Variational multiscale problems and applications to thin …mbaia/thesis-June-23.pdf · Variational...

Variational multiscale problems and

applications to thin films

Margarida Baıa

Department of Mathematical Sciences

Carnegie Mellon University

Pittsburgh, PA 15213, USA

May 2, 2005

Submitted in partial fulfillment of the requirements for the degree of

DOCTOR OF PHILOSOPHY IN MATHEMATICAL SCIENCES

CARNEGIE MELLON UNIVERSITY

Advisor: Irene Fonseca

To the memory of my uncle Manuel.

To my parents Lucılia and Justino

Acknowledgements

First of all, I would like to express my deep gratitude to my advisor, Professor Irene Fonseca,

for her warm support, for all her guidance, and for all that she has taught me during

my PhD studies. I thank her for encouraging me to collaborate with other students and

for encouraging me to participate in many interesting conferences. I am thankful for the

financial support she provided through the Center for Nonlinear Analysis and the National

Science Foundation. These years have been very gratifying.

A big part of the results presented in this thesis were obtained in collaboration with Jean

Francois Babadjian. I thank him for being an excellent and patient collaborator. I thank

his advisor, Professor Gilles Francfort, for several suggestions.

I am extremely grateful to Professor Giovanni Leoni for stimulating discussions and for all

the help he gave me.

I would like to thank the other members of my committee, Professor Leonid Berlyand,

Professor William Hrusa and Professor David Owen for their comments and suggestions on

the first draft of my thesis.

I thank Professor Alexei Novikov for his interest and for several discussions that lead us to

generalize one of the results.

A special appreciation goes to Professor Luıs Magalhaes for his encouragement to pursue a

PhD program in a good foreign university.

I would like to thank Professor Adelia Sequeira for supporting my decision to come to

Carnegie Mellon University.

I thank the Mathematics Department of Instituto Superior Tecnico for granting me the

opportunity to study abroad, and the Department of Mathematical Sciences of Carnegie

Mellon University for its hospitality during these five years.

Further thanks go to my parents, for their understanding and unconditional support.

I consider myself very fortunate to have made a lot of new friends in Pittsburgh and I thank

them all for brightening my stay here. I am grateful for the scholarly help some of them

vi

gave me. In particular, I owe a special appreciation to Massimiliano Morini and Pedro

Santos for the time they spent helping me to clarify some of my mathematical questions.

I am grateful to Enrico Babilio and Bernardo Sousa for their help with computers, and I

thank Luca Deseri and Giuseppe Zurlo for some insights into Continuum Mechanics.

I thank my friends overseas for being there for me. In particular, I thank Pedro Girao for

his strong encouragement.

A well deserved treat goes to Gato for his many hours of company during my late nights of

study.

My research was partially supported by Fundacao para a Ciencia e a Tecnologia under

Grant PRAXIS XXI SFRH\BD \ 1174 \ 2000, Fundo Social Europeu, the Department of

Mathematical Sciences of Carnegie Mellon University and its Center for Nonlinear Analysis,

and the National Science Foundation under Grant DMS-0401763.

Abstract

The main objective of this dissertation is to study the asymptotic behavior of two kinds of

multiple scale problems by Γ-convergence: Relaxation problems involving families of multi-

ple scale integral functionals, and 3D-2D reduction problems for heterogeneous thin domains

with periodic microstructure. Periodicity, standard growth conditions and nonconvexity are

assumed whereas a stronger uniform continuity with respect to the macroscopic variable,

normally required in the existing literature, is avoided.

Key words: Integral functionals, periodic integrands, Γ-convergence, two-scale conver-

gence, quasiconvexity, equi-integrability, dimension reduction, thin films.

LIST OF NOTATIONS

• R := R ∪ ∞;

• [[a]]: Integer part of a;

• ∇i := ∂/∂xi;

• xα := (x1, x2);

• ∇α = (∇1,∇2);

• Rd×N (resp. Qd×N ) ≡ set of real (resp. rational)-valued d×N matrices;

• (ξ|z) with ξ ∈ R3×2 and z ∈ R3: Matrix whose first two columns are those of ξ and

the last one is z;

• B(a, δ) := x ∈ RN : |x− a| < δ, a ∈ RN , δ > 0;

• Q = (0, 1)N ;

• Q(a, δ) := a+ δQ ≡ (a, δ)N , a ∈ RN , δ > 0;

• Q′= (0, 1)2;

• Q′(a, δ) := a+ δQ′, a ∈ R2, δ > 0;

• χA : Characteristic function of a set A;

• A: Closure of A;

• ∂A: Boundary of A;

• A ⊂⊂ B: A ⊂ B, A compact;

x

• LN (E) or |E|: Lebesgue measure of E ⊂ RN ;

• A(Ω): Open subsets of Ω;

• A0(Ω) : Open and bounded subsets of Ω;

• A∞(Ω): Lipschitz subsets of Ω;

• C∞(X; Rd): Rd-valued functions defined in X with derivatives of any order in X

(C∞(X) if d = 1);

• supp(u): Support of u;

• Cc(X; Rd): Rd-valued functions defined in X with compact support in X (Cc(X) if

d = 1);

• C∞c (X; Rd) := C∞(X; Rd) ∩ Cc(X; Rd);

• C0(X; Rd) := Cc(X; Rd) with respect to the supremum norm;

• C∞0 (X; Rd) := C∞(X; Rd) ∩ C0(X; Rd);

• Cper(Q; Rd): Q- periodic continuous functions defined in RN with values in Rd (Cper(Q)

if d = 1);

• W 1,pper(kQ; Rd): W 1,p-closure of all kQ- periodic and C1-functions defined on RN with

values in Rd (W 1,pper(kQ) if d = 1);

• Lp(X,µ) or Lp(X,µ; Rd): Usual scalar and vectorial Lebesgue spaces(

Lp(X; Rd) if

µ = LN or even Lp(X) if also d = 1)

;

• ess supx∈X

|u(x)| = ||u||L∞(X);

• W 1,p(X; Rd) or W 1,p(X) if d = 1: Usual Sobolev spaces;

• W 1,p(ω; R3): u ∈ W 1,p(Ω; R3) such that ∇3u(x) = 0 for a.e. x ∈ Ω, Ω := ω × I,

I := (−1, 1);

• ... Weak convergence in Lp or W 1,p;

• ⋆ ... Weak⋆ convergence in the sense of measures; also weak⋆ convergence in L∞ or

in W 1,∞;

xi

• s.l.s.c: Sequential lower semicontinuous;

• s.w.l.s.c on W 1,p(Ω; Rd) and s.w⋆.l.s.c on W 1,∞(Ω; Rd): s.l.s.c with respect to the

weak or weak⋆ convergence of W 1,p(Ω; Rd) and W 1,∞(Ω; Rd);

• slsc I: Sequential lower semicontinuous envelope of I;

• swlsc I: Sequential lower semicontinuous envelope of I with respect to a weak topol-

ogy;

• Qf : Quasiconvex envelope of f ;

• Ker(T ): Kernel of an operator T ;

• Γ(Lp(Ω))-limit: Γ-convergence with respect to the usual metric in Lp(Ω; Rd);

• limk,m,n

:= limk

limm

limn

, with obvious generalizations.

CONTENTS

1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

Part I Preliminaries and Previous results 9

2. Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.1 A short review of Measure Theory . . . . . . . . . . . . . . . . . . . . . . . 11

2.1.1 Measures and integration . . . . . . . . . . . . . . . . . . . . . . . . 11

2.1.2 Radon measures and Vitali’s Covering Theorem . . . . . . . . . . . . 16

2.1.3 Decomposition and differentiation of measures . . . . . . . . . . . . 19

2.1.4 Weak⋆ convergence of measures . . . . . . . . . . . . . . . . . . . . . 21

2.2 Sobolev spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

2.2.1 Definition and main properties . . . . . . . . . . . . . . . . . . . . . 22

2.2.2 Extension, approximation and traces . . . . . . . . . . . . . . . . . . 23

2.2.3 Compactness and Poincare inequalities . . . . . . . . . . . . . . . . . 24

2.2.4 Weak convergence and decomposition lemmas for sequences of gradi-

ents and of scaled-gradients . . . . . . . . . . . . . . . . . . . . . . . 26

2.3 An overview of the Direct Method of the Calculus of Variations . . . . . . . 29

2.3.1 The basic notions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

2.3.2 Convex and quasiconvex functions: main properties . . . . . . . . . 32

Contents xiv

2.3.3 Lower semicontinuity characterization for integral functionals defined

on Sobolev spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

2.4 Integral representation of nonlinear local functionals defined on Sobolev spaces 39

2.5 Γ-convergence of a family of functionals . . . . . . . . . . . . . . . . . . . . 42

2.5.1 The notion of Γ-convergence and main results . . . . . . . . . . . . . 43

2.5.2 The Direct Method of Γ-convergence for a class of integral functionals 47

2.6 Two-Scale Convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

2.6.1 Generalized Riemann-Lebesgue Lemmas . . . . . . . . . . . . . . . . 55

2.6.2 The notion of two-scale convergence and some properties . . . . . . 56

3. Variational problems in periodic homogenization: Previous results . . . . . . . . 57

3.1 Pure periodic (iterated) homogenization . . . . . . . . . . . . . . . . . . . . 57

3.1.1 The case where Iε(u) =

∫

Ωf(x

ε,∇u

)

dx . . . . . . . . . . . . . . . 57


∫

Ωf(

x,x

ε, u,∇u

)

dx . . . . . . . . . . . . . 62


∫

Ωf(

x,x

ε,x

ε2,∇u

)

dx . . . . . . . . . . . . 63

3.2 Thin films with periodic microstructure in the nonlinear membrane theory . 64

3.2.1 The case Wε(u) =

∫

ΩW

(

x,xαε,∇αu

∣

∣

∣

1

ε∇3u

)

dx . . . . . . . . . . . 67


∫

ΩW

(

x,x

ε,xαε2,∇αu

∣

∣

∣

1

ε∇3u

)

dx . . . . . . . . . 69

Part II Main results 71

4. Γ-convergence of functionals with periodic integrands . . . . . . . . . . . . . . . . 73

4.1 An approach by 2-scale convergence . . . . . . . . . . . . . . . . . . . . . . 73

4.1.1 Properties of the homogenized density . . . . . . . . . . . . . . . . . 76

4.1.2 Proof of the main result . . . . . . . . . . . . . . . . . . . . . . . . . 86

4.2 Multiple scale functionals . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

Contents xv


4.2.2 Main result when the integrands do not depend on the macroscopic

variable . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

4.2.3 The general case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122

4.2.4 Some remarks in the convex case . . . . . . . . . . . . . . . . . . . . 130

5. Application to thin films . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133

5.1 Thin films with periodic microstructure in the in-plane direction . . . . . . 133


5.1.2 Main result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140

5.2 When heterogeneities are allowed also in the transverse direction . . . . . . 149


5.2.2 Main result when the integrands do not depend on the macroscopic

variable . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152

5.2.3 The general case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159

6. Generalizations and further work . . . . . . . . . . . . . . . . . . . . . . . . . . . 169

Appendix 173

A Auxiliary lemmas for periodic homogenization . . . . . . . . . . . . . . . . . 175

B Continuous extension results for the applications to thin films . . . . . . . . 175

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186

LIST OF FIGURES

1.1 Domains with periodic structure: Examples . . . . . . . . . . . . . . . . . . 3

3.1 Two different behaviors under compression . . . . . . . . . . . . . . . . . . 61

3.2 Cylindrical thin domain of thickness ε . . . . . . . . . . . . . . . . . . . . . 65

3.3 Rescaled domain of unit thickness . . . . . . . . . . . . . . . . . . . . . . . 66

3.4 Thin domain with periodic structure in the in-plane direction . . . . . . . . 69

1. INTRODUCTION

The main objective of this dissertation is to study the effective behavior of elastic (thin)

bodies with multiple scales and periodic microstructure. This study is undertaken from a

variational point of view through an asymptotic analysis based on Γ-convergence arguments.

The asymptotic analysis of media with multiple scale of homogenization is referred to in

the literature as Reiterated Homogenization.

Roughly speaking, the aim of homogenization theory is to describe the behavior of micro-

scopically heterogeneous composite physical structures by means of homogeneous structures

with global characteristics equivalent to the initial ones. In many physical situations the

heterogeneities are very small in comparison with the region in which the structure is to

be studied and the heterogeneities are evenly distributed, so that they can be modelled by

a periodic distribution of period a small parameter. In practice, one is interested in the

global behavior of these structures when the heterogeneities are very, very small. From

the mathematical point of view, we are led to characterizing the asymptotic behavior of

(systems of) ordinary or partial differential equations with oscillating periodic coefficients

of period a small parameter ε, as ε tends to zero.

A well-known model problem in periodic homogenization, used frequently to describe ther-

mal as well as electrical or linear elasticity properties in a periodic composite medium has

as underlying the following linear second-order partial differential equation

−div(

A(x

ε

)

∇uε)

= g on Ω. (1.1)

Here Ω is the (material) domain in RN (N > 1), A is a scalar or tensor-valued function with

periodic coefficients, and uε and g are scalar or vector-valued functions in some appropriate

functional spaces. One wishes to know the asymptotic behavior of the solutions uε as ε→ 0.

1. Introduction 2

They converge, under appropriate hypotheses, to a solution of an “homogenized” differential

equation of the form

−div(Ahom(∇u)) = g on Ω.

Starting with the use of asymptotic expansions methods (see Bensoussan, Lions and Papan-

icolau [14], Jikov, Kozlov and Oleinik [58] and Sanchez-Palencia [72]) adapted to the study

of periodic problems like (1.1), homogenization techniques evolved toward more general sit-

uations through the concepts of G-convergence due to Spagnolo (see [74]), of H-convergence

due to Murat and Tartar (see [68] [67], and [76]), of Γ-convergence due to De Giorgi (see [38]

and [40]), and of two-scale convergence due to Nguetseng (see [61], [69] and [70]), further

developed by Allaire and Briane (see [4] and [5]), and generalized by many other authors.

We refer to the book of Cioranescu and Donato [32] for an introduction to homogenization

and for an overview of different homogenization methods.

From a variational point of view, for instance in the context of elasticity, the theory of

periodic homogenization rests on the study of a family of minimum problems

min

∫

Ωfε(x, u(x),∇u(x)) dx+

∫

Ωug dx : u = ϕ on ∂Ω

, (1.2)

where the functions fε (the elastic density energy) are increasingly oscillating in the first

variable as ε tends to zero, and u (the deformation), g (the density of applied body forces)

and ϕ are scalar or vector-valued functions in some Sobolev space. In the example (1.1)

if A = (Aij) and u is a scalar function, fε(x,∇u) =∑

Aij(xε )∇iu∇ju, where ∇i = ∂/∂xi.

More general minimum problems can be considered but in this Introduction we restrict to

this case for simplicity. The homogenization of the family of minimum problems (1.2) leads

to an “effective homogenized minimum problem” (not depending on ε)

min

∫

Ωfhom(x, u(x),∇u(x)) dx+

∫

Ωug dx : u = ϕ on ∂Ω

(1.3)

such that a sequence of minimizers of (1.2) converges, as ε tends to zero, to a limit u, which

is a minimizer of (1.3). The fundamental property of De Giorgi’s notion of Γ-convergence,

and its main link to the other homogenization techniques, is that, under certain growth and

compactness properties on fε and some regularity on g, it implies a sequence of minimizers

of (1.2) has this convergence property.

Due to the the properties of Γ-convergence (see Theorem 2.5.11 and Propositions 2.5.6 and

2.5.13 below), the convergence of minimizers (or almost minimizers) of (1.2) to minima of

1. Introduction 3

(1.3) can be derived from the Γ-convergence of the family

Iε(u) =

∫

Ωfε(x, u(x),∇u(x)) dx (1.4)

to the homogenized functional

Ihom(u) =

∫

Ωfhom(x, u(x),∇u(x)) dx.

This functional provides the macroscopic, or average description, of the periodic body by

capturing the limiting behavior of the equilibrium states of Iεε. The effective energy

density fhom is to be determined.

In this work we seek to approximate, in a Γ-convergence sense, the behavior of elastic (thin)

bodies whose microstructure is periodic of period ε and ε2 (Figure. 1.1).

ε

ε ε

Fig. 1.1: Domains with periodic structure: Examples

ε

ε

ε2ε

To describe this behavior, let us introduce some notation. We identify Rd×N (resp. Qd×N )

with the set of real (resp. rational)-valued d × N matrices, with d,N > 1. For ξ ∈ R3×2

and z ∈ R3, let (ξ|z) denote the matrix whose first two columns are those of ξ and the last

one is z. Let xα := (x1, x2), with x1, x2 ∈ R and let ∇α = (∇1,∇2). We will consider two

families of energies:

Iε(u) =

∫

Ωf(

x,x

ε,x

ε2,∇u(x)

)

dx

for Ω ⊂ RN , and

Wε(u) :=

∫

ΩW

(

x,x

ε,xαε2,∇αu(x)

∣

∣

∣

1

ε∇3u(x)

)

dx,

1. Introduction 4

for Ω := ω × I, ω ⊂ R2 and I := (−1, 1).

The functional Iε(u) can be interpreted as the energy of a deformation u of an elastic

body whose microstructure is periodic of period ε and ε2. Similarly, as it will be seen

later (Section 3.2), the functional Wε(u) can be interpreted as the rescaled energy of a

deformation u of a cylindrical thin film of thickness ε whose microstructure is periodic of

period ε in the in-plane direction xα, and periodic of period ε2 in all directions. The variable

x is called the macroscopic or slow variable, whereas the variables y = x/ε and z = x/ε2

(respectively z = xα/ε2) are called the microscopic or fast variables. Roughly speaking, the

dependence of the energy on x captures its macroscopic variation while the dependence on

y and z captures its microscopic or local variations. One could generalize the study to a

higher number of scales by iterating the argument.

The overall plan of this dissertation in the ensuing chapters is as follows. Chapter 2 is of

introductory nature. Its objective is to set up the basic notations and background results

that are used later. The aim of Chapter 3 is to present the previous developments in the

asymptotic analysis of this type of problems. This serves as a motivation for our work and

highlights the novelties. Chapter 4 is a collection of two works, obtained in collaboration

with J-F Babadjian [10] and with I. Fonseca [12], respectively. We consider Ω ⊂ RN (N > 1)

open and bounded, 1 1. We set Q := (0, 1)N and LN

stands for the Lebesgue measure in RN .

Main problem of Chapter 4 (Theorem 4.2.1): Characterize the behavior, as ε tends to zero,

of a family of integral functionals defined on Lp(Ω; Rd) by

Iε(u) :=

∫

Ωf(

x,x

ε,x

ε2,∇u(x)

)

dx if u ∈W 1,p(Ω; Rd),

∞ otherwise,

(1.5)

where the integrand f : Ω × RN × RN × Rd×N → R satisfies:

- f(x, · , · , · ) is continuous for a.e. x ∈ Ω;

- f( · , y, z, ξ) is LN -measurable for all (y, z, ξ) ∈ RN × RN × Rd×N ;

- f(x, · , z, ξ) is Q-periodic for all (z, ξ) ∈ RN ×Rd×N and for a.e. x ∈ Ω; f(x, y, · , ξ) is

Q-periodic for all (y, ξ) ∈ RN × Rd×N and for a.e. x ∈ Ω;

1. Introduction 5

- there exists β > 0 such that for all (y, z, ξ) ∈ RN × RN × Rd×N and for a.e. x ∈ Ω

1

β|ξ|p − β 6 f(x, y, z, ξ) 6 β(1 + |ξ|p).

