure associated toð�nÞ.

Consider now the following sequence of functionals ðJnÞ, Jn : L2� �R

defined by

Jnð�Þ ¼

Z�

f ðx; �nðxÞ; �ðxÞÞ dx; ð4:3:1Þ

where f : �� ð��RÞ�R satisfies the following hypotheses:

(H1) f and @f =@ are Caratheodory integrands: measurable in x 2 � andcontinuous in ð�; Þ 2 ��R.

(H2) there exists C > 0: 1C j j

2� f ðx; �; Þ � Cð1þ j j2Þ, for all ðx; �; Þ

2 ��R.(H3) f is convex in .


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Remark 4.3.2. As a consequence of the previous hypotheses we also have, forsome constant C > 0 and for all ðx; �; Þ 2 ��R,

@f

@ ðx; �; Þ

�� Cð1þ j jÞ: ð4:3:2Þ

Theorem 4.3.3. Under hypothesis (H1), (H2) and (H3) the sequence ðJnÞ,defined in (4.3.1), �-converges to the following functional J : L2

ð�Þ�R,

Jð�Þ :¼ infw2HR

�wd�x¼v

Z��

f ðx; �;wðx; �ÞÞ d�ðx; �Þ;

where H :¼ L2�ð��Þ.

Proof. In what follows C will denote any positive constant eventually chang-ing from line to line.

In view of Corollary 8.12 and of Proposition 8.17, in [DM], in order toprove that the functional J is the �-limit of the sequence ðJnÞ, it is enough toshow that the �-lower limit of any subsequence ðJnkÞ of ðJnÞ coincides with J.In what follows we will deal with the sequence ðJnÞ, but the same argumentholds for any subsequence.

We will divide the proof into two parts, representing, to simplify,H :¼ L2

ð�Þ.1st part : we prove that for all � 2 H and for all sequences ð�nÞ such that

�n * � weakly in H, lim infn!þ1 JnðvnÞ � Jð�Þ.Let � 2 H and let ð�nÞ be such that �n * � weakly in H. Let ð�nkÞ be

a subsequence of ð�nÞ satisfying

lim infn!þ1

Jnð�nÞ ¼ limk!þ1

Jnkð�nkÞ:

Using Theorem 3.2 we may also suppose that ð�nkÞ ð�nkÞ-converges to somew 2 H, with

R� wðx; �Þd�xð�Þ ¼ �ðxÞ, for a.e. x 2 �. In order to simplify the

notations we assume, with no loss of generality, that ð�nkÞ coincides with thewhole sequence ð�nÞ.

Let now ðwkÞ be a sequence in L2ð�;Cð�ÞÞ such that kwk � wkH < 1=k,

and define wknðxÞ :¼ wkðx; �nðxÞÞ. Using the fact that f is convex, we have

for all n 2 N and for all k 2 N:

Z�

f ðx; �n; �nÞ dx �

Z�

f ðx; �n;wknÞ dxþ

Z�

@f

@ ðx; �n;w

knÞð�n � w

knÞ dx:


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Therefore

Jnð�nÞ � JnðwknÞ þ

Z�

@f

@ ðx; �n;w

knÞð�n � w

knÞ dx: ð4:3:3Þ

In view of (4.3.2) ð@f =@ Þ ðx; �;wkðx; �ÞÞ is in L2ð�;Cð�ÞÞ and, using the

ð�nÞ-convergence of ð�nÞ,

limn!þ1

@f

@ ðx; �n;w

knÞ�n dx ¼

Z��

@f

@ ðx; �;wkðx; �ÞÞwðx; �Þ d�ðx; �Þ:

Also by (4.3.2), using Proposition 3.3 with ðx; �; Þ ¼ @f =@ ðx; �; Þ ,we obtain

limn!þ1

Z�

@f

@ ðx; �n;w

knÞw

kn dx ¼

Z��

@f

@ ðx; �;wkðx; �ÞÞwkðx; �Þ d�ðx; �Þ:

Analogously by Proposition 3.3, in view of (H2),

limn!þ1

JnðwknÞ ¼

Z��

f ðx; �;wkðx; �ÞÞ d�ðx; �Þ:

Inequality (4.3.3) yields then:

lim infn!1

Jnð�nÞ �

Z��

f ðx; �;wkðx; �ÞÞ d�ðx; �Þ

þ

Z��

@f

@ ðx; �;wkÞðw� wkÞ d�:

ð4:3:4Þ

But, due to (4.3.2) and since ðwkÞ is bounded in H we haveZ��

@f

@ ðx; �;wkÞðw� wkÞ d�

�� C 1

k: ð4:3:5Þ

Using hypotheses (H1) and (H2), u 2 L2�ð��Þ� f ðx; �; uÞ 2 L1

ð��Þ isa Nemitsky operator and consequently, since lim

k!þ1kwk � wkH ¼ 0, we get

limk!þ1

Z��

f ðx; �;wkðx; �ÞÞ d�ðx; �Þ ¼

Z��

f ðx; �;wðx; �ÞÞ d�ðx; �Þ: ð4:3:6Þ

Passing to the limit in (4.3.4) as k goes to þ1 and using estimate (4.3.5) andconvergence (4.3.6) we obtain

lim infn!1

Jnð�nÞ �

Z��

f ðx; �;wðx; �ÞÞ d�ðx; �Þ � JðvÞ:


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2nd part: we prove that for all � 2 H there exists a sequence ð ��nÞ, ��n * �weakly in H, such that lim infn Jnð ��nÞ ¼ Jð�Þ.

Let � 2 H and let ðwkÞ be a minimizing sequence for

infw2HR

�wd�x¼�

Z��

f ðx; �;wðx; �ÞÞ d�ðx; �Þ:

For almost every x 2 � we haveR� w

kðx; �Þ d�xð�Þ ¼ �ðxÞ. In view of the

coercivity hypothesis (H2) the sequence ðwkÞ is bounded in H. For each k,let wwk 2 L2

ð�;Cð�ÞÞ satisfy kwk � wwkkH < ð1=kÞ.We claim that

limk!þ1

Z��

f ðx; �; wwkðx; �ÞÞ d�ðx; �Þ ¼ Jð�Þ ð4:3:7Þ

and

limk!þ1

Z��

wwkðx; �ÞðxÞ d�ðx; �Þ ¼

Z�

�ðxÞðxÞ dx; ð4:3:8Þ

for every 2 H. In fact convergence (4.3.8) follows from inequality

Z��

ð wwkðx; �Þ � wkðx; �ÞÞðxÞ d�ðx; �Þ

�� k wwk � wkkHkkH � 1

kkkH;

while in order to establish assertion (4.3.7) we remark that the growth con-dition (4.3.2) yields

j f ðx; �; Þ � f ðx; �; 0Þj � Cð1þ j j þ j 0jÞj � 0j:

Assertion (4.3.7) is, then, a consequence of the following estimates

Z��

�� ½ f ðx; �;wkðx; �ÞÞ

�

Z��

Cð1þ jwkðx; �Þj þ j wwkðx; �ÞjÞjwkðx; �Þ � wwkðx; �Þj d�ðx; �Þ �C

k;

where we have used Holder inequality, the fact that the sequences ðwkÞ andð wwkÞ are bounded in H and the definition of ð wwkÞ.

We finally construct the sequence ð ��nÞ by using a diagonalization pro-cedure, similar to the one used in the proof of Proposition 3.4. ConsiderwwknðxÞ :¼ wwkðx; �nðxÞÞ. Let ðjÞ be a countable dense family in L2

ð�Þ andnormalize it. For all k 2 N and for all j 2 N, by Proposition 3.3, we have,


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as n goes to þ1,

Z�

f ðx; �nðxÞ; wwknðxÞÞ dx!

Z��

f ðx; �; wwkðx; �ÞÞ d�ðx; �Þ;Z�

wwknðxÞjðxÞ dx!

Z��

wwkðx; �ÞjðxÞ d�ðx; �Þ:

Then, for all j 2 N, k 2 N and for all � > 0, there exists n0 ¼ n0ð j; k; �Þ suchthat for all n > n0ð j; k; �Þ and i � j we have

Z�

f ðx; �nðxÞ; wwknðxÞÞ dx�

Z��

f ðx; �; wwkðx; �ÞÞ d�ðx; �Þ

�� < �;

wwknðxÞiðxÞ dx�

Z��

wwkðx; �ÞiðxÞ d�ðx; �Þ

�� < �:

Taking � ¼ 1=k, j ¼ k and defining an increasing sequence ðnkÞ such thatnk > n0ðk; k; 1=kÞ, we obtain

Z�

f ðx; �nkðxÞ; wwknkðxÞÞ dx�

Z��

f ðx; �; wwkðx; �ÞÞ d�ðx; �Þ

�� < 1

k;

wwknkðxÞiðxÞ dx�

Z��

wwkðx; �ÞiðxÞ d�ðx; �Þ

�� < 1

k;

for all i � k. Passing to the limit as k goes to þ1 and using convergencies(4.3.7) and (4.3.8) it follows that

Z�

f ðx; �nk; wwknkðxÞÞ dx! Jð�Þ

and

Z�

wwknkðxÞiðxÞ dx!