This kind of asymptotic problems can be seen as a generalization of the Iterated Homoge-

nization Theorem for linear integrands, proved by Bensoussan, Lions and Papanicolau [14],

in which the homogenized operator is derived by a formal two-scale asymptotic expansion

method. In the Γ-convergence setting it is customary to assume that

|f(x, y, z, ξ) − f(x′, y′, z, ξ)| 6 ω(|x− x′| + |y − y′|)[

b(z) + f(x, y, z, ξ)]

, (1.6)

for some b ∈ L1loc(R

N ) and some continuous positive real function ω with ω(0) = 0 (see

Braides and Defranceschi [19], Braides and Lukkassen [21] and Lukkassen [60]). As remarked

by Allaire (Section 5 in [4]), the natural regularity on f for the integral (1.5) to be well

defined is not clear. The measurability of the function x 7→ f(x, x/ε, x/ε2, ξ), for fixed ξ, is

assured whenever f is continuous in its second and third variables. The originality of this

work is that we do not require any strong uniform continuity hypotheses on f with respect

to the first and second variables. In particular, we will recover the results of Fonseca and

Zappale in [51], where the authors were also able to weaken hypothesis (1.6) in the convex

case when f = f(y, z, ξ), but by using multiscale arguments that however cannot be adapted

to the nonconvex setting.

As previous results show, the natural candidate for the Γ(Lp(Ω))-limit functional of the fam-

ily Iεε is the functional obtained by iterating twice the homogenization formula derived

for functionals of the type∫

Ωf(

x,x

ε,∇u(x)

)

dx.

Therefore, we expect that

Ihom(u) :=

∫

Ωfhom(x,∇u(x)) dx if u ∈W 1,p(Ω; Rd),

∞ otherwise,

(1.7)

where fhom is defined, for all ξ ∈ Rd×N and for a.e. x ∈ Ω, by

fhom(x, ξ) := limT→∞

infφ

1

TN

∫

(0,T )N

fhom(x, y, ξ + ∇φ(y)) dy : φ ∈W 1,p0

(

(0, T )N ; Rd)

,

1. Introduction 6

and where, for all (y, ξ) ∈ RN × Rd×N and for a.e. x ∈ Ω,

fhom(x, y, ξ) := limT→∞

infφ

1

TN

∫

(0,T )N

f(x, y, z, ξ + ∇φ(z)) dz : φ ∈W 1,p0

(

(0, T )N ; Rd)

.

The analysis we do calls for a new look into the following auxiliary problem.

Auxiliary problem (Theorem 4.1.1): Study the asymptotic behavior of the functional Iε :

Lp(Ω; Rd) → R, where R := R ∪ ∞, given by

Iε(u) :=

∫

Ωf(

x,x

ε,∇u(x)

)


∞ otherwise.

(1.8)

This analysis undertaken in Section 4.1 by means of two-scale convergence arguments allows

us to weaken the hypotheses considered in previous works (see Braides [15], Braides and

Defranceschi [19] and Braides and Lukkassen [21]), and consequently derive (1.7).

Chapter 5 collects parts of two joint works with J-F Babadjian [9, 10]. Here we study the

asymptotic behavior of cylindrical heterogeneous thin domains whose microscopic hetero-

geneities vary periodically. More precisely, given ω ⊂ R2 open and bounded, we consider

thin microstructures of the form Ωε := ω× (−ε, ε), whose heterogeneities are periodic of pe-

riod ε in the in-plane direction and of period ε2 in all directions. Two simultaneous features

occur in this case: a reiterated homogenization and a dimension reduction phenomena. As

is usual, in order to study this asymptotic problem we rescale the thin body into a reference

domain of unit thickness Ω := ω × (−1, 1) (see e.g. Acerbi, Buttazzo and Percivale [2], Le

Dret and Raoult [55]), and we study the rescaled family of functionals defined on Ω, whose

dependence on ε turns out to be explicit in the transverse derivative.

Main problem of Chapter 5 (Theorem 5.2.4): Characterize the behavior as ε tends to zero

of the family of rescaled functionals defined on Lp(Ω; Rd) by

Wε(u) :=

∫

ΩW

(

x,x

ε,xαε2,∇αu(x)

∣

∣

∣

1

ε∇3u(x)

)

dx if u ∈W 1,p(Ω; R3),

∞ otherwise,

(1.9)

where W : Ω × R3 × R2 × R3×3 → R satisfies:

1. Introduction 7

- W (x, · , · , · ) is continuous for a.e. x ∈ Ω;

- W ( · , · , · , ξ) is L3 ⊗ L3 ⊗ L2-measurable for all ξ ∈ R3×3;

- yα 7→ W (x, yα, y3, zα, ξ) is Q′-periodic for all (zα, y3, ξ) ∈ R3 × R3×3 and for a.e. x ∈Ω, where Q′ := (0, 1)2;

- (zα, y3) 7→W (x, yα, y3, zα, ξ) is Q -periodic for all (yα, ξ) ∈ R2×R3×3 and for a.e. x ∈Ω, where Q := (0, 1)3;

− there exists β > 0 such that for all (y, zα, ξ) ∈ R3 × R2 × R3×3 and for a.e. x ∈ Ω

1

β|ξ|p − β 6 W (x, y, zα, ξ) 6 β(1 + |ξ|p).

We identify W 1,p(ω; R3) with the set of functions u ∈W 1,p(Ω; R3) such that D3u(x) = 0 for

a.e. x ∈ Ω. Under the above hypotheses, the Γ(Lp(Ω))-limit of the family Wεε is given

by the functional

Whom(u) :=

2

∫

ωW hom(xα,∇αu(xα)) dxα if u ∈W 1,p(ω; R3),

∞ otherwise,

where W hom is defined, for all ξ ∈ R3×2 and for a.e. xα ∈ ω, by

W hom(xα, ξ) := limT→∞

infφ

1

2T 2

∫

(0,T )2×IWhom(xα, y3, yα, ξ + ∇αφ(y)|∇3φ(y)) dy :

φ ∈W 1,p((0, T )2 × I; R3), φ = 0 on ∂(0, T )2 × I

(1.10)

and where, for all (yα, ξ) ∈ R2 × R3×3 and for a.e. x ∈ Ω,

Whom(x, yα, ξ) := limT→∞

infφ

1

T 3

∫

(0,T )3W (x, yα, z3, zα, ξ + ∇φ(z)) dz :

φ ∈W 1,p0 ((0, T )3; R3)

.

Our main contribution is that we are able to homogenize this material in the reducing

direction. As far as we know, there have not been previous results in this direction (see

Braides, Fonseca and Francfort [20] and Shu [75]). Let us outline the idea for the derivation

of the formula (1.10). In a first step, since the volume of Ωε is of order ε and ε2 ≪ ε, we

1. Introduction 8

can think of ε as being a fixed parameter and let ε2 tend to zero. At this point, dimension

reduction is not occurring, and (1.9) can be seen as a single one-scale homogenization

problem, as in (1.5). This leads to the stored energy density Whom(x, yα, ξ). In a second

step, Whom(x, yα, ξ) is used as the integrand for the following reduction dimension problem.

Auxiliary problem (Theorem 5.1.1): Characterize the asymptotic behavior of a family of

functionals Iε : Lp(Ω; R3) → R given by

Iε(u) :=

∫

ΩW

(

x,xαε,∇αu(x)

∣

∣

∣

1

ε∇3u(x)

)


∞ otherwise.

This problem will be studied in Section 5.1. Two features differentiate our approach from

what is available in most of the literature on the subject (see Braides, Fonseca and Francfort

[20] and Shu [75]; see also Chapter 3). The first feature is the use of a two-scale convergence

argument as in problem (1.8). The second feature is a decoupling argument, motivated by

the work in Babadjian and Francfort [11], to take into account the different nature of the

variables that appear in the structure of the limit functional.

We note that if Ω is assumed to be Lipschitz, as p > 1, the Γ-limit of the previous functionals

for u ∈ W 1,p(Ω; Rd) would be the same if the weak W 1,p-topology had been considered in

place of the strong Lp-topology. For p = 1 our argument fails to characterize this Γ-limit

for u ∈ W 1,1(Ω; Rd), either with the strong L1-topology or with the weak W 1,1-topology,

since sequences whose gradients are bounded in L1 are not necessarily relatively compact

in W 1,1(Ω; Rd). They are relatively compact only in the space of functions of bounded

variation.

We finally remark that, from the applications point of view, it would be interesting to

prove similar results to the ones addressed in Chapters 4 and 5 for integrands that are only

measurable with respect to (some of) the oscillating variables. This is what is relevant in

the case of mixtures. In Chapter 6 we conclude with some generalizations in this direction

and we address some open problems for future research.

Part I

PRELIMINARIES AND PREVIOUS RESULTS

2. PRELIMINARIES

The purpose of this introductory chapter is to give a survey of the concepts and results that

are used throughout this dissertation. Almost all these results are stated without proofs as

they can be readily found in the references given below.

2.1 A short review of Measure Theory

In this section we recall well known results in Measure Theory (see e.g Ambrosio, Fusco and

Pallara [7], Evans and Gariepy [46] and Fonseca and Leoni [47], as well as the bibliography

therein; see also Brezis [29] and Dugundji [44] for a reference on functional analysis and

topological notions).

2.1.1 Measures and integration

A measurable space is a pair (X,M) where X is a nonempty set and M is a σ-algebra in

X. A set E ⊂ X is said to be measurable if E ∈ M. If X is a topological space and if not

otherwise said, then M is taken to be the Borel σ-algebra in X, that we denote by B(X),

i.e. the smallest σ-algebra that contains all open subsets of X.

Definition 2.1.1. (Measure) A measure on (X,M) is a set function µ : M → [0,∞] such

that µ(∅) = 0 and µ is σ-additive, i.e.

µ(

∞⋃

n=0

En

)

=∞∑

n=0

µ(En)

for any sequence Enn of pairwise disjoint elements of M.

The triple (X,M, µ) is called a measure space; it is said to be σ-finite if X is the union of

an increasing sequence of sets with finite µ-measure, and it is said to be finite if µ(X) <∞.

2. Preliminaries 12

Definition 2.1.2. (Borel measure) Let X be a topological space and let (X,M, µ) be a

measure space. The measure µ is said to be Borel if B(X) ⊆ M.

Remark 2.1.3. Any measure µ on (X,M) is monotone with respect to set inclusion and

continuous along monotone sequences, that is, if Enn is an increasing sequence of sets

(respectively a decreasing sequence of sets with µ(E1) finite), then

µ(

∞⋃

n=1

En

)

= limn→∞

µ(En), resp. µ(

∞⋂

n=1

En

)

= limn→∞

µ(En).

Definition 2.1.4. Let (X,M, µ) be a measure space.

i) A set N ⊂ X is said to be µ-negligible if there exists E ∈ M such that N ⊂ E and

µ(E) = 0.

ii) A property P (x), depending on the point x ∈ X, is said to hold µ-a.e. (or simply

a.e. ) in X if the set where P fails is a µ-negligible set.

Proposition 2.1.5. Let (X,M, µ) be a measure space and let Mµ be the collection of all

the subsets of X of the form F = E ∪N , with E ∈ M and N µ-negligible. Then Mµ is a

σ-algebra and it is called the µ-completion of M.

Definition 2.1.6. (µ-measurable set) Let (X,M, µ) be a measure space. A set E ⊂ X is

said to be µ-measurable if E ∈ Mµ.

Given (X,M, µ) a measure space the measure µ extends to Mµ by setting, for F as above,

µ(F ) = µ(E), and µ is said to be complete if M = Mµ. Throughout this work any Borel

measure is tacitly understood to be extended to its completion.

We denote by LN the usual Lebesgue measure in RN as well as its restriction to B(RN ). The

set LN stands for the σ-algebra of all Lebesgue measurable sets (that is L

N = B(RN )LN ).

Given E ∈ LN , we will write indifferently LN (E) or |E|.

Definition 2.1.7. Let (X,M) and Y be, respectively, a measurable and a topological

space. Let u : X → Y .

i) (Measurable function) The function u is said to be M-measurable, or simply measur-

able, if u−1(B) ∈ M for every open set B ⊂ Y.

ii) (Borel function) Assuming that X is also a topological space, the function u is said

to be Borel if u−1(B) ∈ B(X) for every open set B ⊂ Y.

2. Preliminaries 13

Definition 2.1.8. (µ- measurable function) Let (X,M, µ) and Y be, respectively, a mea-

sure and topological space. A function u : X → Y is said to be µ-measurable if it is

Mµ-measurable.

The following theorem provides conditions guaranteeing the existence of a measurable se-

lection of a given multifunction. It can be found in Castaing and Valadier [30] (Theorem

III.30) and it is important for the analysis undertaken in Subsection 4.2.4 below.

Theorem 2.1.9. Let (X,M, µ) be a finite complete measure space, and let Y be a complete

and separable metric space. Let F : X → C ⊂ Y : C 6= ∅ and C is a closed set be a

multifunction such that (x, y) ∈ X × Y : y ∈ F (x) ∈ M ⊗ B(Y ).1 Then there exists a

sequence of measurable functions un : X → Y such that

F (x) = un(x) : n ∈ N

for µ a.e. x ∈ X.

Let (X,M, µ) be a measure space and let us denote by Lp(X,µ), with 1 6 p 6 ∞ , the usual

Lebesgue spaces, that is the set (of the equivalence classes) of all µ-measurable functions

u : X → R such that the (Lebesgue) integral

||u||Lp(X,µ) :=

(∫

X|u|p dµ

)1/p

<∞

for p <∞, or

||u||L∞(X,µ) := infC ∈ [0,∞] : |u(x)| 6 C for µ-a.e. x ∈ X <∞.2

We abbreviate Lp(X,µ) by Lp(X) when this will cause no confusion (e.g. when µ is the

Lebesgue measure in RN ). Is is well known that Lp(X,µ) is a Banach space with the norm

|| · ||Lp(X,µ) for 1 6 p 6 ∞ (Hilbert when p = 2); it is a reflexive space for 1 < p < ∞, and

in this case its dual space is (identified with) Lq(X) with q = p/(p − 1). If (X,M, µ) is

σ-finite then L∞(X,µ) is the dual space of L1(X,µ); if, in addition, X is separable3 then

Lp(X,µ) is separable for 1 6 p <∞.

1 The set M⊗B(Y ) denotes the usual product σ-algebra of M and B(Y ).2 ||u||L∞(X,µ) is sometimes called the essential supremum of u and written ||u||L∞(X,µ) = ess sup

x∈X

|u(x)|;

usual convention: inf∅ = ∞.3 We recall that a measurable space (X,M) is said to be separable if there exists a sequence Enn ⊂ M

such that the smallest σ-algebra that contains all the sets En is M. If X is a metric space and M is a Borel

σ-algebra, then (X,B(X)) is a separable space.

2. Preliminaries 14

Lemma 2.1.10. (Chebyshev inequality) Let (X,M, µ) be a measure space. If u ∈ Lp(X;µ),

with 1 6 p <∞, then for any t > 0

µ(x ∈ X : |u(x)| > t) 61

tp

∫

X|u|p dµ.

We assume that the reader is familiar with the properties of integrals, measurable functions

and Lp-spaces. For the sake of completeness we state here the fundamental convergence

results in the theory of integration on abstract measure spaces.

Theorem 2.1.11. (Levi’s Theorem or Monotone Convergence Theorem) Let (X,M, µ) be

a measure space, and let un : X → R be an increasing sequence of µ-measurable functions.

Assume that un > v for any n ∈ N, with v ∈ L1(X,µ), then

limn→∞

∫

Xun dµ =

∫

Xlimn→∞

un dµ.

Lemma 2.1.12. (Fatou’s Lemma) Let (X,M, µ) be a measure space and let un : X → R

be a sequence of µ-measurable functions.

i) If there exists v ∈ L1(X,µ) such that un > v for any n ∈ N, then∫

Xlim infn→∞

un dµ 6 lim infn→∞

∫

Xun dµ.

ii) If there exists v ∈ L1(X,µ) such that un 6 v for any n ∈ N, then∫

Xlim supn→∞

un dµ > lim supn→∞

∫

Xun dµ.

As a consequence we get the following result.

Corollary 2.1.13. Let (X,M, µ) be a measure space and let un ⊂ Lp(X,µ) with 1 6

p < ∞, be such that un → u µ-a.e. as n → ∞, for some function u ∈ Lp(X,µ). Then

||un − u||Lp(X,µ) → 0 if and only if ||un||Lp(X,µ) → ||u||Lp(X,µ).

Theorem 2.1.14. (Dominated Convergence Theorem) Let u, un : X → R be µ-measurable

functions, and assume that un → u µ-a.e. as n→ ∞. If∫

Xsupn∈N

|un| dµ <∞

then

limn→∞

∫

Xun dµ =

∫

Xu dµ.

2. Preliminaries 15

The following variant of the Dominated Convergence Theorem can be found in Evans and

Gariepy [46] and it will be of use in the sequel.

Proposition 2.1.15. Let v, vn ∈ L1(X,µ) and let u, un be µ-measurable, for n ∈ N.

Suppose that |un| 6 vn for all n ∈ N, and that un and vn converge µ-a.e. to u and v,

respectively. If in addition

limn→∞

∫

Xvn dµ =

∫

Xv dµ,

then

limn→∞

∫

X|un − u|dµ = 0.

Theorem 2.1.16. (Fubini-Tonelli Theorem) Let (X1,M1, µ1) and (X2,M2, µ2) be two σ-

finite measure spaces. Then, there is a unique positive σ-finite measure µ on (X1×X2,M1⊗M2) such that

µ(E1 × E2) = µ1(E1)µ2(E2)

for all E1 ∈ M1 and E2 ∈ M2.

i) (Tonelli) In addition, for any measurable function u : X1 ×X2 → [0,∞] we have that

x→∫

X2

u(x, y)dµ2(y) and y →∫

X1

u(x, y)dµ1(x)

are M1-measurable and M2-measurable respectively, and∫

X1×X2

u dµ =

∫

X1

(∫

X2

u(x, y)dµ2(y)

)

dµ1(x) (2.1)

=

∫

X2

(∫

X1

u(x, y)dµ1(x)

)

dµ2(y).

ii) (Fubini) If u ∈ L1(X1 ×X2;µ), then u(x, ·) ∈ L1(X2, µ2) for µ1-a.e. x ∈ X1, u(·, y) ∈L1(X1, µ1) for µ2-a.e. y ∈ X2, the a.e. defined functions

x→∫

X2


X1

u(x, y)dµ1(y)

are in L1(X1, µ1) and L1(X2, µ2), respectively, and equality (2.1) holds.

We present here another version of Fubini-Tonelli’s Theorem which deals with complete

measures and is relevant to Lebesgue integration on RN .

2. Preliminaries 16

Theorem 2.1.17. Suppose that (X1,M1, µ1) and (X2,M2, µ2) are two complete and σ-

finite measure spaces. If u : X1 × X2 → [0,∞] is µ-measurable (where µ is the measure

given by theorem 2.1.16)4, then the functions

x→∫

X2


X1

u(x, y)dµ1(y)

are respectively µ1-measurable and µ2-measurable and

∫

X1×X2

u dµ =

∫

X1

(∫

X2

u(x, y)dµ2(y)

)

dµ1(x)

=

∫

X2

(∫

X1

u(x, y)dµ1(x)

)

dµ2(y).

2.1.2 Radon measures and Vitali’s Covering Theorem

Definition 2.1.18. (Radon Measure on (X,M)) Let X be a topological space. A Radon

measure on a measurable space (X,M) is a Borel measure, finite on compact sets, and such

that for every open set E ⊂ X

µ(E) = supµ(K) : K ⊂ E, K compact (inner-regularity),

and for every set E ∈ M

µ(E) = supµ(A) : E ⊂ A, A open (outer-regularity).

Remark 2.1.19. If X is a locally compact Hausdorff space the following two properties hold.5

i) (see Theorem 2.7 in Rudin [71]) Let E,K ⊂ X be an open and compact set, re-

spectively, with K ⊂ E. Then there is an open set A with compact closure such

that

K ⊂ A ⊂ A ⊂ E.

4 By Definition 2.1.6 this means measurable with respect to the completion σ-algebra of M1 ⊗M2; if µ is

a Borel measure then it is understood to be extended to (M1 ⊗M2)µ. In the case of the Lebesgue measure

B(RN ) = B(R)N-times⊗ · · · ⊗ B(R).