Z�

�ðxÞiðxÞ dx;

for all i 2 N. The last convergence still holds for the non normalized initialdense family i, by multiplying it by the norm of i, for each i. Since ð wwkÞis bounded in H then ð wwknkÞ is bounded in L2

ð�Þ and, by a density argument,we obtain the last convergence for all 2 L2. Defining ��n :¼ wwknk if n ¼ nkand ��n :¼ � if not, the sequence ð ��nÞ satisfies, then, lim infn!þ1 Jnð ��nÞ ¼ Jð�Þand ��n ! � weakly in H. œ


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5. DISCRETIZATION OF YOUNG MEASURES.

NUMERICAL EXAMPLES

This section presents a method, inspired by the Monte-Carlo method,for numerical approximation of Young measures. Given a sequence of func-tions, it allows us to compute approximations of the density of the associatedYoung measure. This method can also be used to compute the weak limit ofthe sequence obtained by composition of the given one with a continuousfunction. Other methods of discretization of Young measures are describedin [R].

Approximations of scale limits can be obtained by our method, since theð�nÞ-limit of a sequence ð�nÞ is given by the Young measure associated to theduble sequence ð�n; �nÞ (see the proof of Theorem 3.2).

We describe the numerical method first for a Young measure associatedto a function and then for a Young measure associated to a sequence offunctions.

Let us consider a function u : ½a; b��½�; �� and its associated Youngmeasure �u, concentrated along the graph of u. For each k 2 N we considerthe partition a ¼ x0 < x1 < . . . xk ¼ b of ½a; b�. For simplicity we assume thek intervals to be equal. We define the approximation �ku of the measure �uas follows

�ku :¼m1ð½a; b�Þ

k

Xkl¼1

�ðxl;uðxlÞÞ: ð5:1Þ

The approximation is justified since, for each interval I :¼ I1 � I2 �½a; b� � ½�; ��, we have that �kuðIÞ ! �uðIÞ as k goes to þ1. Infact �uðIÞ ¼ m1ðI1 \ u

�1ðI2ÞÞ yields the convergence of the following

Riemann sum

�kuðIÞ ¼m1ð½a; b�Þ

k

Xkl¼1

�xl ðI1 \ u�1ðI2ÞÞ ! �uðIÞ:

Consider now a sequence of functions ðunÞ and � its associated Youngmeasure. We compute a discretization �k;m of � by averaging a large numberm of Young measures associated to the terms of the sequence. The discretizedYoung measure is then given by the formula

�k;m :¼1

m

Xmn¼1

�kun ¼m1ð½a; b�Þ

km

Xkl¼1

Xmn¼1

�ðxl;unðxlÞÞ: ð5:2Þ


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For every interval I :¼ I1 � I2 � ½a; b� � ½�; ��, we have that �k;mðIÞ ! �ðIÞas k goes to þ1 and the order of the terms taken into account is sufficientlylarge.

In order to visualize the density of the measure �k;m, we dividethe domain ½a; b� � ½�; �� in small rectangles. For each rectangle we countthe number p of points ðxl; unðxlÞÞ in it. On each rectangle we assume that themeasure has constant density, proportional to p, and we represent it byshading the rectangle with a gray level (darker regions meaning higher den-sity). To visualize a weak limit, we join the discretized points obtained byintegration.

We tested the method with simple examples, in order to comparethe approximations with the corresponding exact Young measures. InExample 5.1 we consider the sequence ðsinðn�xÞÞ; in Example 5.2 we con-sider ðsinð�xÞ þ sinðn�xÞÞ. Finally, in Example 5.3, we add a perturbation�n to the previous sequence and study its influence on the obtainedapproximations.

The error is controled by computing the L1 distance between the densityof the exact Young measure and the density of the approximated Youngmeasure.


Figure 5.1.

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Example 5.1. Consider the sequence �n : ½0; 2� ! ½�1; 1�, defined by�n :¼ sinðn�xÞ. The Young measure � ¼ ð�xÞ associated to the sequenceðsinðn�xÞÞ is given by

h’; �xi ¼1

2

Z 2

0

’ðsinðn � yÞÞ dy ¼1

�

Z 1

�1

’ðzÞffiffiffiffiffiffiffiffiffiffiffiffiffi1� z2

p dz;

for all function ’ in Cð½�1; 1�Þ. Figure 5.1 represents the density of the exactYoung measure �. The weak limit of ðsinðn�xÞÞ is the constant function 0, asindicates the value of the barycenter of �x, hz; �xi ¼1�

R 1

�1ðzÞ=ðffiffiffiffiffiffiffiffiffiffiffiffiffi1� z2

pÞ dz ¼ 0, for a.e. x 2 ½0; 2�.