5 A topological space X is said to be Hausdorff if given two distinct points x and y, there are disjoint

open sets E1 and E2 such that x ∈ E1 and y ∈ E2; X is said to be locally compact if for each x ∈ X there

is an open set E containing x such that E is compact. Metric spaces are Hausdorff spaces and RN with the

usual metric is Hausdorff and locally compact.

2. Preliminaries 17

ii) Let µ be a Radon measure on X and let E be an open set of X. Then

µ(E) = supµ(K) : K ⊂ E, K compact = supµ(A) : A ⊂⊂ E,A open,

where A ⊂⊂ E means that A is a compact set with A ⊂ E.

Given a topological space X we denote by A(X) the family of all its open subsets. The

following lemma provides sufficient conditions for a set function Π : A(X) → [0,∞) to

be the restriction of a Radon measure on A(X). It is close in spirit to De Giorgi-Letta’s

criterion (see [39]) and it is of importance to apply the Direct Method of Γ-convergence (see

Chapters 4 and 5 below) as well as for the use of relaxation methods that strongly rely on

the structure of Radon measures.

Lemma 2.1.20. (see Fonseca and Maly [48]; also Fonseca and Leoni [47]) Let X be a locally

compact Hausdorff space, let Π : A(X) → [0,∞), and let µ be a finite Radon measure µ on

X satisfying

i) (nested-subadditivity) Π(D) ≤ Π(D\B) + Π(C) for all B,C,D ∈ A(X) with B ⊂⊂C ⊂ D;

ii) Given D ∈ A(X), for all ǫ > 0 there exists Dε ∈ A(X) such that Dε ⊂⊂ D and

Π(D\Dε) ≤ ε;

iii) Π(X) ≥ µ(X);

iv) Π(D) ≤ µ(D) for all D ∈ A(X).

Then Π = µ|A(X).

Proof. Fix D ∈ A(X). We start by proving that the inequality Π(D) 6 µ(D) holds. Let

ε > 0 and by condition ii) choose Dε ∈ A(X) such that Dε is a compact set, Dε ⊂ D and

Π(D\Dε) ≤ ε. As X is a locally compact Hausdorff space, we can find Cε ∈ A(X) such

that Dε ⊂ Cε ⊂ Cε ⊂ D (see Remark 2.1.19).

By hypotheses i), ii) and iv) we have

Π(D) 6 Π(D \Dε) + Π(Cε) 6 ε+ µ(Cε) 6 ε+ µ(D),

2. Preliminaries 18

DDεCε

and then letting ε→ 0 it follows that

Π(D) 6 µ(D).

To prove the reverse inequality, using the inner regularity property of the measure µ (see

Remark 2.1.19), for every ε > 0 we may find B ∈ A(X) with B ⊂⊂ D and such that

µ(D) 6 ε+ µ(B).

Therefore

µ(D) 6 ε+ µ(X) − µ(X \B)

and, consequently, by iii) and the previous step

µ(D) 6 ε+ Π(X) − Π(X \B).

Hence by i) we have

µ(D) 6 ε+ µ(D)

and therefore letting ε→ 0 we get µ(D) 6 Π(D).

Given a ∈ RN and δ > 0 we denote by B(a, δ) := x ∈ RN : |x− a| < δ.

Theorem 2.1.21. (Vitali’s Covering Theorem) (see Braides and Defranceschi [19]) Let

Ω ⊂ RN be a bounded open set with N > 1, and let F be a family of closed subsets of Ω. If

there exists a positive number M > 1 such that for each F ∈ F , B(x, δ) ⊆ F ⊆ B(x,Mδ)

for some x ∈ Ω and δ > 0, and if

infdiamF : x ∈ F, F ⊂ F = 0

for a.e. x ∈ Ω, then there exists a disjoint countable subfamily Fjj of F such that∣

∣

∣Ω \⋃

j

Fj

∣

∣

∣ = 0.

2. Preliminaries 19

2.1.3 Decomposition and differentiation of measures

Definition 2.1.22. Let (X,M, µ) and (X,M, ν) be two measure spaces. The measure ν

is said to be absolutely continuous with respect to µ, and we write ν << µ, if for every

E ∈ M the following implication holds:

µ(E) = 0 ⇒ ν(E) = 0.

The measures µ and ν are said to be mutually singular, and we write ν ⊥ µ, if there exists

E ∈ M such that µ(E) = 0 and ν(X \ E) = 0.

Theorem 2.1.23. (Lebesgue-Radon-Nikodym Theorem) Let (X,M, µ) and (X,M, ν) be

two σ-finite measure spaces. Then there exists a unique pair of measures νa and νs such

that νa << µ, νs ⊥ µ and ν = νa + νs. Moreover, there is a unique measurable function

u : X → [0,∞] such that for all E ∈ M

νa(E) =

∫

Eu dµ.

The decomposition ν = νa + νs, where νa << µ and νs ⊥ µ, is called the Lebesgue decom-

position of ν with respect to µ. In the case where ν << µ, Theorem 2.1.23 says that

ν(E) =

∫

Eu dµ

for all E ∈ M. This result is known as the Radon-Nikodym Theorem, and u is called the

Radon-Nikodym derivative of ν with respect to µ, u = dν/dµ.

Theorem 2.1.24. (General version of the Besicovitch derivation Theorem)(see Proposition

2.2 in Ambrosio and Dal Maso [6]) Let µ and ν be two Radon measures on RN and let

ν = νa + νs be the Lebesgue decomposition of ν with respect to µ (dνa = udµ). There exists

a Borel set E ⊂ RN , with µ(E) = 0, such that, for every x ∈ RN \ E and C ⊂ RN open

bounded convex set containing the origin, the limit

limδ↓0

ν(x+ δC)

µ(x+ δC)

exists, is finite, and coincides with u(x).

Let X be a topological space and let µ be a measure on X. A function u is said to be in

L1loc(X,µ) if u ∈ L1(E, µ) whenever E ⊂⊂ X. Theorem 2.1.24 implies the following result.

2. Preliminaries 20

Proposition 2.1.25. (Lebesgue Differentiation Theorem) Let µ be a Radon measure on

RN and let u ∈ L1loc(R

N , µ). Then for a.e. x ∈ RN

limδ↓0

1

µ(B(x, δ))

∫

B(x,δ)|u(y) − u(x)| dµ(y) = 0, (2.2)

and in particular

u(x) = limδ↓0

1

µ(B(x, δ))

∫

B(x,δ)u(y) dµ(y).

Definition 2.1.26. Any point x where (2.2) holds is called a Lebesgue point of u.

Theorem 2.1.27. (see Theorem 2.8 in Fonseca and Muller [49]) Let µ be a Radon measure

on RN and u ∈ L1loc(R

n, µ). Then there exists a Borel set E ⊂ RN , with µ(E) = 0, such

that for every x ∈ RN \ E

limδ↓0

1

µ(x+ δC)

∫

x+δC|u(y) − u(x)| dµ(y) = 0, (2.3)

for every C ⊂ RN open bounded convex set containing the origin.

Definition 2.1.28. (Signed measure) A signed measure on a measurable space (X,M) is

a set function µ : M → [−∞,∞] such that µ(∅) = 0, µ takes at most one of the values ∞or −∞, and for any family Enn of pairwise disjoint elements of M

µ(

∞⋃

n=0

En

)

=∞∑

n=0

µ(En).6

In particular a measure is a signed measure.

Definition 2.1.29. (Total variation of a signed measure) Let (X,M) be a measurable

space and let µ : M → [−∞,∞] be a signed-measure. Its total variation |µ| : M → [0,∞]

is defined by

|µ|(E) := sup

∞∑

n=0

|µ(En)| : En ∈ B(M) pairwise disjoint, E =∞⋃

n=0

En

.

If µ is a positive measure then µ = |µ|, and if µ is a signed-measure then |µ| is a measure.

The signed measure µ is said to be σ-finite if |µ| is a σ-finite measure on X; µ is said to be

a (signed) Radon measure if its total variation |µ| is a Radon measure. Finally, given two

signed measures µ and τ on a measurable space (X,M), µ is absolutely continuous with

respect to τ (respectively mutually singular) if |µ| << |τ | (respectively |µ| ⊥ |τ |), and an

analog of the Lebesgue-Radon-Nikodym Theorem holds for signed-measures.

6 The absolutely convergence of this series is understood.

2. Preliminaries 21

2.1.4 Weak⋆ convergence of measures

Let X be a locally compact Hausdorff space, and let Cc(X) denote the set of continuous

functions with compact support on X. We denote by C0(X) the completion of Cc(X) with

respect to the supremum norm.7 By the Riesz-Representation Theorem the dual of the

Banach space C0(X) is the space of finite (signed) Radon measures µ : B(RN ) → R. This

characterization leads to the following notion of convergence of a sequence of finite (signed)

Radon measures.

Definition 2.1.30. (Weak⋆ convergence of measures) Let µnn be a sequence of finite

(signed) Radon measures on X. This sequence is said to weak⋆ converge to a finite (signed)

Radon measure µ on X, and we write µn⋆ µ, if for all φ ∈ C0(X)

limn→∞

∫

Xφdµn =

∫

Xφdµ.

Proposition 2.1.31. (Weak⋆ compactness property) Let X be a σ-compact metric space.8

Then every sequence µnn of finite (signed) Radon measures on X with supn∈N

|µn|(X) <∞has a weak⋆ converging subsequence.

Proposition 2.1.32. Let X be a locally compact Hausdorff space and let µnn be a se-

quence of finite (signed) Radon measures on X such that µn⋆ µ. Then

i) if K ⊂ X is compact

µ(K) ≥ lim supn→∞

µn(K);

ii) if A ⊂ X is open

µ(A) ≤ lim infn→∞

µn(A);

iii) if A ⊂⊂ X is open and µ(∂A) = 0

µ(A) = limn→∞

µn(A).

7 The support of u is by definition supp (u) := x ∈ X : u(x) 6= 0; the function u is said to have compact

support in X if supp (u) ⊂⊂ X.8 A metric space is said to be σ-compact if it is the union of a countable collection of compact subsets;

for instance RN is a σ-compact metric space.

2. Preliminaries 22

2.2 Sobolev spaces

The aim of this section is to give the main properties needed throughout the text on weak

derivatives and Sobolev spaces. We refer to the books of Adams [3], Brezis [29], Evans and

Gariepy [46], Fonseca and Leoni [47], Giusti [57], and Ziemer [77] for a detailed analysis on

this topic.

2.2.1 Definition and main properties

Let Ω be an open subset of RN with N > 1, and let 1 6 p 6 ∞. In the sequel W 1,p(Ω)

(respectively W 1,ploc (Ω)) stands for the usual Sobolev space, that is the space of functions

u ∈ Lp(Ω) (respectively Lploc(Ω)) with weak derivatives of order one in Lp(Ω) (respectively

Lploc(Ω)).9 For any u ∈W 1,p(Ω) we set ∇u := (∇1u, ...,∇Nu). The space W 1,p0 (Ω) stands for

the closure of C∞0 (Ω) in W 1,p(Ω) for 1 < p <∞. W 1,∞

0 (Ω) denotes the closure of C∞0 (Ω) in

the weak⋆ topology of W 1,∞(Ω). It is well known that W 1,p(Ω) is a Banach space (Hilbert

for p = 2) when endowed with the norm

||u||W 1,p(Ω) :=

(

||u||pLp(Ω) +N∑

i=1

||∇iu||pLp(Ω)

)1/p

for 1 6 p <∞; for p = ∞ the norm is given by

||u||W 1,∞(Ω) := ||u||L∞(Ω) +N∑

i=1

||∇iu||L∞(Ω).10

Since for 1 < p < ∞ the space W 1,p(Ω) is reflexive, for each bounded sequence unn ⊂W 1,p(Ω), with 1 < p < ∞, there exists a subsequence unk

k ⊂ W 1,p(Ω) and u ∈ W 1,p(Ω)

such that unk u in W 1,p(Ω) (see Theorem III.27 in Brezis [29]). For 1 6 p < ∞ the

space W 1,p(Ω) is separable. Finally we remark that the W 1,p(Ω)-weak limit of a sequence

in W 1,p0 (Ω) still belongs to W 1,p

0 (Ω) since a convex subset of a Banach space is closed with

respect to the weak topology if and only if it is closed with respect to the strong topology.

9 For any i ∈ 1, ..., d we set ∇i := ∂/∂xi. Given u, v ∈ Lploc(Ω) we recall that v is said to be the

ith-derivative of u and we write ∇iu = v, provided∫

Ω

u∇iϕ dx = −

∫

Ω

vϕ dx

for all functions ϕ ∈ C∞c (Ω).

10 ||u||W1,p(Ω) ≡ ||u||Lp(Ω) + ||∇u||Lp(Ω).

2. Preliminaries 23

For d > 1 we denote by

Lp(Ω; Rd) :=

u : Ω → Rd : ui ∈ Lp(Ω) for all i ∈ 1, ..., d

;

W 1,p(Ω; Rd) :=

u ∈ Lp(Ω; Rd) : ∇ju ∈ Lp(Ω; Rd) for j = 1, ..., N

,

where ∇ju := (∇ju1, ...,∇jud). If u ∈W 1,p(Ω; Rd) we write ∇u := (∇1u|...|∇Nu).

The following result gives a sufficient condition for a function to belong to W 1,p(Ω). It is a

straightforward consequence of the definition of the space and of the properties above.

Proposition 2.2.1. Let Ω ⊂ RN be open, and let unn be a sequence in W 1,p(Ω) converg-

ing in Lp(Ω) to some function u. Then

i) if 1 6 p 6 ∞ and there exists a function g ∈ Lp(Ω; RN ) such that ∇un → g in

Lp(Ω; RN ), then u ∈W 1,p(Ω) and g = ∇u;

ii) if 1 < p 6 ∞ and the sequence ∇unn is bounded in Lp(Ω; RN ) then u ∈ W 1,p(Ω)

and ∇un ∇u in Lp(Ω; RN ) (weak⋆ if p = ∞).11

2.2.2 Extension, approximation and traces

Theorem 2.2.2. (Extension Theorem) Let Ω be an open bounded subset of RN with Lip-

schitz boundary and let 1 6 p 6 ∞. Let Ω be any open set such that Ω ⊂⊂ Ω. Then there

exist a bounded linear (extension) operator

E : W 1,p(Ω) →W 1,p(RN )

such that Eu = u in Ω and supp(Eu) ⊂ Ω for all u ∈W 1,p(Ω).

Theorem 2.2.3. (Approximation by smooth functions) Let Ω be an open subset of RN and

let 1 6 p <∞. Then

i) (Meyers-Serrin) C∞(Ω) ∩W 1,p(Ω) is dense in W 1,p(Ω);

ii) if, in addition, Ω is a bounded Lipschitz set, then the restriction to Ω of functions in

C∞c (RN ) is dense in W 1,p(Ω).

11 If p = 1 the function u is in BV (Ω), the space of functions with bounded variation, but not necessarily

in W 1,1(Ω).

2. Preliminaries 24

Next we recall a Trace Theorem that makes it possible to assign “boundary values” along

∂Ω to a function u ∈W 1,p(Ω).

Theorem 2.2.4. (Trace Theorem) Let Ω ⊂ RN be a bounded set with Lipschitz boundary,

and let 1 6 p < ∞. Then there exists a bounded linear operator T : W 1,p(Ω) → Lp(∂Ω)

such that

i) Tu = u⌊∂Ω if u ∈ C(Ω);

ii) ||Tu||Lp(∂Ω) 6 C||u||W 1,p(Ω) for each u ∈W 1,p(Ω), for some constant C = C(N, p,Ω).

Moreover Ker(T ) = W 1,p0 (Ω).

2.2.3 Compactness and Poincare inequalities

Theorem 2.2.5. (Sobolev-Rellich-Kondrachov Theorem) Let Ω be an open bounded set in

RN with Lipschitz boundary and let 1 6 p <∞. Then

i) for 1 6 p < N and 1 6 q < p⋆ = NpN−p

W 1,p(Ω) ⊂ Lq(Ω),

and the imbedding is compact;12

ii) if p = N then for every 1 6 q <∞

W 1,p(Ω) ⊂ Lq(Ω),

and the imbedding is compact;

iii) if p > N then

W 1,p(Ω) ⊂ C(Ω),

and the imbedding is compact.

Remark 2.2.6. Under hypotheses of Theorem 2.2.5:12 Recall that given X and Y two Banach spaces, X ⊂ Y , the space X is said to be compactly embedded in

Y if ||x||Y 6 C||x||X for some constant C, and if each bounded sequence in X has a convergent subsequence

in Y .

2. Preliminaries 25

a) condition iii) is still true for p = ∞;

b) the imbedding W 1,p(Ω) ⊂ Lp(Ω) is compact for every 1 6 p 6 ∞ (when p = ∞ this

follows from Morrey’s inequality and the Arzela-Ascoli Theorem; see Brezis [29]);

c) if un u in W 1,p(Ω) with 1 6 p 6 ∞ (⋆ if p = ∞) then

If 1 6 p < N , un → u in Lq(Ω) for every 1 6 q < p⋆;

if p = N , un → u in Lq(Ω) for every 1 6 q <∞;

if N < p 6 ∞, then un → u in L∞(Ω).

In particular un → u in Lp(Ω) for every 1 6 p 6 ∞.

Remark 2.2.7. All the conclusions of Theorem 2.2.5 hold in W 1,p0 (Ω) for a general open

bounded set Ω ⊂ RN .

Proposition 2.2.8. (Poincare-type inequalities)

i) (Poincare inequality) Let Ω be an open set in RN with finite width, that is, a subset

of RN that lies between two parallel hyperplanes, and let 1 6 p <∞. Then there exist

a constant C (depending only on p, N and the distance between the two planes) such

that for all u ∈W 1,p0 (Ω)

∫

Ω|u|p dx 6 C

∫

Ω|∇u|p dx.13

ii) Let Ω be a bounded and connected subset of RN with Lipschitz boundary. Then there

exists a constant C = C(N, p,Ω) such that

||u− uΩ||Lp(Ω) 6 C||∇u||Lp(Ω),

for all u ∈W 1,p(Ω) with 1 6 p 6 ∞, where uΩ = 1|Ω|

∫

Ωu dx.

13 Thus ‖u‖W1,p and ||∇u||Lp(Ω) are equivalent in the space W 1,p0 (Ω). In particular, this result is true for

any open bounded set of RN .

2. Preliminaries 26

2.2.4 Weak convergence and decomposition lemmas for sequences of

gradients and of scaled-gradients

We conclude this section by recalling two decomposition lemmas that will allow us to char-

acterize the Γ-limit of the functionals studied in Chapters 4 and 5 by considering recovering

sequences whose gradients have equi-integrability properties. We start with a short review

of equi-integrability. Throughout this part we assume that Ω is an open bounded subset of

RN with N > 1.

Definition 2.2.9. (Equi-integrability) A sequence of functions unn ⊂ L1(Ω) is said to

be equi-integrable if for all ε > 0 there exist δ > 0 such that

supn∈N

∫

E|un| dx < ε

whenever E ⊂ Ω with |E| < δ.

Proposition 2.2.10. A sequence unn ⊂ L1(Ω) is equi-integrable if and only if one of the

two following conditions holds:

i) for every ε > 0 there exists a constant M > 0 such that for all n ∈ N∫

x∈Ω: |un|>M|un| dx 6 ε;

ii) (De la Vallee Poussin Criterion) there exists an increasing and continuous function

ϕ : [0,∞) → [0,∞] satisfying

limt→∞

ϕ(t)

t= ∞

and such that∫

Ωϕ(|un|) dx 6 1 for all n ∈ N.

Theorem 2.2.11. (Dunford-Pettis Theorem) A sequence unn ⊂ L1(Ω) is weakly compact

in L1(Ω) if and only if

i) unn is bounded in L1(Ω),

ii) unn is equi-integrable.

In particular, if un u in L1(Ω) then unn is equi-integrable. In fact the following

characterization holds.

2. Preliminaries 27

Proposition 2.2.12. A sequence un u in L1(Ω) if and only if

i) supn∈N

||un||L1(Ω) <∞,

ii)

∫

Cun dx→

∫

Cu dx for any cube C ⊂ RN ,

ii) unn is equi-integrable.

Definition 2.2.13. (p-equi-integrability) A sequence unn ⊂ Lp(Ω), with 1 < p < ∞, is

said to be p-equi-integrable if |un|pn is equi-integrable.

Theorem 2.2.14. (Vitali’s Theorem) Let 1 < p <∞. A sequence unn ⊂ Lp(Ω) converges

strongly to u in Lp(Ω) if and only if

i) unn converges to u in measure14,

ii) unn is p-equi-integrable.

In general un u in Lp with 1 < p <∞ does not imply that unn is p-equi-integrable.

Example. Let un(x) =√nχ[0, 1

n](x) with x ∈ R, where χ[0, 1

n] denotes the characteristic

function of the interval[

0, 1n

]

. Then un 0 in L2(R) but |un|2n is not equi-integrable.

0 1/n

n 1/2

The following result characterizes weak convergent sequences in Lp(Ω) for 1 < p 6 ∞.

Proposition 2.2.15. A sequence un u in Lp(Ω) with 1 ε| → 0, as n → ∞, for every ε > 0.

2. Preliminaries 28

As a consequence of next theorem each sequence with bounded gradients in Lp, for 1 <

p < ∞, admits a subsequence that can be decomposed as a sum of a sequence with p-

equi-integrable gradients and a remainder that converges to zero in measure. This property

turns out to be an important tool for the asymptotic analysis of integral functionals relying

on localization arguments.

Theorem 2.2.16. (Decomposition Lemma) (see Fonseca and Leoni [47]; see also Fonseca,

Muller and Pedregal [50] and Kristensen [56]) Let 1 < p < ∞ and assume that ∂Ω is

Lipschitz, and that un v0 in W 1,p(Ω; Rd). Then, there exists a subsequence unkk of

unn and a sequence vkk ⊂W 1,∞(RN ; Rd) such that

i) vk v0 in W 1,p(Ω; Rd),

ii) vk = v0 in a neighborhood of ∂Ω,

iii) ∇vkk is p-equi-integrable,

iv) limk→∞

LN (x ∈ Ω : vk(x) 6= unk(x)) = 0.

The analog of this Theorem for sequences of scaled-gradients in a cylindrical domain of R3

will be relevant for the applications to thin films.

Theorem 2.2.17. (Theorem 1.1 in Bocea and Fonseca [22]) Let Ω := ω × (−1, 1), where

ω ⊂ R2 is an open bounded set with Lipschitz boundary. Let εnn be a sequence of positive

real numbers converging to zero, and let unn be a bounded sequence in W 1,p(Ω; R3), with

1 < p <∞, satisfying

supn∈N

∫

Ω

∣

∣

∣

∣

(

∇αun

∣

∣

∣

1

εn∇3un

)∣

∣

∣

∣

p

dxα dx3 <∞.

Suppose further that un v0 in W 1,p(Ω; R3) and 1εn∇3vn b in Lp(Ω; R3). Then there

exists a subsequence unkk of unn and a sequence vkk ⊂W 1,∞(Ω; R3) such that

i) vk v0 in W 1,p(Ω; R3);

ii) 1εnk

D3vk b in Lp(Ω; R3);

iii)(

Dαvk∣

∣

1εnk

D3vk

)

kis p-equi-integrable;

iv) limk→∞

L3(x ∈ Ω : vk(x) 6= unk(x)) = 0.

2. Preliminaries 29

2.3 An overview of the Direct Method of the Calculus of

Variations

A typical problem in the Calculus of Variations is to minimize an integral functional (energy)

of the form

I(u) =

∫

Ωf(x, u(x),∇u(x)) dx (2.4)

in a class of admissible functions u : Ω → Rd with d > 1 (deformations in the context of

elasticity) where Ω is an open bounded set in RN with N > 1 (reference configuration) and

the integrand is some Borel function f : Ω×Rd×Rd×N → R satisfying appropriate growth

and coercivity conditions. The objective of this section is to give an overview of Tonelli’s

Direct Method of the Calculus of Variations. This is the most classical way of proving the

existence of a minimizer for this kind of variational problems.

2.3.1 The basic notions

We start by recalling the notions of semicontinuity and coercivity on a topological space

X, the two main ingredients of Tonelli’s Direct Method of the Calculus of Variations. For

our purposes in this work it is sufficient to use the sequential versions of these topological

notions and we refer to Dal Maso [35] and Fonseca and Leoni [47] for a more detailed

description of the properties presented here.

Definition 2.3.1. (Sequential lower semicontinuity) A function I : X → [−∞,∞] is said

to be sequentially lower semicontinuous, s.l.s.c for short, if, whenever unn is a sequence

converging to u

I(u) 6 lim infn→∞

I(un).

Example. Let X be a Banach space then || · ||X is sequentially lower semicontinuous for the

weak topology of X.

The supremum of a family of s.l.s.c functions is s.l.s.c while the infimum of a finite fam-

ily of s.l.s.c functions is s.l.s.c A function I is sequentially upper semicontinuous if -I is

sequentially lower semicontinuous.

Definition 2.3.2. (Sequential coercivity) We say that I : X → [−∞,∞] is sequentially

2. Preliminaries 30

coercive if for any t ∈ R the closures of the sublevel sets of I

u ∈ X| I(u) 6 t

are sequentially compact in X.15

Example 1.14 in Dal Maso [35]. If X is a reflexive normed space (e.g. W 1,p(Ω; Rd) with

1 < p < ∞) and I(u) → ∞ as ||u||X → ∞, then I is sequentially coercive with respect to

the weak topology of X.

The Direct Method of the Calculus of Variations can be summarized in the following result.

Theorem 2.3.3. (Weierstrass Theorem) Let I : X → [−∞,∞] be sequentially coercive and

s.l.s.c. Then I attains its minimum in X.

Proof:

Step 1. (if I 6≡ ∞) take a minimizing sequence unn of I in X:

infu∈X

I(u) = limn→∞

I(un) <∞;

Step 2. (compactness property) as I is sequentially coercive, this sequence has a

convergent subsequence unkk;

Step 3. as I is sequentially lower semicontinuous, the limit u0 of the subsequence

unkk is a minimum point of I on X since

I(u0) 6 limk→∞

I(unk) = inf

u∈XI(u) 6 I(u0).

In general, the topology of X needs to be weak enough to ensure that the previous com-

pactness argument hold. In the applications of this method to functionals of the form (2.4),

X is typically a Sobolev Space endowed with the weak or weak⋆ topology.

Typical minimization problem for the functional (2.4). To study:

minu∈A

I(u)

15 Recall that a subset K of X is said to be sequentially compact in X if every sequence in K has a

subsequence which converges (with respect to the topology of X) to a point of K. In a metric space

sequential compactness and compactness are equivalent notions.

2. Preliminaries 31

where

A =

u ∈W 1,p(Ω; Rd) : u− ϕ ∈W 1,p0 (Ω), ϕ ∈W 1,p(Ω)

or

A =

u ∈W 1,p(Ω; Rd) :

∫

Ωu = C

,

C ∈ R, given. In this example, if 1 C|ξ|p−C for

some positive constant C, then by virtue of Poincare inequalities - modulo some regularity

in the domain - and by the reflexivity of the space W 1,p(Ω; Rd), the functional I given

in (2.4) is sequentially coercive with respect to the weak topology of W 1,p(Ω; Rd) in the

admissible (weakly closed) class A. Generally, the main difficulty here is to ensure the

sequential lower semicontinuity of the functional I.

To simplify, we will write s.w.l.s.c on W 1,p(Ω; Rd) and s.w⋆.l.s.c on W 1,∞(Ω; Rd) when we

refer to functionals sequentially lower semicontinuous with respect to the weak or weak⋆

convergence of W 1,p(Ω; Rd) and W 1,∞(Ω; Rd), respectively.

When sequentially lower semicontinuous properties fail, one usually tries to relax this con-

dition.

Definition 2.3.4. (Sequential lower semicontinuous envelope) The sequential lower semi-

continuous envelope (or relaxed functional) of I : X → [−∞,∞] is defined by

slsc I := supG : X → [−∞,∞], G sequentially lower semicontinuous, G 6 I.

The functional slsc I is sequentially lower semicontinuous (it coincides with I if the function

I is sequentially lower semicontinuous). If I is sequentially coercive, so is slsc I. The next

theorem describes the limits of minimizing sequences of a functional I, not necessarily

sequentially lower semicontinuous, in terms of minimum points of slsc I.

Theorem 2.3.5. Assume that the function I : X → [−∞,∞] is sequentially coercive. Then

slsc I has a minimum point in X and

minu∈X

slsc I(u) = infu∈X

I(u).

If X satisfies the first axiom of countability, that is if every point x ∈ X has a countable base

of open sets, then it is possible to derive a characterization of slsc I in terms of sequences.

2. Preliminaries 32

Proposition 2.3.6. Suppose that X satisfies the first axiom of countability. Let I : X →[−∞,∞] and let u ∈ X. Then slsc I(u) is characterized by the following properties

i) for every sequence unn converging to u in X

slsc I(u) 6 lim infn→∞

I(un),

ii) there exists a sequence unn converging to u in X such that

slsc I(u) > lim supn→∞

I(un).

As a consequence, if X is metrizable then

slsc I(u) = infn∈N

lim infn→∞

I(un), un → u in X

.

We turn back to the study of functionals of the type (2.4). In the scalar case (N =

1 or d = 1), and under some standard growth, coercivity and regularity conditions on f ,

the convexity of f(x, s, ·) turns out to be a necessary and sufficient condition for the s.w.l.s.c

of I in W 1,p(Ω; Rd) (s.w⋆.l.s.c if p = ∞). In the vectorial case (N > 1 and d > 1) this

condition (still sufficient) is no longer necessary - quasiconvexity is.

2.3.2 Convex and quasiconvex functions: main properties

We refer to Fonseca and Leoni [47] for the proofs of the results presented here; see also

Braides and Defranceschi [19], Dacorogna [34], Evans and Gariepy [46], as well as the

references therein. Throughout this part N, d > 1.

Definition 2.3.7. (Convex function) A function f : RN → R is said to be convex (resp.

strictly convex) if

f(θξ + (1 − θ)η) 6 θf(ξ) + (1 − θ)f(η)

(resp. <) for all ξ, η ∈ RN and θ ∈ (0, 1).

Definition 2.3.8. The subdifferential ∂f of a convex function f : RN → R at a point

ξ ∈ RN is a set-valued function characterized by the property that η ∈ ∂f(ξ) if and only if

for all γ ∈ RN

f(γ) > f(ξ)+ < η, γ − ξ > .

2. Preliminaries 33

For all points ξ ∈ RN the set ∂f(ξ) is nonempty and convex. Moreover, f is differentiable

at a point ξ if and only if ∂f(ξ) contains a single element, which is then ∇f(ξ).

Theorem 2.3.9. (see Theorem 1 in section 6.3 of Evans and Gariepy [46]) Let f : RN → R

be a convex function. Then f is locally Lipschitz in RN , and there exists a constant C

(depending only on N) such that for every ball B(ξ, r) ⊂ RN , r > 0,

supη∈B(

ξ, r2

)

|f(η)| 6C

|B(ξ, r)|

∫

B(ξ,r)|f(η)| dη.

As a consequence of Theorem 2.3.9 and Rademacher’s Theorem16, if f is convex then it is

differentiable almost everywhere. For all points ξ where f is differentiable

f(η) > f(ξ) + ∇f(ξ) · (η − ξ) (2.5)

for all η ∈ RN , which expresses the geometrical fact that the graph of f lies above its

tangent hyperplane at the point ξ. If f is convex and satisfies 0 6 f(ξ) 6 C(1 + |ξ|p), for

some C > 0, 1 6 p <∞ and for all ξ ∈ RN , then

|f(ξ) − f(η)| 6 C(1 + |ξ|p−1 + |η|p−1)|ξ − η|, (2.6)

for all ξ and η in RN .

Theorem 2.3.10. (Theorem 1 in section 6.3 of Evans and Gariepy [46]) Let f : RN → R

be a convex function. Then for each ball B(ξ, r) ⊂ RN , r > 0, we have

ess supη∈B(

ξ, r2

)

|∇f(η)| 6C

r|B(ξ, r)|

∫

B(ξ,r)|f(η)| dη.

Theorem 2.3.11. (Jensen inequality) (see also Lemma 23.2 in Dal Maso [35]) A function

f : RN → R is convex if and only if given any measure µ on a measurable space (E,M),

with E ⊂ RN containing at least two distinct points and such that µ(E) = 1 (probability

measure on E), and given any g ∈ L1(E, µ; RN ) then

f(

∫

Eg dµ

)

6

∫

Ef(g) dµ.

16 Rademacher’s Theorem (see section 3.1.2 in Evans and Gariepy [46]): Every locally Lipschitz function

f : RN → R is differentiable a.e.

2. Preliminaries 34

Definition 2.3.12. (Convex envelope) The convex envelope (or convexification) of a func-

tion f : RN → R is the function Cf : RN → [−∞,∞] defined by

Cf(ξ) := supg(ξ) : g : RN → R, g convex, g 6 f.17

Clearly Cf = f if the function f is convex. We now recall Morrey’s notion of quasiconvexity.

Definition 2.3.13. (Quasiconvex function; Morrey [65]) A Borel measurable function f :

Rd×N → R is quasiconvex at a point ξ ∈ Rd×N if

f(ξ) 6

∫

Ωf(ξ + ∇φ(η)) dη (2.7)

for every φ ∈W 1,∞0 (Ω; Rd) (for which the integral (2.7) is well defined), and for every open

bounded set Ω ⊂ RN with |∂Ω| = 0. The function f is said to be quasiconvex if it is

quasiconvex at every ξ ∈ Rd×N .

Remark 2.3.14.

i) A quasiconvex function is locally Lipschitz.

ii) Using the Vitali’s Covering Theorem (Theorem 2.1.21), it can be seen that it suffices

to check (2.7) for one fixed open bounded set Ω of RN , for instance Ω = Q = (0, 1)N

(see also Remark 5.15 in Braides and Defranceschi [19]).

iii) In view of Jensen’s inequality (Theorem 2.3.11), if a function f : Rd×N → R is convex

then it is quasiconvex. Indeed for every ξ ∈ Rd×N

f(ξ) = f

(∫

Qξ + ∇φ(η) dη

)

6

∫

Qf (ξ + ∇φ(η)) dη, ∀φ ∈W 1,∞

0 (Q). (2.8)

The converse is not generally true (e.g. f(ξ) = |det ξ| with ξ ∈ R2×2; see for instance

Fonseca and Leoni [47]), but both notions are equivalent in the scalar case N = 1 or

d = 1.

Proposition 2.3.15. (Marcellini [64]) If f : Rd×N → R is quasiconvex and if it satisfies

the growth condition 0 6 f(ξ) 6 C(1 + |ξ|p) for all ξ ∈ Rd×N and some 1 6 p < ∞, then

for all ξ, η ∈ Rd×N

|f(ξ) − f(η)| 6 C(1 + |ξ|p−1 + |η|p−1)|ξ − η|. (2.9)

17 Usual convention: sup∅ = −∞.

2. Preliminaries 35

Definition 2.3.16. (Quasiconvex envelope) The quasiconvex envelope (or quasiconvexifi-

cation) of a function f : Rd×N → R is the function Qf : Rd×N → [−∞,∞] defined by

Qf(ξ) := supg(ξ) : g : Rd×N → R, g quasiconvex, g 6 f.

Remark 2.3.17. Clearly Qf = f if the function f is quasiconvex. In general Cf 6 Qf 6 f.

As a consequence of this remark and the fact that the function ξ → |ξ|p, ξ ∈ Rd×N , is

convex for p > 1, we get the following useful lemma.

Lemma 2.3.18. If f : Rd×N → R satisfies

|ξ|pC

− C 6 f(ξ) 6 C(1 + |ξ|p)

for some positive constant C and for all ξ ∈ Rd×N with 1 6 p <∞, then for all ξ ∈ Rd×N

|ξ|pC

− C 6 Qf(ξ) 6 C(1 + |ξ|p).

Theorem 2.3.19. If f : Rd×N → R is locally bounded and Borel measurable then for any

open bounded set Ω ⊂ RN with |∂Ω| = 0

Qf(ξ) = inf

1

|Ω|

∫

Ωf(ξ + ∇φ(η)) dη : φ ∈W 1,∞

0 (Ω; Rd)

.

In addition the function Qf is quasiconvex.

Next we recall a stronger notion called W 1,p-quasiconvex introduced by Ball and Murat

in [13] that allows the competing functions φ to belong to the Sobolev space W 1,p0 (Ω; Rd)

rather than to the smaller space W 1,∞0 (Ω; Rd).

Definition 2.3.20. (W 1,p-quasiconvexity) Let f : Rd×N → R be a Borel measurable func-

tion, and let 1 6 p 6 ∞. The function f is said to be W 1,p-quasiconvex at ξ ∈ Rd×N

if

f(ξ) 6

∫

Ωf(ξ + ∇φ(η)) dη (2.10)

for every φ ∈W 1,p0 (Ω; Rd) (for which the integral (2.10) is well defined) and for every open

bounded set Ω ⊂ RN with |∂Ω| = 0. The function u is said W 1,p-quasiconvex if it is

W 1,p-quasiconvex at every ξ ∈ Rd×N .

Remark 2.3.21.

2. Preliminaries 36

i) As before, it is enough to check inequality (2.10) for Ω = Q.

ii) When p = ∞ we recover the notion of quasiconvexity.

iii) It f is W 1,p-quasiconvex then f is W 1,q-quasiconvex for all q with p 6 q 6 ∞. Thus

W 1,1-quasiconvexity is the strongest condition and W 1,∞-quasiconvexity the weakest.

Proposition 2.3.22. (see Proposition 2.4 in Ball and Murat [13]) Let f : Rd×N → R be a

Borel function and assume that

f(ξ) 6 C(1 + |ξ|p)

for all ξ ∈ Rd×N , where C is a positive constant and 1 6 p < ∞. Then f is W 1,p-

quasiconvex if and only if f is quasiconvex.

Lemma 2.3.23. If f : Rd×N → R is a locally bounded and Borel measurable function such

that

f(ξ) 6 C(1 + |ξ|p)

for some positive constant C and for all ξ ∈ Rd×N , with 1 6 p < ∞, then, for any open

bounded set Ω ⊂ RN with |∂Ω| = 0,

Qf(ξ) = inf

1

|Ω|

∫

Ωf(ξ + ∇φ(η)) dη : φ ∈W 1,p

0 (Ω; Rd)

.

There are very few explicit examples in the literature of quasiconvex envelopes due to the

nonlocal character of its definition. A very interesting example in the nonlinear elasticity

setting is the Saint-Venant-Kirchhoff stored energy function.

Example. (Le Dret and Raoult [54]) Let

f(ξ) =µ

8(1 + ν)

∣

∣ξT ξ − I3∣

∣

2+

µν

8(1 + ν)(1 − 2ν)

(

|ξ|2 − 3)2,

where I3 denotes the identity matrix in R3, and µ > 0 and 0 6 ν < 12 are respectively the

Young modulus and the Poisson ratio of a hyperelastic material. Then

Qf(ξ) = Φ(α1(ξ), α2(ξ), α2(ξ)),

where 0 6 α1(ξ) 6 α2(ξ) 6 α3(ξ) are the singular values of the matrix ξ, i.e. the square

roots of the eigenvalues of ξT ξ, and where

Φ(α1, α2, α2) := µ8

[

(α23 − 1)+

]2+ µ

8(1−ν2)

[

(α22 + να2

3 − (1 + ν))+]2

+ µν8(1−ν2)(1−2ν)

[

((1 − ν)α21 + ν(α2

2 + α23) − (1 + ν))+

]2.