Figure 5.2 shows the results of our method with 200 terms of thesequence sinðn�xÞ for x 2 ½0; 2�; visualizing the density of the dicretizedYoung measure in the rectangle ½0:0; 1:9999� � ½�1:0; 1:0� and also thegraph of its barycenter. We used k ¼ 1500 points xi and a grid of 50� 50rectangles. The error was equal to 0.1049.

Example 5.2. Consider the sequence �n : ½0; 2� ! ½�2; 2�, defined by�n :¼ sinð�xÞ þ sinðn�xÞ. The Young measure associated to ð�nÞ is given by:

h’;�xi ¼1

�

Z 1þsinð�xÞ

�1þsinð�xÞ

’ðzÞffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� ðz� sinð�xÞÞ2

q dz;

for all functions ’ in Cð½�2; 2�Þ.


Figure 5.2.

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Applying the method with 1500 terms of the sequence ðsinð�xÞþsinðn�xÞÞ we obtain the density of the discretized Young measure representedin Figure 5.3. The results were compared with the exact Young measureobtaining the error of 0:2811. As expected, the graph of the barycenterpractically coincides with the graph of the function sinð�xÞ, i.e., the weaklimit of ðsinð�xÞ þ sinðn�xÞÞ. The approximations obtained for the secondand third moments of the Young measure were represented in the samefigure.

Example 5.3. Consider also the following sequence whose oscillating beha-vior is governed by a perturbation �n:

�n :¼ sinð�xÞ þ sinðn�xÞ þ �nðxÞ:


second moment

barycenter

third moment

Figure 5.3.

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The function �n takes random values such that their absolute value isless then a certain constant � (0 < � < 1). Assume that the values of �n areuniformly distributed in the interval ½��; ��, that is the Young measure �associated to ð�nÞ has constant density with respect to the Lebesgue measure:h�x; ’2i ¼

12�

R �� ’2ð�Þ d�; for all ’2 in Cð½��; ��Þ:

It is reasonable to assume that the sequences ð�nÞ and ðsinð�xÞþ sinðn�xÞÞ are independent, and obtain the Young measure � associatedto ð�nÞ

h ;�xi ¼1

2��

Z 1þsinð�xÞ

�1þsinð�xÞ

Z zþ�

z��

ð yÞ dydzffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� ðz� sinð�xÞÞ2

q ;

for all function in Cð½�2� �; 2þ ��Þ. By interchanging the integrals weobtain

h ;�xi ¼1

2��

Z �1þ�þsinð�xÞ

�1��þsinð�xÞ

� ð yÞðarcsinðyþ �Þ � arcsinð�1þ sinð�xÞÞÞ dy

þ

Z 1��þsinð�xÞ

�1þ�þsinð�xÞ

ð yÞðarcsinðyþ �Þ � arcsinð y� �ÞÞ dy

þ

Z 1þ�þsinð�xÞ

1��þsinð�xÞ

ð yÞðarcsinð1þ sinð�xÞÞ � arcsinð y� �ÞÞ dy

:

Note that the Young measure associated to ð�nÞ has finite density in allpoints (the measure has no concentrations any more).

We tested how this perturbation influences the density and the bary-center of the discretized Young measure, by taking 1500 terms.

Figure 5.4. represents the results obtained for a random perturbation�n 2 ½�0:15; 0:15�, while the exact Young measure is represented in Figure 5.5.The error was equal to 0:0722.

Considering a random perturbation in ½�0:5; 0:5� we obtained theresults in Figure 5.8. The exact Young measure is represented in Figure 5.9and the error is equal to 0:0675.

Figures 5.6 and 5.7 show the discretized and respectively exact Youngmeasures for a random perturbation �n 2 ½�0:25; 0:25�; The error was0.0691.

Considering a random perturbation in [�0.5, 0.5] we obtained theresults in Figure 5.8. The exact Young measure is represented in Figure 5.9and the error is equal to 0.0675.

We observe that, as the amplitude of the random perturbation increases,the density of the discretized Young measure gets more diffuse. The darkregions appearing in Figure 5.3, corresponding to concentrations, disappear


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Figure 5.4.

Figure 5.5.

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Figure 5.6.

Figure 5.7.

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Figure 5.8.

Figure 5.9.

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in Figures 5.4, 5.6 and 5.8, which confirms that, in the presence of a pertur-bation, the Young measure has finite density in all points.

In Figures 5.4, 5.6 and 5.8 we represented also the barycenter of thediscretized Young measure that is, an approximation of sinð�xÞ, the weaklimit of the sequence.

ACKNOWLEDGMENTS

This research work was supported by JNICT-PRAXIS XXI, FEDER-PRAXIS/2/2.1/MAT/125/94, PRAXIS-FEDER/3/3.1/CTM/10/94 and JNICT-MENESR 97/166.

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scale convergence in homogenization

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