2. Preliminaries 37

Here (·)+ stands for the positive part.

We conclude this Subsection with the following result.

Lemma 2.3.24. (see Dal Maso, Fonseca, Leoni and Morini [36]) Let Ω ⊂ RN with N > 1

be an open set, and let f : Ω × Rd×N → R with d > 1 be a lower semicontinuous function

which satisfies the following hypotheses:

i) for every x ∈ Ω the function f(x, ·) is continuous in Rd×N ;

ii) there exist two locally bounded functions a, b : Ω → [0,∞), a lower semicontinuous

function c : Ω → (0,∞), and a constant 1 < p < ∞ such that for every (x, ξ) ∈Ω × Rd×N

c(x)|ξ|p − b(x) 6 f(x, ξ) 6 a(x)(1 + |ξ|p).

For every x ∈ Ω let Qf(x, ·) be the quasiconvexification of the function f(x, ·). Then

Qf is lower semicontinuous on Ω × Rd×N .

2.3.3 Lower semicontinuity characterization for integral functionals

defined on Sobolev spaces

We start by recalling the definition of an important class of integrands.

Definition 2.3.25. (Caratheodory integrand) Let Ω ⊂ RN with N > 1 be an open set, and

let B be a Borel set of Rl with l > 1. A function f : Ω×B → R is said to be a Caratheodory

integrand if

i) x→ f(x, ξ) is measurable for every ξ ∈ B,

ii) ξ → f(x, ξ) is continuous for almost all x ∈ Ω.

In this work we will deal with Caratheodory integrands f : Ω× Rl → R with l > 1, and we

will use the following characterization.

Theorem 2.3.26. (Scorza-Dragoni Theorem) (see Ekeland and Teman [45]) Let Ω ⊂ RN

with N > 1 be an open set. A function f : Ω × Rl → R, with l > 1, is Caratheodory if

and only if given a compact set K ⊂ Ω and a positive number ε, there exists a compact set

Kε ⊂ K such that LN (K \Kε) 6 ε and the restriction of f to Kε × Rl is continuous.

2. Preliminaries 38

The following result shows that every Caratheodory integrand is (equivalent to) a Borel

function.

Proposition 2.3.27. (see Proposition 3.3 in Braides and Defranceschi [19] or Ekeland and

Teman [45]) Let Ω ⊂ RN with N > 1 be an open set, and let B be a Borel set of Rl with

l > 1. Every Caratheodory integrand f : Ω × B → R is (equivalent to) a Borel function,

that is there exists a Borel function g : Ω ×B → R such that f(x, ·) = g(x, ·) for a.e. x.

As a consequence of the Dominated Convergence Theorem (Proposition 2.1.15) a useful

result follows.

Lemma 2.3.28. Let Ω ⊂ RN with N > 1 be an open bounded set, and let f : Ω × Rd ×Rd×N → R with d > 1 be a Caratheodory integrand. Let 1 6 p <∞ and assume that

|f(x, s, ξ)| 6 C(1 + |ξ|p)

for a.e. x ∈ RN and for all (s, ξ) ∈ Rd × Rd×N , for some positive constant C. Then the

functional I : W 1,p(Ω; Rd) → R defined by

I(u) :=

∫

Ωf(x, u(x),∇u(x)) dx

is continuous for the strong topology of W 1,p(Ω; Rd).

The following theorem shows that quasiconvexity is a necessary and sufficient condition for

s.w.l.s.c on the Sobolev spaces W 1,p(Ω; Rd) (s.w⋆.l.s.c if p = ∞) for integral functionals

I : W 1,p(Ω; Rd) → R,

I(u) :=

∫

Ωf(x, u(x),∇u(x)) dx. (2.11)

Theorem 2.3.29. (see Statement II.5 in Acerbi and Fusco [1]; see also Morrey [65]) Let

Ω ⊂ RN with N > 1 be an open set, and let f : RN × Rd × Rd×N → R with d > 1 be a

Caratheodory integrand satisfying

i) 0 ≤ f(x, s, ξ) ≤ a(x) + C(|s|p + |ξ|p) with 1 6 p <∞, for every (x, s, ξ) ∈ RN × Rd ×Rd×N , where C > 0 and a is a non-negative function in L1

loc(RN );

ii) 0 ≤ f(x, s, ξ) ≤ b(x) + c(s, ξ), for every (x, s, ξ) ∈ RN ×Rd×Rd×N , where b is a non-

negative function in L1loc(R

N ), and c is a non-negative and locally bounded function

on Rd × Rd×N , if p = ∞.

2. Preliminaries 39

Then the functional I given in (2.11) is s.w.l.s.c on W 1,p(Ω; Rd) [or s.w⋆.l.s.c on W 1,∞(Ω; Rd)

if p = ∞] if and only if f(x, s, ·) is quasiconvex for every x ∈ RN and for every s ∈ Rd.

We conclude this part with a relaxation theorem obtained in Acerbi and Fusco [1] (see

Statement III.7; see also Dacorogna [34]). We will denote the sequential lower semicontin-

uous envelope of I with respect to weak topology of W 1,p (weak⋆ topology if p = ∞) by

swlsc I.

Theorem 2.3.30. Let Ω ⊂ RN with N > 1 be an open set, and let f : RN×Rd×Rd×N → R

with d > 1 be a Caratheodory integrand such that

i) 0 ≤ f(x, s, ξ) ≤ a(x) + C(|s|p + |ξ|p), p ≥ 1;

ii) 0 ≤ f(x, s, ξ) ≤ b(x) + c(s, ξ), p = ∞,

for every (x, s, ξ) ∈ RN×Rd×Rd×N , where C is a non-negative constant, b is a non-negative

function in L1loc(R

N ) and c is a locally bounded and non-negative function on Rd × Rd×N .

Then

swlsc I(u) =

∫

ΩQf(x, u(x),∇u(x)) dx,

where I is the functional given in (2.11) and Qf(x, s, ξ) stands for the quasiconvexification

of f(x, s, .) at ξ.

2.4 Integral representation of nonlinear local functionals

defined on Sobolev spaces

In this section we recall two integral representation theorems for local functionals depending

on Sobolev functions and on open sets, that are useful in relaxation and Γ-convergence

theories.

The first theorem was obtained by Buttazzo and Dal Maso and gives abstract conditions

under which a local functional I admits an integral representation of the form

I(u,A) =

∫

Af(x,∇u(x)) dx

for some Caratheodory integrand f (see Theorem 1.1 in [26] and references therein).

2. Preliminaries 40

Theorem 2.4.1. Let Ω be an open subset of RN with N > 1. Let I : W 1,p(Ω; Rd)×A(Ω) →R, with d > 1 and 1 6 p 6 ∞, satisfies the following properties

i) I is local on A(Ω), i.e. I(u,A) = I(v,A) whenever A ∈ A(Ω), u, v ∈ W 1,p(Ω; Rd)

and u = v a.e. on A;

ii) I is a measure on A(Ω), i.e. for every u ∈W 1,p(Ω; Rd) the set function I(u, .) is the

restriction to A(Ω) of a finite Radon measure;

iii) I satisfies a growth condition of order p, i.e. when p < ∞ there exist a ∈ L1(Ω) and

b > 0 such that for every A ∈ A(Ω) and every u ∈W 1,p(Ω; Rd),

|I(u,A)| 6

∫

A[a(x) + b|∇u|p] dx,

and when p = ∞ for every r > 0 there exist ar ∈ L1(Ω) such that

|I(u,A)| 6

∫

Aar(x) dx,

for every A ∈ A(Ω) and every u ∈W 1,∞(Ω; Rd) with |∇u| 6 r a.e. in A;

iv) I is translation invariant, i.e. for every A ∈ A(Ω), u ∈W 1,p(Ω; Rd), c ∈ Rd,

I(u+ c, A) = I(u,A);

v) for every A ∈ A(Ω), the function I(·, A) is s.w.l.s.c on W 1,p (s.w⋆.l.s.c if p = ∞).

Then, there exists a function f : Ω × Rd×N → R such that

a) for every A ∈ A(Ω) and every u ∈ W 1,p(Ω; Rd) the integral representation formula

holds

I(u,A) =

∫

Af(x,∇u(x)) dx;

b) f is a Caratheodory integrand;

c) f(x, z) satisfies a growth condition of order p, that is, when p < ∞ there exist a ∈L1(Ω) and b > 0 such that

|f(x, z)| 6 a(x) + b|z|p,

2. Preliminaries 41

for a.e. x ∈ Ω and for all z ∈ Rd×N , and when p = ∞ for every r > 0 there exists

ar ∈ L1(Ω) such that

|f(x, z)| 6 ar(x)

for a.e. x ∈ Ω and for all z ∈ Rd×N with |z| 6 r.

Remark 2.4.2.

• Conditions a), b), c) imply i), ii), iii), iv) but not v). Nevertheless, the integral repre-

sentation theorem does not hold if we drop hypothesis v) (see examples in Buttazzo

and Dal Maso [26]).

• Conditions a), b), c) and v) imply that for a.e. x ∈ Ω the function ξ → f(x, ξ) is

quasiconvex (see Theorem 2.3.29).

The second theorem was derived in Bouchitte, Fonseca, Leoni and Mascarenhas [24] (Theo-

rem 1.1), and gives abstract conditions under which a local functional I admits an integral

representation of the form

I(u,A) =

∫

Af(x, u(x),∇u(x)) dx

for some Borel function f (see references in [24]; see also Buttazzo and Dal Maso [27] where

under additional uniform continuity hypothesis they derive a similar result in terms of a

Caratheodory integrand).

Theorem 2.4.3. Let Ω ⊂ RN with N > 1. Let I : W 1,p(Ω; Rd) × A(Ω) → [0,∞], with

1 < p <∞, be a functional satisfying hypotheses

(i) I(u; ·) is the restriction to A(Ω) of a Radon measure;

(ii) I(u;A) = I(v;A) whenever u = v LN -a.e. on A ∈ A(Ω);

(iii) I(·;A) is s.l.s.c with respect to the L1(Ω; Rd)-topology;

(iv) there exists C > 0 such that

1

C

∫

A|∇u|p dx 6 I(u;A) 6 C

∫

A(1 + |∇u|p) dx.

2. Preliminaries 42

Then, for every u ∈W 1,p(Ω; Rd) and A ∈ A(Ω) we have

I(u;A) =

∫

Af(x, u,∇u) dx

where f is the Borel function given by

f(x, s, ξ) = lim supε→0

m(s+ ξ(· − x));Q(x, ε))

εN

for all (x, s, ξ) ∈ Ω × Rd × Rd×N , and where for (v,A) ∈W 1,p(Ω; Rd) ×A∞(Ω)

m(v;A) := infI(w;A) with w = v in a neighborhood of ∂A,

and Q(x, ε) := x + εQ for x ∈ Ω and ε > 0. Here A∞(Ω) denotes the class of Lipschitz

subdomains of Ω.

2.5 Γ-convergence of a family of functionals

The aim of this section is to recall the notion of Γ-convergence introduced by De Giorgi and

afterwards applied to a large number of different problems in the Calculus of Variations

(see De Giorgi and Franzoni [40]; see also De Giorgi and Dal Maso [38]). Γ-convergence

plays an important role in situations where, to understand properties of equilibrium states

of a given functional, it is necessary to study the behavior of a family of minimum problems

depending on a small parameter ε:

minu∈X

Iε(u). (2.12)

A typical example concerns periodic homogenization problems. Here one is led to consider

functionals of the type

Iε(u) =

∫

Ωfε(x, u(x),∇u(x)) dx (2.13)

where the integrands fε are increasingly oscillating in the first variable as the parameter ε

goes to zero.

In general to characterize the solutions of problem (2.12) one is led to consider an “effective

minimum problem” (not depending on ε)

minu∈X

I(u),

2. Preliminaries 43

that captures the relevant behavior of minimizers and for which a solution can be obtained

more easily. The effectiveness of Γ-convergence is linked to the possibility of obtaining

convergent sequences from minimizers (or almost-minimizers) of problem (2.12). This is

possible in the class of integral functionals (2.13) under suitable conditions on the integrands

fε.

We give the abstract definition of Γ-convergence and we collect some of its most useful

properties for the analysis of functionals of the type (2.13). We refer to Braides [18] and

Dal Maso [35] for a comprehensive treatment and detailed bibliography on this subject.

2.5.1 The notion of Γ-convergence and main results

Throughout this part (X, d) denotes a metric space.

Definition 2.5.1. (Γ-convergence of a sequence of functionals) Let Inn be a sequence

of functionals defined on X with values on R. The functional I : X → R is said to be the

Γ-lim inf (resp. Γ-lim sup) of Inn with respect to the metric d if for every u ∈ X

I(u) = infun

lim infn→∞

In(un) : un ∈ X, un → u in X

(resp. lim supn→∞

).

In this case we write

I = Γ-lim infn→∞

In(

resp. I = Γ-lim supn→∞

In)

.

Moreover, the functional I is said to be the Γ-lim of Inn if


In = Γ-lim supn→∞

In,

and in this case we write

I = Γ-limn→∞

In.

Lemma 2.5.2. Let In, I : X → R. Then I = Γ- limn→∞

In if and only if for every u in X

the following conditions hold:

i) for all sequences unn converging to u in X,

I(u) 6 lim infn→∞

In(un); (2.14)

2. Preliminaries 44

ii) (recovering sequence) there exists a sequence unn converging to u in X such that

I(u) > lim supn→∞

In(un). (2.15)

Due to this result, Definition 2.5.1 can be also given pointwise: The family of functionals

Inn Γ-converges to the value I(u) at a point u ∈ X if inequalities (2.14) and (2.14) hold;

in this case we write I(u) = Γ- limn→∞

In(u). Thus, I is the Γ-lim of Inn with respect to

the metric d if and only if I(u) = Γ- limn→∞

In(u) for all u ∈ X.

For every ε > 0 let Iε be a functional over X with values on R, Iε : X → R.

Definition 2.5.3. (Γ-convergence of a family of functionals)

A functional I : X → R is said to be the Γ- lim inf (resp. Γ-lim sup or Γ-lim) of Iεε with

respect to the metric d, as ε→ 0, if for every sequence εn ↓ 0,


Iεn

(

resp. I = Γ-lim supn→∞

Iεn or I = Γ- limn→∞

Iεn

)

,

and we write

I = Γ-lim infε→0

Iε(

resp. I = Γ-lim supε→0

Iε or I = Γ-limε→0

Iε)

.

The expressions “Γ-converge to I” or “is the Γ-lim of Iε” are used interchangeably.

Remark 2.5.4.

i) If a family of functionals Γ-converges so does every sub-family (and to the same limit).

ii) As a consequence of Proposition 2.3.6, if Iε ≡ I then Iεε Γ-converges to slscI(hence a constant family may Γ-converges to a limit different from the constant itself).

iii) Obviously, the notions of Γ-convergence and pointwise or uniform convergence do not

coincide.

Examples (see Dal Maso [35]). Let X = R with the usual metric.

• Let Iε(u) := uε exp[−2

(

uε

)2]. Then Iεε converges pointwise to 0 but it Γ-

converges in R to the function

2. Preliminaries 45

I(u) :=

−12 exp[12 ] if u 6= 0,

0 if u = 0.

• Let Iε(u) := sin(

uε

)

. Then Iεε Γ-converges in R to the constant function

I := −1 but it does not converge pointwise, except at u = 0.

Proposition 2.5.5. If I = Γ-lim infε→0

Iε (or Γ- lim supε→0

) then I is lower semicontinuous (with

respect to the metric d). Consequently, if I = Γ- limε→0

Iε then I is lower semicontinuous.

The following result studies the Γ-limit of a continuous perturbation of a family of func-

tionals.

Proposition 2.5.6. Let G : X → R be a continuous functional and let Iεε be a family of

functionals on X. Then

i) Γ- lim infε→0

(Iε + G) = Γ- lim infε→0

Iε + G;

ii) Γ- lim supε→0

(Iε + G) = Γ- lim supε→0

Iε + G.

In particular, if Iεε Γ-converges to I in X, then Iε + Gε Γ-converges to I + G in X.

The next result states that a metric space (X, d) satisfies the Urysohn property with respect

to Γ-convergence.

Proposition 2.5.7. Given I : X → R and εn ↓ 0, I = Γ- limn→∞

Iεn if and only if for every

subsequence εnjj there exists a further subsequence εjkk such that Iεjk

k Γ-converges

to I.

If, in addition, (X, d) is a separable metric space then the following compactness property

holds.

Theorem 2.5.8. Each sequence εn ↓ 0 has a subsequence εnjj ≡ εjj such that Γ-

limj→∞

Iεjexists.

Remark 2.5.9.

i) If the space X is not separable the conclusion of Theorem 2.5.8 may fail (see a coun-

terexample in Braides and Defranceschi [19]).

2. Preliminaries 46

ii) We remark that, as a consequence of Theorem 2.5.8 and Proposition 2.5.7, if (X, d)

is separable then to conclude that Γ- limε→0

Iε = I it suffices to prove that given εn ↓ 0

there exists a subsequence εnjj ≡ εjj such that I = Γ- lim inf

j→∞Iεj

.

Definition 2.5.10. A family of functionals Iεε is said to be equi-coercive if for every real

number λ there exists a compact set Kλ in X such that for each sequence εn ↓ 0,

u ∈ X : Iεn(u) 6 λ ⊆ Kλ for every n ∈ N.

Example. Let Ω be an open, bounded subset of RN with N > 1. Let Iε and fε be given as

in (2.13), and assume that C|ξ|p − C 6 fε(x, s, ξ) for some positive constant C and some

1 < p < ∞. Then by Poincare inequality the family of functionals Iε is equi-coervice on

W 1,p0 (Ω) with respect to the strong topology of Lp.

As mentioned before, one of the most important properties of Γ-convergence, and the reason

why this kind of convergence is so important in the asymptotic analysis of variational

problems, is that under appropriate compactness properties it implies the convergence of

(almost) minimizers of a family of equi-coercive functionals to the minimum of the limiting

functional (the minimum exists by virtue of Weierstrass’s Theorem, Theorem 2.3.3). More

precisely, we have the following result.

Theorem 2.5.11. (Fundamental Theorem of Γ-convergence) If Iεε is a family of equi-

coercive functionals on X and if

I = Γ- limε→0

Iε,

then the functional I has a minimum on X and

minu∈X

I(u) = limε→0

infu∈X

Iε(u).

Moreover, given εn ↓ 0 and unn a converging sequence such that

limn→∞

Iεn(un) = limn→∞

infu∈X

Iεn(u), (2.16)

then its limit is a minimum point for I on X.

If (2.16) holds, then unn is said to be a sequence of almost-minimizers for I.

Theorem 2.5.12. Let Iεε be a family of functionals on X. The following equalities hold

i) Γ- lim infε→0

Iε = Γ- lim infε→0

slsc Iε,

2. Preliminaries 47

ii) Γ- lim supε→0

Iε = Γ- lim supε→0

slsc Iε,

where slsc Iε denotes the sequential lower semicontinuous envelope of Iε (see Definition

2.3.4). In particular, Iεε Γ-converges to I if and only if slsc Iεε Γ-converges to I.

2.5.2 The Direct Method of Γ-convergence for a class of integral

functionals

The purpose of this part is to give an overview of the Direct Method of Γ-convergence, first

outlined by De Giorgi and later used by many other authors (see De Giorgi [37], Dal Maso

and De Giorgi [38]; see also Braides [17] and Dal Maso [35]). It is an important tool to

obtain the representation of the Γ-limit of a family of functionals, and is used in Chapters

4 and 5. We restrict ourselves to a Lp-setting and discuss the Direct Method for

Iε(u) :=

∫

Ωfε (x, u(x),∇u(x)) dx

with u ∈W 1,p(Ω; Rd), where Ω ⊂ RN with N > 1 is an open set, and fε : Ω×Rd×Rd×N →R, with d > 1, is a family of Borel functions satisfying standard coercivity and growth

conditions. In what follows, we will use the notation Γ(Lp(Ω))-limit to refer to the Γ-

convergence with respect to the usual metric in Lp(Ω; Rd) for 1 < p < ∞. Denoting by

I = Γ(Lp(Ω))- limε→0

Iε (if it exists), the main questions are:

• Does there exist a Borel function f : RN × Rd × Rd×N → R such that for all u ∈W 1,p(Ω; Rd)

I(u) =

∫

Ωf(x, u(x),∇u(x)) dx ?

• Is it possible to derive f explicitly, or, at least, to determine some of its properties?

The main steps to arrive at an answer are:

i) Consider the dependence of the integrals on the integration set (localization), that

is, consider the family of functionals Iε : Lp(Ω; Rd) × A(Ω) → R defined for u ∈W 1,p(Ω; Rd) by

Iε(u,A) =

∫

Afε (x, u(x);∇u(x)) dx,

2. Preliminaries 48

and then study the functional

I(u,A) = Γ(Lp(Ω))- limε→0

Iε(u,A).

ii) Examine the dependence of I on the variable A in order to prove that the set function

A → I(u,A) is the trace of a Radon measure for every u ∈ W 1,p(Ω; Rd), using, for

instance, Lemma 2.1.20 above or another De Giorgi’s-Letta type of argument.

We remark that, in general, the main difficulty in applying this lemma is to prove the

(nested) subadditivity of I(u, ·):

I(u,D) ≤ I(u,D\B) + I(u,C),

DC

B

for all B,C,D ∈ A(Ω) with B ⊂⊂ C ⊂ D. By the definition of I, this means that we

should be able to construct recovering sequences for I(u,D) from recovering sequences

for I(u,D\B) and I(u,C) by some matching process. For families of functionals that

satisfy standard growth and coerciveness conditions, this procedure is possible by a

slicing argument introduced by De Giorgi in [37] described in Theorem 2.5.13 below.

iii) If the functional I has good continuity or convexity properties, the next step is to use

convenient integral representation theorems (for instance Theorems 2.4.1 and Theorem

2.4.3) to prove that the measure I(u,A) can be written in the form

I(u,A) =

∫

Af(x, u(x),∇u(x)) dx.

Due to Remark 2.5.9, one can use this strategy to prove both the existence of the Γ-limit

of the family Iεε and its integral representation:

Step 1. Establish a compactness result that guarantees each sequence Iεnn, with εn ↓ 0,

has a subsequence Γ(Lp(Ω))-converging to an abstract limit functional.

Step 2. Establish an integral representation result.

Step 3. Prove the representation formula is well defined, i.e. does not depend on the subse-

quence.

2. Preliminaries 49

This method involves testing on the linear functions only (Step 3), as opposed to working

with general recovering sequences (Lemma 2.5.2). In addition, the two first steps can be

carried out in a systematic way for a large class of situations, due to the properties of

Γ-convergence.

The next result is an important tool for the application of Γ-convergence to variational

problems for functionals of the form Iεε with Dirichlet boundary conditions. It states

that, under appropriate hypotheses on the integrand fε, the boundary conditions do not

affect the limit functional. In particular, it provides appropriate conditions to accomplish

point ii) above.

Proposition 2.5.13. Let Ω be an open and bounded subset of RN and let p > 1. Assume

that fε : Ω × Rd × Rd×N → R is a family of Caratheodory integrands such that

1

C|ξ|p − C 6 fε(x, s, ξ) 6 C(1 + |s|p + |ξ|p) (2.17)

for a.e. x ∈ Ω and for all (s, ξ) ∈ Rd×Rd×N , for some constant C > 0. Consider the family

of functionals Iε : Lp(Ω; Rd) → [0,∞] defined by

Iε(u) :=

∫

Ωfε (x, u(x),∇u(x)) dx if u ∈W 1,p(Ω; Rd),

∞ otherwise,

and for each ϕ ∈W 1,p(Ω; Rd) define the functional

Gϕε (u) :=

Iε(u) if u− ϕ ∈W 1,p0 (Ω; Rd),

∞ otherwise.

If Iεε Γ(Lp(Ω))-converges to a functional I, then the family of functionals Gϕε ε Γ(Lp(Ω))-

converges to the functional

Gϕ(u) :=

I(u) if u− ϕ ∈W 1,p0 (Ω; Rd),

∞ otherwise.

2. Preliminaries 50

Proof. (for an alternative proof see Proposition 11.7 in Braides and Defranceschi [19]; see

also Theorem 21.1 in Dal Maso [35]) We start by remarking that there is no loss of generality

in assuming that f is positive. If not, we may replace f by f+C that, in view of hypotheses

(2.17), is positive. As mentioned before, the proof relies on De Giorgi’s slicing argument

introduced in [37]. Let u ∈W 1,p(Ω) and let εn ↓ 0. If u− ϕ 6∈W 1,p0 (Ω; Rd) then, by (2.17),

Γ- limn→∞

Gϕεn(u) = ∞.

Let us assume that u−ϕ ∈W 1,p0 (Ω; Rd). The conclusion will follow if we can find a sequence

wnn ⊂W 1,p(Ω) with wn = u on ∂Ω such that

I(u) = limn→∞

∫

Ωfεn (x,wn, Dwn) dx (2.18)

holds. Let vnn ⊂W 1,p(Ω; Rd) be a sequence such that ||u− vn||Lp(Ω) →n→∞

0 and

I(u) = limn−→∞

Iεn(vn).

Set

β0 := supn

∫

Ω(1 + |Dvn|p) dx <∞ (by (2.17)),

and define for n ∈ N

Kn :=

∣

∣

∣

∣

∣

[[

1

||vn − u|| 12 Lp(Ω)

]]∣

∣

∣

∣

∣

,

Mn :=∣

∣

∣

[[

√

Kn

]]∣

∣

∣ ,

and finally

Ωn :=

x ∈ Ω : dist(x, ∂Ω) <Mn

Kn

,

where [[·]] stands for the integer part function.

We observe that by definition Kn ↑ ∞ and LN (Ωn) ↓ 0 as n→ ∞. For each n,subdivide Ωn (*)

into Mn disjoint subsets

Ωin :=

x ∈ Ωn : dist(x, ∂Ω) ∈[

i

Kn,i+ 1

Kn

]

, i = 0, ...,Mn − 1,

and choose in ∈ 0, ...,Mn − 1 such that∫

Ωinn

(1 + |Dvn|p) dx 6

∫

Ωin

(1 + |Dvn|p) dx

2. Preliminaries 51

/Mn Kn

nK/i

nK/i+1

Ω

Ω n

Ω ni

for all i = 0, ...,Mn − 1. Then

Mn

∫

Ωinn

(1+|Dvn|p) dx 6

∫

Ωn

(1+|Dvn|p) dx =

Mn−1∑

i=0

∫

Ωin

(1+|Dvn|p) dx 6

∫

Ω(1+|Dvn|p) dx 6 β0,

or, equivalently,∫

Ωinn

(1 + |Dvn|p) dx 6β0

Mn.

Let φn ∈ C∞0 (Ω) be such that 0 6 φn 6 1, ||Dφn||∞ 6 Kn,

φn :=

1 if dist(x, ∂Ω) > in+1Kn

,

0 if dist(x, ∂Ω) 6 inKn,

nK/i

nK/i+1n

φ

Ω

1

0

and define wn := φnvn + (1 − φn)u ∈ W 1,p(Ω; Rd). Clearly wn → u strongly in Lp(Ω; Rd),

2. Preliminaries 52

wn = u in Ω\Kn, with Kn :=

x ∈ Ω : dist(x, ∂Ω) > inKn

and |Ω\Kn| → 0. Moreover

I(u) = limn→∞

Iεn(vn)

> lim supn→∞

∫

Ω∩

x: dist(x,∂Ω)>in+1Kn

fεn (x,wn, Dwn) dx.

Consequently,


∫

Ωfεn (x,wn, Dwn) − C lim inf

n→∞

∫

Ωn∩

x: dist(x,∂Ω)< inKn

(1 + |u|p + |Du|p) dx

−Cβ lim infn→∞

∫

Ωinn

(1 + |vn|p + |Dvn|p) dx− Cβ lim infn→∞

|Kn|p∫

Ωinn

|vn − u|p dx

−Cβ lim infn→∞

∫

Ωinn

(|u|p + |Du|p) dx,

where C is the constant given in hypothesis (2.17) and β is some positive constant. Then


∫

Ωfεn (x,wn, Dwn) − Cββ0 lim inf

n→∞

1

Mn− Cβ lim inf

n→∞||vn − u||

p2

Lp(Ω)

= lim supn→∞

Iεn(wn),

Accordingly (2.18) holds, that is

I(u) = limn→∞

∫

Ωfεn (x,wn, Dwn) dx.

The next result shows that in many situations we can assume, without loss of generality,

that our functionals Iε have good convexity properties. More precisely, as a consequence of

Theorem 2.3.30 and Theorem 2.5.12 we have the following result.

Corollary 2.5.14. Let Ω ⊂ RN be an open bounded set with N > 1. For each ε > 0 let

fε : Ω × Rd × Rd×N → R be a Caratheodory integrand and suppose that

0 ≤ fε(x, s, ξ) ≤ a(x) + C(|s|p + |ξ|p),

2. Preliminaries 53

for a.e. x ∈ Ω and for all (s, ξ) ∈ Rd×Rd×N , with 1 6 p <∞, where C is a positive constant,

and a is a non-negative function in L1loc(R

N ). For each ε > 0 consider the functional Iεdefined in W 1,p(Ω; Rd) by

Iε(u) =

∫

Ωfε(x, u(x),∇u(x)) dx.

Then Iεε Γ(Lp(Ω))-converge to a functional I if and only if the family of functionals Jεdefined in W 1,p(Ω; Rd) by

Jε(u) =

∫

ΩQfε(x, u(x),∇u(x)) dx,

Γ(Lp(Ω))-converge to I, where Qfε(x, s, ξ), stands for the quasiconvexification of fε(x, s, .)

at ξ.

2.6 Two-Scale Convergence

The origins of two-scale convergence are in a paper by Nguetseng [69] (see [61] and also

[70]) concerning the homogenization of linear elliptic problems with periodic coefficients of

the form

−div(A(xε )∇uε)) = g on Ω,

uε = 0 on ∂Ω,

where Ω is some open, bounded and Lipschitz subset of RN with N > 1, uε ∈ W 10 (Ω),

g ∈ L2(Ω), and some ellipticity conditions on the coefficients of A are assumed.

Nguetseng gives an alternative proof of the classical homogenization result previously ob-

tained by two-scale asymptotic expansion and energy methods (see Bensoussan, Lions and

Papanicolau [14] and Tartar [76]), by means of a detailed study of functionals of the form

∫

Ωfn(x)φ

(

x,x

εn

)

dx.

The key point of his argument was to prove that from each bounded sequence fnn in

L2(Ω) there exists a subsequence (still denoted by fnn) such that

∫

Ωfn(x)φ

(

x,x

εn

)

dx→∫

Ω

∫

Qf(x, y)φ(x, y) dy dx,

2. Preliminaries 54

and to derive a similar result for sequences of gradients. Allaire [3] called this two-scale

convergence and developed further properties of this notion as a tool to study more general

homogenization problems. Later this was extended to the notion of n-scale convergence by

Allaire and Briane (see [5] and Lukkassen, Nguetseng and Wall [61]) to study reiterated

homogenization problems of the type

−div(A(xε , ...xεN ;∇uε)) = g on Ω,

uε = 0 on ∂Ω.

Since then two-scale convergence is a well-known tool in the theory of homogenization and

has been generalized by many authors. As explained later, the great advantage of using

two-scale convergence techniques in our work is that it allows us to substantially weaken

the continuity hypothesis required in the current literature when studying homogenization

of integral functionals. The aim of this section is to present in a schematic way the main

properties of two-scale convergence.

Definition 2.6.1. (Periodic function) A function f : RN → R, with N > 1, is

i) Q- periodic if f(·) = f(· + lei) for all l ∈ Z, where e1, ..., eN is the canonical basis

of RN ;

ii) kQ- periodic (or k- periodic), with k ∈ N, if f(k · ) is Q-periodic.

As for notation, we denote by Cper(Q) the Banach space of all Q-periodic continuous func-

tions defined on RN with values in R endowed with the supremum norm, and by W 1,pper(kQ)

the W 1,p-closure of all kQ- periodic and C1-functions defined on RN with values in R

endowed with the W 1,p-norm.

Given Ω an open bounded subset of RN and 1 6 p < ∞, we denote by Lp(Ω;Cper(Q))

(resp. Lp(Ω;W 1,pper(kQ))) the space of all measurable functions f : Ω → Cper(Q) (f : Ω →

W 1,pper(kQ)) such that

∫

Ω||f(x)||pCper(Q) dx <∞,

(

resp.

∫

Ω||f(x)||p

W 1,pper (kQ)

dx <∞)

where

||f(x)||Cper(Q) := supy∈Q

|f(x, y)|

2. Preliminaries 55

(

resp. ||f(x)||pW 1,p

per (kQ)=

∫

kQ|f(x, y)|p dy +

∫

kQ|∇yf(x, y)|p dy <∞

)

.

Clearly a function f ∈ Lp(Ω;Cper(Q)) (resp. Lp(Ω;W 1,pper(kQ))) may be identified with the

function defined on Ω×RN via f(x, y) := f(x)(y) (∇yf denotes its derivative with respect

to the second argument y).

2.6.1 Generalized Riemann-Lebesgue Lemmas

We start by recalling some facts about periodic oscillating functions of the form fε(x) =

f(

x, xε)

, which play an essential role in homogenization theory (see Cioranescu and Donato

[32]). When f does not depend on the first variable, we have the following well known

result.

Lemma 2.6.2. (Riemann-Lebesgue Lemma) Let f ∈ Lploc(RN ) with 1 6 p 6 ∞, and assume

that f is kQ-periodic. For ε > 0 define fε(x) := f(xε ). Then fε f in Lploc(RN ) (weak⋆ if

p = ∞), where f = 1kN

∫

kQf(y) dy.

A generalized version of the Riemann-Lebesgue Lemma holds for functions in Lp(Ω;Cper(Q)).

Lemma 2.6.3. (see Lemma 5.2 in Allaire [4]; see also Bensoussan, Lions and Papanicolaou

[14] and Donato [43] ) Let f ∈ Lp(Ω;Cper(Q)) and let εnn be a fixed sequence of positive

real numbers converging to zero. Then, for every n ∈ N, the function f(·, ·εn

) is measurable

in Ω,∣

∣

∣

∣

∣

∣

∣

∣

f

(

·, ·εn

)∣

∣

∣

∣

∣

∣

∣

∣

Lp(Ω)

6 ||f ||Lp(Ω;Cper(Q)) :=

(∫

Ω||f(x)||pCper(Ω) dx

)1/p

and

limn

∫

Ω

∣

∣

∣

∣

f

(

x,x

εn

)∣

∣

∣

∣

p

dx =

∫

Ω

∫

Q|f(x, y)|p dxdy.

We finish this part with a useful characterization of functions in L1(Ω;Cper(Q)).

Lemma 2.6.4. (see Lemma 5.3 in Allaire [4] ) A function f belongs to L1(Ω;Cper(Q)) if

and only if there exists a subset E ⊂ Ω of measure zero such that

i) the function y → f(x, y) is continuous and Q- periodic for any x ∈ Ω \ E;

2. Preliminaries 56

ii) the function x→ f(x, y) is measurable for any y ∈ Q;

iii) x→ supy∈Q

|f(x, y)| has finite L1(Ω)-norm.

2.6.2 The notion of two-scale convergence and some properties

Let p and q be real numbers such that 1 < p < ∞ and 1p + 1

q = 1, and let εnn be a

sequence of positive numbers converging to zero.

Definition 2.6.5. A sequence of functions fnn in Lp(Ω) is said to two-scale converge to

a limit f ∈ Lp(Ω ×Q), and we will write fn2s f , if

∫

Ωfn(x)φ

(

x,x

εn

)

dx→∫

Ω

∫

Qf(x, y)φ(x, y) dy dx

for all φ ∈ Lq(Ω;Cper(Q)).

Examples. (see e.g. Lukkassen, Nguetseng and Wall [61])

i) If fnLp(Ω)−→ f , then fn

2s f.

ii) If fn2s f , then fn

∫

Qf(·, y) dy in Lp(Ω).

iii) If f ∈ Lp(Ω;Cper(Q)), then fn(·, ·) := f(

·, ·εn

) 2s f.

Lemma 2.6.6. (see e.g. Lukkassen, Nguetseng and Wall [61]) For each sequence fnnbounded in Lp(Ω) there exists a subsequence (still denoted by fnn) and f ∈ Lp(Ω × Q)

such that fn2s f.

For sequences weakly convergent in W 1,p(Ω) the following compactness result holds.

Theorem 2.6.7. (see Allaire [4] or Nguetseng [69] ) Assume that fnn weakly converges

to a function f in W 1,p(Ω). Then fn2s f , and there exist a subsequence (still denoted by

fnn ) and f1 ∈ Lp(Ω;W 1,pper(Q)) such that

∇fn 2s ∇f + ∇yf1.

We finally remark that analogous properties hold for the extended notion of n-scale conver-

gence and we refer to Allaire and Briane [5].

3. VARIATIONAL PROBLEMS IN PERIODIC

HOMOGENIZATION: PREVIOUS RESULTS

In this section we give a brief account of the developments on periodic homogenization of

integral functionals, along with several references, that motivated the present work.

3.1 Pure periodic (iterated) homogenization

We turn our attention to the asymptotic analysis of a family of functionals defined on

Lp(Ω; Rd), with 1 1, by

Iε(u) :=

∫

Ωf(

x,x

ε,x

ε2, u(x),∇u(x)

)


∞ otherwise,

(3.1)

for some open bounded set Ω ⊂ RN with N > 1. The results below provide conditions on

the integrands f under which the Γ-limit of Iεε can be obtained. We set Q := (0, 1)N .

Main problem: Study the Γ(Lp(Ω))- limε→0

Iε.


∫

Ω

f(x

ε,∇u

)

dx

Let d = 1, that is, assume that u is a scalar-valued function, and let f : RN × RN → R be

a measurable function Q-periodic with respect to the first variable, strictly convex and of

class C1 with respect to the second one, and satisfying

i) |ξ − ξ0|p 6 f(x, ξ) 6 C(1 + |ξ|p), for all x, ξ ∈ RN and some ξ0 ∈ RN ;

ii)∣

∣f(x, ξ)1p − f(x, ξ′)

1p∣

∣ 6 C|ξ − ξ′|, for all x, ξ, ξ′ ∈ RN ,

3. Variational problems in periodic homogenization: Previous results 58

for some positive constant C. Then Marcellini [63] showed that the Γ(Lp(Ω))- limit of Iεε,with respect to the strong topology of Lp(Ω) is given by

Ihom(u) =

∫

Ωfhom(∇u(x)) dx

for all u ∈W 1,p(Ω), where fhom is defined by

fhom(ξ) := infφ

∫

Qf(y, ξ + ∇φ(y)) dy, φ ∈W 1,p

per(Q)

. (3.2)

The function fhom is strictly convex and satisfies the same growth conditions as f . We note

that if Ω is assumed to be Lipschitz, then by Proposition 2.2.1, Remark 2.2.6 and the fact

that p > 1, the Γ-limit of the previous functionals for u ∈ W 1,p(Ω) would be the same if

the weak W 1,p-topology had been considered in place of the strong Lp-topology.

To illustrate the main idea in the convex case we give a sketch of Marcellini’s proof.

Step 1. Use a compactness argument due to C. Sbordone [73] to obtain converging (sub)sequences

Iεjj to an integral Γ-limit functional whose integrand fεj is convex.

Step 2. Prove that fεj is independent on the variable x.

Step 3. Observe that by Jensen’s inequality (see (2.8))

fεj(ξ) = inf

∫

Qfεj(ξ + ∇φ(y)) dy : φ ∈W 1,p

0 (Q)

(

resp. φ ∈W 1,p(Q),

∫

Q∇φ = 0

)

.

Step 4. Show that

lim supε→0

minφ

∫

Qf(x

ε, ξ + ∇φ(y)

)

dy : φ ∈W 1,p0 (Q)

6 fhom(ξ)

6 lim infε→0

minφ

∫

Qf(x

ε, ξ + ∇φ(y)

)

dy : φ ∈W 1,p(Q),

∫

Q∇φ = 0

.

Step 4. By Theorem 2.5.11

(

resp. for φ ∈W 1,p(Q),

∫

Q∇φ = 0

)

limj→∞

minφ

∫

Qf

(

x

εj, ξ + ∇φ(y)

)

dy : φ ∈W 1,p0 (Q)


= min

∫

Qfεj(ξ + ∇φ(y)) dy : φ ∈W 1,p

0 (Q)

.

Step 5. Conclude that fεj(ξ) = fhom(ξ).

We refer to Carbone-Sbordone [31] and Cioranescu, Damlamian and De Arcangelis [33] for

similar results in the convex case.

Muller shows in [66] that the homogenized formula (3.2) does not necessarily hold in the

nonconvex case when d > 1, that is, when u is assumed to be a vector-valued function; in

fact for nonconvex f and d > 1 it is necessary to consider variations which are periodic over

an infinite ensemble of cells, instead of considering variations which are periodic over just

the unit cell Q. Under the assumptions that f : RN × Rd×N → R, with d > 1, is a Borel

measurable function not necessarily convex and Q-periodic in the first variable, and

i) there exist C, β > 0 such that

β|ξ|p 6 f(x, ξ) 6 C(1 + |ξ|p); (3.3)

ii) there exists L > 0 such that the p-Lipschitz condition

|f(x, ξ) − f(x, ξ′)| 6 L(

1 + |ξ|p−1 + |ξ′|p−1)

|ξ − ξ′|

holds, Muller proved that

Γ(Lp(Ω))- limε→0

Iε(u) =

∫

Ωfhom(∇u(x)) dx, (3.4)

for all u ∈ W 1,p(Ω; Rd), where fhom satisfies the same growth conditions as f , and is given

by

fhom(ξ) := infT∈N

infφ

1

TN

∫

(0,T )N

f(y, ξ + ∇φ(y)) dy, φ ∈W 1,p0

(

(0, T )N ; Rd)

, (3.5)

or, equivalently,

fhom(ξ) = infT∈N

infφ

1

TN

∫

(0,T )N

f(y, ξ + ∇φ(y)) dy, φ ∈W 1,pper

(

(0, T )N ; Rd)

. (3.6)

The idea of the proof is to establish the lower bound for the Γ-limit for affine limit functions

u and to prove that this lower bound is achieved for an affine recovering sequence; the next

step is to use an approximation argument (for which Ω is required to be Lipschitz) to obtain

the same bounds for a general function u ∈W 1,p(Ω; Rd).

When f is convex, Muller (see Theorem 1.5 in [66]) recovered Marcellini’s result under

weakened growth conditions but assuming more regularity on the domain Ω, showing in

Section 4 of [66] that for convex integrands the expressions (3.5) or (3.6) are equivalent

to (3.2). Muller remarks that when d = 1 this equivalence holds independently of any

convexity assumption, because in this case

infT∈N

infφ

∫

(0,T )N

f(y, ξ + ∇φ(y)) dy, φ ∈W 1,pper

(

(0, T )N)

= infT∈N

infφ

∫

(0,T )N

Cf(y, ξ + ∇φ(y)) dy, φ ∈W 1,pper

(

(0, T )N)

where Cf denotes the convex envelope of f (see Definition 2.3.12). Finally, Muller showed

with the example below that, when d > 1 and f is nonconvex, expressions (3.5) or (3.6) are

not necessarily equivalent to (3.2).

Example. Let f0 : R2×2 → R be given by f0(ξ) := |ξ|4 + h(det ξ), where

h(t) :=

8(1+a)2

t+a − 8(1 + a) − 4 if t > 0,

8(1+a)2

a − 8(1 + a) − 4 − 8(1+a)2

a2 t if t 6 0,

for 0 < a < 12 . Let now R := (0, 1

2) × (0, 1), and define for x ∈ Q := (0, 1)2 and ξ ∈ R2×2

f(x, ξ) := χR(x) + δχQ\R(x)f0(ξ)

where δ is a small positive number. Extend f by Q-periodicity in the first variable and let

ξ = diag(1, c), π4 < c < 1. Muller showed that

infφ

∫

Qf(y, ξ + ∇φ(y)) dy, φ ∈W 1,p

per

(

Q; R2)

> C > 0

and

infT∈N

infφ

1

T 2

∫

(0,T )2f(y, ξ + ∇φ(y)) dy, φ ∈W 1,p

0

(

(0, T )2; R2)

6 Cδ,

for some positive constant C. Thus

infT∈N

infφ

1

T 2

∫

(0,T )2f(y, ξ + ∇φ(y)) dy, φ ∈W 1,p

0

(

(0, T )2; R2)

< infφ

∫

Qf(y, ξ + ∇φ(y)) dy, φ ∈W 1,p

per

(

Q; R2)

,

provided δ is sufficiently small. The function f can be interpreted as the energy density of

a composite material consisting of a strong and a very weak component (see Figure. 3.1),

where δ represents the strength of the weak material. The first block (unit cell) can support

compression while the second block (which shows T × T cells rescaled to the unit cell) can

achieve very low energetic states as T → ∞.

Fig. 3.1: Two different behaviors under compression

In an independent work Braides [16] also treated the vectorial and nonconvex case. Precisely,

using a compactness Γ-convergence result of Fusco [53], Braides proved that equality (3.4)

holds whenever f : RN × Rd×N → R is a Borel measurable function, almost periodic,1

quasiconvex and satisfying the standard growth and coercivity conditions (3.3). Braides

remarks that equality (3.5) is equivalent to

fhom(ξ) := limT→∞

infφ

1

TN

∫

(0,T )N

f(y, ξ + ∇φ(y)) dy, φ ∈W 1,p0 ((0, T )N )

, (3.7)

and that fhom is a quasiconvex function. By Corollary 2.5.14 this result implies that (3.4)

holds under the assumptions that f is a Borel measurable function Q-periodic in the first

variable and such that (3.3) holds (for a more direct proof see Braides and Defranceschi [19]

where the authors derived a compactness result for functionals with this class of integrands

and used the integral representation theorem of Buttazzo and Dal Maso [26]).

1 A function f : RN → R is said to be almost periodic if for every β > 0 there exists Lβ > 0 such that for

every a ∈ RN , there exists τ ∈ a + [0, Lβ ]N such that, for a.e. x ∈ RN , |f(x + τ) − f(x)| < ε. In particular,

if f is Q- periodic it is also almost periodic.



∫

Ω

f(

x,x

ε, u,∇u

)

dx

In [15] Braides studied functionals of the form

Iε(u) =

∫

Ωf(

x,x

ε, u(x),∇u(x)

)

dx,

for scalar-valued u under the assumptions that the integrand f : RN×RN×R×RN → [0,∞),

f = f(x, y, s, ξ), is convex in ξ, and that there exist b ∈ L1loc(R

N ) and a continuous positive

real function ω with ω(0) = 0 such that

|f(x, y, s, ξ) − f(x′, y, s′, ξ)| 6 ω(|x− x′| + |s− s′|)[

b(y) + f(x, y, s, ξ)]

(3.8)

for all x, x′, y, ξ ∈ RN and all s, s′ ∈ Rd. In addition, f is assumed to be measurable, Q-

periodic with respect to y, continuous with respect to the variables x and s, and to satisfy

the growth condition

0 6 f(x, y, s, ξ) 6 a(x)[

b(y) + |s|p + |ξ|p]

for all (x, y, s, ξ) ∈ RN ×RN ×R×RN , and for some continuous function a : RN → [0,∞).

Using a compactness and representation result by Buttazzo and Dal Maso (Theorem 4.4 in

[28]), Braides showed that


Iε(u) =

∫

Ωfhom(x, u(x),∇u(x)) dx,

for all u ∈W 1,p(Ω), where fhom is the convex function given by

fhom(x, s, ξ) := infφ

∫

Qf(x, y, s, ξ + ∇φ(y)) dy, φ ∈W 1,p

per(Q; Rd)

.

In a first step Braides proves the result for integrands of the form f = f(y, ξ) by an argument

similar to the one used by Marcellini. Then he uses this case to deduce the general one by

an argument that takes into account some properties derived in [28].

A sketch of the proof of an analogous result in the vectorial setting for f = f(x, y, ξ) can be

found in Exercise 14.6 of Braides and Defranceschi [19] (convex and nonconvex case) and

also in Theorem 1.3 of Braides and Lukkassen [21] (convex case).

Our goal in Theorem 4.1.1 is to prove a similar result under substantially weaker continuity

hypothesis than (3.8). This will be possible by combining Γ-convergence and two-scale


convergence arguments. We treat the nonconvex case assuming that the integrand f =

f(x, y, ξ) is continuous with respect to the pair (y, ξ), measurable in x, and Q-periodic

as a function of y. We note that the improvement in (3.8) is done at the expense of

requiring continuity in the variable y, as opposed to only measurability as in [15], [19] and

[21]. Recently we were able to prove another version of this result for integrands that are

continuous with respect to the first variable x and measurable with respect to the second

one y (see Chapter 6). This case turns out to be more relevant for the applications to

problems of mixtures. We still use a two-scale argument but the analysis is more delicate.


∫

Ω

f(

x,x

ε,

x

ε2,∇u

)

dx

In Theorem 1.1 of [21] Braides and Lukkassen (see also [60]) study the Γ-convergence of a

family of functionals of the type

Iε(u) =

∫

Ωf(x

ε,x

ε2,∇u(x)

)

dx,

where the integrand f : RN × RN × Rd×N → [0,∞) is periodic in the first two oscillating

variables and satisfies usual coercivity and growth conditions. In addition,

- f(y, · ; ξ) is measurable for all (y, ξ) ∈ RN × Rd×N ;

- f(y, z; ·) is convex for all (y, z) ∈ RN × RN ;

- there exist b ∈ L1loc(R

N ) and a continuous positive real function ω, with ω(0) = 0,

such that

|f(y, z, ξ) − f(y′, z, ξ)| 6 ω(|y − y′|)[

b(z) + f(y, z, ξ)]

(3.9)

for all y, y′, z ∈ RN , and all ξ ∈ Rd×N .

A compactness and integral representation theorem by Fusco [53], an analogous argument

to the one used by Marcellini, and the reiteration of the homogenization formula (3.2) are

used in the proof. They showed that


Iε(u) =

∫

Ωfhom(∇u(x)) dx

for all u ∈W 1,p(Ω; Rd), where fhom is defined by

fhom(ξ) := infφ

∫

Qfhom(y; ξ + ∇φ(y)) dy : φ ∈W 1,p

per(Q; Rd)

,


and

fhom(y; ξ) := infφ

∫

Qf(y, z; ξ + ∇φ(z)) dz : φ ∈W 1,p

per(Q; Rd)

The analysis has been extended to the case of nonconvex integrands (see Theorem 22.1 in

Braides and Defranceschi [19]) and to the case where f depends explicitly on the macro-

scopic variable x, as in (3.1) (see Remark 22.8 of Braides and Defranceschi [19]), under the

strong uniform continuity condition (3.9) or (1.6), respectively. Using techniques of mul-

tiscale convergence and restricting the argument to the convex and homogeneous case (no

dependence on the variable x), Fonseca and Zappale were able to recover these results with

weaker continuity conditions than (3.9). Namely, they only required f to be continuous

(see Theorem 1.9 in [51]).

As these results seem to show, it is not clear what is the natural regularity on f for the

integral (3.1) to be well defined. Motivated by Theorem 4.1.1 we treat in Theorem 4.2.1

the case where the integrand f satisfies the following conditions:

- f(x, · , · , · ) is continuous for a.e. x ∈ Ω;

- f( · , y, z, ξ) is measurable for all (y, z, ξ) ∈ RN × RN × Rd×N ;

- f(x, · , z, ξ) is Q-periodic for all (z, ξ) ∈ RN ×Rd×N and for a.e. x ∈ Ω; f(x, y, · , ξ) is

Q-periodic for all (y, ξ) ∈ RN × Rd×N and for a.e. x ∈ Ω;

- there exists β > 0 such that for all (y, z, ξ) ∈ RN × RN × Rd×N and for a.e. x ∈ Ω

1

β|ξ|p − β 6 f(x, y, z, ξ) 6 β(1 + |ξ|p).

We recover Theorem 1.9 in Fonseca and Zappale [51].

We point out that our analysis follows the lines of the one in Braides and Defranceschi [19]

(Theorem 22.1 and Remark 22.8), and that our main contribution is to use arguments that

allow us to weaken the strong uniform continuity hypothesis (1.6).

3.2 Thin films with periodic microstructure in the

nonlinear membrane theory

Reduction dimension arguments are used variationally to study minimization problems over

domains whose dimension, in one or more directions, is small compared with the dimension

in the other direction. Membranes are 3-dimensional continuum bodies with a reference

configuration with cylindrical shape, such that the height of the cylinder - the thickness -

is small in comparison with the other dimensions. This feature suggests the possibility of

deriving 2-dimensional models in membrane theory from the full 3-dimensional theory. The

idea is to regard the thickness of these thin cylindrical bodies as a small parameter ε and

then to study the asymptotic behavior as ε goes to zero.

Starting from the works of Acerbi, Buttazzo and Percivale [2], Γ-convergence has become

an important tool to do this asymptotic analysis in nonlinear elasticity. We briefly discuss

the main approach to this study.

Let ω be an open and bounded subset of R2. For each 0 < ε ≪ 1 define Ωε := ω × (−ε, ε)and denote Σε := ω × −ε, ε (Figure. 3.2).

εΣ

εΣ

εε

Ω ε

ω

−

+

Fig. 3.2: Cylindrical thin domain of thickness ε

We assume that the body is pinned on the lateral boundary, that is v(x) = x on ∂ω×(−ε, ε),for all its admissible deformations, and that it is subjected to the action of regular surface

traction densities g(ε) on Σε, and regular dead loads f(ε). The total energy of this body

under the action of this forces is the difference between the elastic energy and the work of

external forces. More precisely,

W(ε)(v) :=

∫

Ωε

W (ε)(x,Dv) dx−∫

Ωε

f(ε) · v dx−∫

Σε

g(ε) · v dS,

for v ∈ V(ε) := v ∈W 1,p(Ωε; R3) : v(x) = x on ∂ω × (−ε, ε).

Main problem: To study

limε→0

minv∈V(ε)

W(ε)(v)

by means of Γ-convergence. As usual, in order to study this problem as ε → 0 we rescale

the ε-thin body into a reference domain of unit thickness (see e.g. Acerbi, Buttazzo and

Percivale [2], Anzellotti, Baldo and Percivale [8], Le Dret and Raoult [55], Braides, Fonseca

and Francfort [20]), so that the resulting energy will be defined on a fixed body, while the


dependence on ε turns out to be explicit in the transverse derivative. For this, we consider

the change of variables

Ωε → Ω := ω × I, (x1, x2, x3) 7→(

x1, x2,1

εx3

)

,

−

−

+

+

εε

Ωε

3

α

α =1,2

x

x

ω1

1

ΩΣ

Σ

Σ

Σε

ε

ω

Fig. 3.3: Rescaled domain of unit thickness

and define u(xα, x3/ε) = v(xα, x3) on the rescaled cylinder Ω, where I := (−1, 1) and

xα := (x1, x2) is the in-plane variable. We denote Σ := ω × −1, 1 (Figure. 3.3).

It is well known that membrane theory arises at the order ε of a formal asymptotic expansion

(see Fox, Raoult and Simo [52]), provided that the body forces are of order 1 and the surface

loadings are of order ε. Since this energy is of order ε, we divide the total energy by ε and

in addition we assume that

f(ε)(xα, εx3) = f(xα, x3),

g(ε)(xα, εx3) = ε g(xα, x3),

where f ∈ Lp′(Ω; R3), g ∈ Lp

′(Σ; R3) (1/p + 1/p′ = 1). If Wε(xα, x3; ·) = W (ε)(xα, εx3; ·),

for fixed ε minimizing W(ε) on V(ε) is equivalent to minimizing

Wε(u) :=W(ε)(v)

ε=

∫

ΩWε

(

x,∇αu(x)∣

∣

∣

1

ε∇3u(x)

)

dx−∫

Ωf · u dx−

∫

Σg · u dS

on Vε := u ∈ W 1,p(Ω; R3) : u(x) = (xα, εx3) on ∂ω × I. We recall that for ξ ∈ R3×2 and

z ∈ R3, (ξ|z) denote the matrix whose first two columns are those of ξ and the last one is

z; ∇α = (∇1,∇2).

Main step: To study

Γ(Lp(Ω))-limε→0

Wε.

∫

Ω

W

(

x,xα

ε,∇αu

∣

∣

∣

1

ε∇3u

)

dx

The motivation to study this case comes from the works of Braides, Fonseca and Francfort

[20], of Babadjian and Francfort [11] and from Theorem 4.1.1.

Starting point: In [20] Braides, Fonseca and Francfort derived a homogenization result for

energiesWε(x; ξ) = W (x3, xα/ε, ξ). Namely, under the hypotheses thatW is a Caratheodory

function satisfying standard coerciveness and growth conditions, they proved that if

Wε(u) =

∫

ΩW

(

x3,xαε,∇αu

∣

∣

∣

1

ε∇3u

)

dx

then


Wε(u) = 2

∫

ωWhom(∇αu) dxα

for all u ∈W 1,p(ω; R3) ≡

u ∈W 1,p(Ω; R3) : D3u(x) = 0 for a.e.x ∈ Ω

, where

Whom(ξ) := limT→+∞

infφ

1

2T 2

∫

(0,T )2×IW(

y3, yα, ξ + ∇αφ(y)|∇3φ(y))

dy

φ ∈W 1,p((0, T )2 × I; R3), φ = 0 on ∂(0, T )2 × I

. (3.10)

Their analysis is based on a compactness property for a family of energies of the form

Wε(x, ξ). We state here this result since it is important for our applications in Chapter 5.2

Theorem 3.2.1. (Theorem 2.5 in [20]) Let ω be an open bounded subset of R2 and let

Ω := ω × I. Let Wε : Ω × R3×3 → [0,∞) be a family of Caratheodory functions such that

for a.e. x ∈ Ω and any ξ ∈ R3×3

β|ξ|p 6 Wε(x, ξ) 6 C(1 + |ξ|p), 0 < β 6 C, 1 0 define Wε : Lp(Ω; R3) ×A(ω) → [0,∞) by

Wε(u;A) :=

∫

A×IWε

(

x,Dαu(x)∣

∣

∣

1

εD3u(x)

)

dx if u ∈W 1,p(A× I; R3),

∞ otherwise.

2 In [20] the authors prove a more general result for thin films with varying profiles. For our purposes it

is enough to present this simpler case.


Let

Wε(u;A) := infuε

lim infε→0

Wε(uε;A) : uε → u in Lp(A× I; R3)

.

Then every sequence εn ↓ 0 admits a subsequence εnjj ≡ εjj such that Wεj( · ;A) is the

Γ(Lp(A × I))-limit of Wεj( · ;A)j for all A ∈ A(ω). Further there exists a Caratheodory

function Wεj : ω × R3×2 → R such that

Wεj(u;A) = 2

∫

AWεj(xα, Dαu(xα)) dxα, (3.11)

for all A ∈ A(ω) and all u ∈W 1,p(A; R3).

The proof of Theorem 3.2.1 is based on the Direct Method of Γ-convergence. In a first step

the authors derive a useful version of Proposition 2.5.13 that will allow us the matching of

recovering sequences in the lateral boundary of open sets A × I, with A ⊂ ω. Its proof is

based on De Giorgi’s slicing argument and the possibility of considering cut-off functions

independent on the transverse direction of the thin film.

Lemma 3.2.2. (Lemma 2.6 in [20]) Under the hypotheses of Theorem 3.2.1 on Wε, let A ∈A(ω), u ∈W 1,p(A; R3), and let εj ↓ 0 be a sequence for which Γ(Lp(A× I))- lim

j→∞Wεj

(u;A)

exists. Then there exists wjj ⊂ W 1,p(A × I; R3) and a sequence of compact sets of A,

Kjj, such that

Γ(Lp(A× I))- limj→∞

Wεj(u;A) = lim

j→∞Wεj

(wj , A)

and wj = u in (A \Kj) × I.

In Chapter 5 we study the asymptotic behavior of a heterogeneous ε-thin film whose mi-

crostructure oscillates on a scale that is comparable to that of the thickness of the domain

(see Figure 3.4). We propose in Theorem 5.1.1 to establish a dimensional reduction and ho-

mogenization result, where both scales are identical, by adding in the stored energy density

an explicit dependence on the in-plane variable xα. Namely, we assume that

Wε(x; ξ) = W(

x,xαε, ξ)

.

We seek to find the Γ(Lp(Ω))-limit of the following family of energies

Wε(u) :=

∫

ΩW

(

x,xαε,∇αu(x)

∣

∣

∣

1

ε∇3u(x)

)

dx


ε

ε

Fig. 3.4: Thin domain with periodic structure in the in-plane direction

for u ∈ W 1,p(Ω; R3), and some function W : Ω × R2 × R3×3 → R whose hypotheses will

be introduced later(

in this case in the reference configuration Ωε we have W(ε)(x, ξ) =

W (xα,xε , ξ)

)

.

Two features differentiate our approach from what is available in most of the literature in

the subject (see Shu [75] and Braides, Fonseca and Francfort [20]). The first one is the

use of two-scale techniques as in Theorem 4.1.1. The second one is based on a decoupling

argument used by Babadjian and Francfort [11] to derive a nonlinear membrane model for

stored energy densities of the form W (x, ξ) generalizing the case where Wε(x, ξ) ≡W (x3, ξ)

that was studied in Braides, Fonseca and Francfort [20]. This decoupling procedure is

necessary to take into account the different nature of the two variables yα and xα that

appear in the structure of the limit functional (see (5.24) below).


∫

Ω

W

(

x,x

ε,xα

ε2,∇αu

∣

∣

∣

1

ε∇3u

)

dx

In Theorem 5.2.1 we want to study the asymptotic analysis of ε-thin elastic bodies whose

microstructure is periodic of period ε in the in-plane direction and periodic of period ε2 in

all directions. To take into account these heterogeneities, our goal is to study the sequence

of energies

Wε(x; ξ) = W(

x,x

ε,xαε2, ξ)

(

in this case in the reference configuration Ωε we have W(ε)(x, ξ) = W (xα,x3ε ,

xα

ε ,xε2, ξ))

.

We seek to find the Γ(Lp(Ω))-limit of the following family of energies

Wε(u) :=

∫

ΩW

(

x,x

ε,xαε2,∇αu(x)

∣

∣

∣

1

ε∇3u(x)

)

dx,

for u ∈W 1,p(Ω; R3), and for some function W : Ω×R3 ×R2 ×R3×3 → R whose hypotheses

will be introduced later. As a corollary, we derive a homogenization result for heterogeneous


ε-thin films of periodic structure of period ε in the in-plane variable xα and of period ε2 in

the transverse direction x3. That is, we derive an homogenization formula for a family of

energies

Wε(u) :=

∫

ΩW

(

x,x

ε,∇αu(x)

∣

∣

∣

1

ε∇3u(x)

)

dx

for u ∈W 1,p(Ω; R3). Integral functionals of the form

∫

ΩW

(

xαε2,∇αv

∣

∣

∣

1

ε∇3v

)

dx,

have been studied in Shu [75] (Theorem 5) under different length scales for the film thickness

and the material microstructure. As far as we know there have no been previous results

that allow for homogenization in the transverse direction of the film.

Part II

MAIN RESULTS

4. Γ-CONVERGENCE OF FUNCTIONALS WITH

PERIODIC INTEGRANDS

The main goal of this chapter is to characterize the asymptotic behavior of a family of

multiple scale integral functionals whose integrands have periodicity properties.

From now on, unless otherwise specified, C will denote a generic constant, and for every

a ∈ RN and δ > 0 we write Q(a, δ) := a + δQ ≡ (a, δ)N , where Q := (0, 1)N . Throughout

this chapter Ω stands for an open bounded set in RN with N > 1.

4.1 An approach by 2-scale convergence

This section is devoted to proving the following result.

Theorem 4.1.1. Let f : Ω × RN × Rd×N → R be a function such that

(H1) f(x, ·, ·) is continuous a.e. x ∈ Ω;

(H2) f(·, y, ξ) is LN -measurable for all y ∈ RN and all ξ ∈ Rd×N ;

(H3) f(x, ·, ξ) is Q-periodic for a.e. x ∈ Ω and for all ξ ∈ Rd×N ;

(H4) there exist a real number p > 1 and a constant β > 0 such that

|ξ|pβ

− β 6 f(x, y, ξ) 6 β(1 + |ξ|p),

for a.e. x ∈ Ω, for all y ∈ RN and all ξ ∈ Rd×N .

For each ε > 0 define the functional Iε : Lp(Ω; Rd) → [0,∞] by

4. Γ-convergence of functionals with periodic integrands 74

Iε(u) :=

∫

Ωf(

x,x

ε,∇u(x)

)


∞ otherwise.

(4.1)

If u ∈ Lp(Ω; Rd) then

Ihom(u) := Γ(Lp(Ω))- limε→0

Iε(u) =

∫


∞ otherwise,

(4.2)

where the integrand fhom is given by


infφ

1

TN

∫

(0,T )N

f(x, y, ξ + ∇φ(y)) dy, φ ∈W 1,p0 ((0, T )N ; Rd)

(4.3)

for a.e. x ∈ Ω and for all ξ ∈ Rd×N . It turns out that fhom is (equivalent to) a Caratheodory

function and satisfies p-coercivity and p-growth conditions similar to those of f . Moreover

fhom(x, · ) is quasiconvex for a.e. x ∈ Ω.

Theorem 4.1.1 was obtained in collaboration with I. Fonseca [12], and the main idea of its

proof is to use the Direct Method of the Calculus of Variations (see Section 2.5.2) combined

with the integral representation theorem of Buttazzo and Dal Maso (Theorem 2.4.1) to

derive the existence of Γ-converging (sub)sequences to an abstract integral functional. Then

the idea is to use arguments of two-scale convergence to derive an upper bound for the

integrand of this functional. To get the other bound we use the fact that, under hypotheses

(H1)-(H4), the integrand f is “uniformly continuous up to a small error”. Indeed, since

f is a Caratheodory integrand, Scorza-Dragoni’s Theorem (Theorem 2.3.26) implies that

the restriction of f to K × RN × Rd×N is continuous, for some compact set K ⊂ Ω whose

complement has arbitrarily small Lebesgue measure. Then the periodicity of f with respect

to its second variable leads f to be uniformly continuous on K × RN × B, for some closed

ball B of Rd×N of sufficiently large radius. Finally, to ensure that the energy remains

arbitrarily small on the complement of K and on the set of x’s such that the gradient of

the deformation does not belong to B, we use the Decomposition Lemma (Theorem 2.2.16)

which allows us to select minimizing sequences with pth-equi-integrable gradients. Thus, in

view of the p-growth character of the integrand, the energy over sets of arbitrarily small

Lebesgue measure tends to zero.

Like for quasiconvex envelopes, there are very few explicit examples of homogenized densities

in the literature. A classical explicit derivation of the function fhom for elliptic operators in

the homogeneous case, that is, for integrands f that do not depend on the variable x, can

be found in De Giorgi and Spagnolo [41]. We present a classical example that can be found

in the book of Dal Maso [35] (see references therein for more examples).

Example. Let N = 1 and let f(y, ξ) := a(y)|ξ|p for (y, ξ) ∈ R2 and 1 < p <∞, where a is a

measurable and Q-periodic function, such that for all y ∈ R

β 6 a(y) 6 C, 0 < β 6 C.

Then fhom(ξ) = ahom|ξ|p where

ahom :=

(

∫ 1

0

(

1

a

)p/p−1)1−p

.

Remark 4.1.2. By hypothesis (H1) and (H2) the integrand f is of Caratheodory-type and

this ensures that f(x, ·, ·) is a Borel function for a.e. x ∈ Ω (see Proposition 2.3.27; in

particular the integral in (4.1) is well defined). Moreover, by hypothesis (H4) replacing f by

f+β we may assume that f is nonnegative almost everywhere. As a consequence of these two

remarks, in the sequel, without loss of generality, we may assume that f is a positive Borel

function such that hypotheses (H1), (H3) and (H4) hold for every (x, y, ξ) ∈ Ω×RN×Rd×N .

Two more remarks are worthy of note (see Muller [66] and Braides and Defranceschi [19];

see also Lemma 4.1.5 and Lemma 4.1.10 below). First, it can be seen that for a.e. x ∈ Ω

and for all ξ ∈ Rd×N

fhom(x, ξ) = infT∈N

infφ

1

TN

∫

(0,T )N


(4.4)

and


infφ

1

TN

∫

(0,T )N

f(x, y, ξ + ∇φ(y)) dy, φ ∈W 1,pper((0, T )N ; Rd)

, (4.5)

Secondly, we observe that under the additional hypothesis that f(x, y, ·) is convex for a.e.

x and for all y (in which case (H1) is equivalent to requiring that f(x, ·, ξ) is continuous for

a.e. x and for all ξ), equalities (4.4) and (4.5) simplify to read, respectively,

fhom(x, ξ) = infφ

∫

Qf(x, y, ξ + ∇φ(y)) dy, φ ∈W 1,p

0 ((0, 1)N ; Rd)

(4.6)


and

fhom(x, ξ) = infφ

∫

Qf(x, y, ξ + ∇φ(y)) dy, φ ∈W 1,p

per((0, 1)N ; Rd)

. (4.7)

Moreover, by hypothesis (H1) and by Lemma 2.3.28, equalities (4.3)-(4.7) hold if the ad-

missible test functions are taken in any smooth dense subset of W 1,p0 ((0, T )N ; Rd) and

W 1,pper((0, T )N ; Rd), respectively.

As a consequence of Theorem 4.1.1, Proposition 2.5.13 and Theorem 2.5.11 we get the

convergence of (almost) minimizers of Iεε.

Corollary 4.1.3. Under hypotheses (H1)-(H4) the functional Ihom defined in (4.2) has a

minimum on Vϕ := u ∈W 1,p(Ω) : u− ϕ ∈W 1,p0 (Ω), ϕ ∈W 1,p(Ω) and

minu∈Vϕ

Ihom(u) = limε→0

infu∈Vϕ

Iε(u).

Moreover, given two sequences εn ↓ 0 and unn ⊂ Vϕ) such that

limn→∞

Iεn(un) = limn→∞

infu∈Vϕ

Iεn(u),

then (up to subsequence) unn is W 1,p-weakly convergent to a minimum for the functional

Ihom on Vϕ.

4.1.1 Properties of the homogenized density

In this section we turn our attention to the main properties of the function fhom for later use

in the proof of Theorem 4.1.1. Most of these properties can be deduced from previous works

(see references below), however we present alternative proofs for the sake of completeness.

By Remark 4.1.2 we restrict our analysis to the case where f is a positive Borel function

such that hypotheses (H1), (H3) and (H4) hold for every (x, y, ξ) ∈ Ω × RN × Rd×N .

We start by showing that the limit in (4.3) is well defined. The argument is analogous

to that used in Bouchitte, Fonseca and Mascarenhas [23] and relies on Lemma A.1 in the

Appendix.

Lemma 4.1.4. For all (x, ξ) ∈ Ω × Rd×N there exists

limT→∞

infφ

1

TN

∫

(0,T )N


. (4.8)


Proof. (see also Braides and Defranceschi [19]) Let (x, ξ) ∈ Ω × Rd×N and let

S(A) := infφ

∫

Af(x, y, ξ + ∇φ(y)) dy : φ ∈W 1,p

0 (A; Rd)

for A ∈ A(RN ). Under the above assumptions on f the function S : A(RN ) → [0,∞) is

well defined, and it satisfies the hypotheses of Lemma A.1 with T = ZN and M = 1. Hence

we conclude that the limit

limT→∞

S((0, T )N )

TN,

or equivalently (4.8), exists.

We now want to show that fhom is a Caratheodory integrand (see Definition 2.3.25) so that

the functional Ihom in (4.2) is well defined. We start by noting the following result.

Lemma 4.1.5. Let fhom, fhom : Ω × Rd×N → [0,∞) be defined, respectively, by

fhom(x, ξ) := infT∈N

infφ

1

TN

∫

(0,T )N


and

fhom(x, ξ) := infT∈N

infφ

1

TN

∫

(0,T )N

f(x, y, ξ + ∇φ(y)) dy, φ ∈W 1,pper((0, T )N ; Rd)

,

for all (x, ξ) ∈ Ω × Rd×N. Then the relations fhom = fhom = fhom hold.

Proof. Let (x, ξ) ∈ Ω × Rd×N . We first show that fhom(x, ξ) = fhom(x, ξ). It is clear

that fhom(x, ξ) > fhom(x, ξ). To prove the other inequality, fixed δ > 0 let S ∈ N and

ϕ ∈W 1,p0 ((0, S)N ; Rd) be such that

fhom(x, ξ) + δ >1

SN

∫

(0,S)N

f(x, y, ξ + ∇ϕ(y)) dy. (4.9)

Extend ϕ periodically to RN with period S. Using Riemann-Lebesgue’s Lemma (Lemma

2.6.2)

1

SN

∫

(0,S)N

f(x, y, ξ + ∇ϕ(y)) dy = limε→0

1

SN

∫

(0,S)N

f(

x,y

ε, ξ + ∇ϕ

(y

ε

))

dy

= limε→0

εN

SN

∫

(

0,Sε

)Nf(x, z, ξ + ∇θε(z)) dz, (4.10)


where θε(z) := 1εϕ(εz) ∈W 1,p

0

(

(

0, Sε)N

; Rd)

. Therefore from (4.9) and (4.10)

fhom(x, ξ) + δ > limε→0

infθ

εN

SN

∫

(0,Sε)N

f(x, z, ξ + ∇θ(z)) dz, θ ∈W 1,p0

((

0,S

ε

)N; Rd

)

= fhom(x, ξ).

Letting δ → 0 we conclude that

fhom(x, ξ) > fhom(x, ξ).

Finally we show that fhom(x, ξ) = fhom(x, ξ) (see also Braides [19] and Muller [66] for an

alternative justification). It is clear that fhom(x, ξ) > fhom(x, ξ). To verify the opposite

inequality, fix δ > 0 and take S ∈ N and a smooth function ϕ ∈ W 1,pper

(

(0, S)N ; Rd)

such

that

fhom(x, ξ) + δ >1

SN

∫

(0,S)N

f(x, y, ξ + ∇ϕ(y)) dy. (4.11)

By hypothesis (H3) the function f(x, ·, ξ + ∇ϕ(·)) is (0, S)N -periodic, and thus

1

SN

∫

(0,S)N

f(x, y, ξ + ∇ϕ(y)) dy = limε→0

∫

Qf(

x,y

ε, ξ + ∇ϕ

(y

ε

))

dy

= limε→0

∫

Qf(

x,y

ε, ξ + ∇ψε(y)

)

dy, (4.12)

where ψε(y) := εϕ(

yε

)

. For each ε > 0 define

Qε :=

y ∈ Q : dist(y, ∂Q) > ε

.

Take θε ∈ C∞c (Q) with θε ∈ [0, 1] such that θε = 1 on Qε and ||∇θε||L∞ 6 Cε−1. Then

limε→0

∫

Qf(

x,y

ε, ξ + ∇(θεψε)(y)

)

dy = limε→0

∫

Qε

f(

x,y

ε, ξ + ∇ψε(y)

)

dy, (4.13)

since by the p-growth condition in (H4) we have∫

Q\Qε

f(

x,y


)

dy

6 C

∫

Q\Qε

(

1 + |ξ|p + |∇ψε(y)|p + ε−p|ψε(y)|p)

dy

= C[

|Q \Qε| +∫

Q\Qε

∣

∣

∣∇ϕ(y

ε

)∣

∣

∣

pdy +

∫

Q\Qε

∣

∣

∣ϕ(y

ε

)∣

∣

∣

pdy]

→ 0.


Hence by (4.11)-(4.13), defining φε(y) := 1ε (θεψε)(εy) ∈W 1,p

0

((

0, 1ε

)N; RN

)

, we obtain


∫

Qf(

x,y


)

dy

> limε→0

εN∫

(

0, 1ε

)Nf(

x, y, ξ + ∇φε(y))

dy

> fhom(x, ξ).

Letting δ → 0 we get fhom(x, ξ) > fhom(x, ξ).

The measurability of fhom follows as a consequence.

Lemma 4.1.6. The function fhom(·, ξ) is measurable for all ξ ∈ Rd×N .

Proof. Let ξ ∈ Rd×N . By Lemma 4.1.5 we can write


infφ∈ST

fT, φ(x)

where

fT, φ(x) :=

∫

Tf(x, y, ξ + ∇φ(z)) dz

for x ∈ Ω, and ST is a countable subset of C∞c ((0, T )N ; Rd) dense in W 1,p

0 ((0, T )N ; Rd). By

Tonelli’s Theorem the functions fT, φ are measurable, and so is fhom(·, ξ) as the infimum of

a countable family of measurable functions.

Our next objective is to show that fhom(x, ·) is continuous for all x ∈ Ω. We are not able

to prove this directly unless f satisfy a p-Lipschitz condition as in (2.9). As quasiconvex

functions satisfy inequality (2.9) (see Proposition 2.3.15), the first step will be to show that

fhom = (Qf)hom where Qf : Ω × RN × Rd×N → R denotes the usual quasiconvexification

of f with respect to the last variable ξ, that we know to be quasiconvex in ξ (see Theorem

2.3.19). We remark that

Qf(x, y, ξ) = infφ

∫

Qf(x, y, ξ + ∇φ(z)) dz : φ ∈W 1,p

0 (Q; Rd)

(4.14)

for all (x, y, ξ) ∈ Ω × RN × Rd×N (see Lemma 2.3.23) and that consequently Qf satisfies

conditions (H3) and (H4). The following properties of Qf are of interest for the argument

that follows.


Lemma 4.1.7. We have that

i) Qf(x, ·, ·) is continuous for all x ∈ Ω;

ii) Qf(·, y, ξ) is measurable for all (y, ξ) ∈ RN × Rd×N ;

iii) (Qf)hom(x, ξ) := lim infT→∞

inf

1TN

∫

(0,T )N

Qf(x, y, ξ + ∇φ(y)) dy, φ ∈W 1,p0 ((0, T )N ; Rd)

= limT→∞

inf

1TN

∫

(0,T )N

Qf(x, y, ξ + ∇φ(y)) dy, φ ∈W 1,p0 ((0, T )N ; Rd)

for all (x, ξ) ∈ Ω × RN × Rd×N .

iv) (Qf)hom(x, ξ) = fhom(x, ξ) for all (x, ξ) ∈ Ω × Rd×N .

Proof. i) The upper semicontinuity of Qf(x, ·, ·) for x ∈ RN follows from equality (4.14) and

hypothesis (H1), while its lower semicontinuity can be obtained by an argument analogous

to that of Lemma 2.3.24.

ii) The proof is identical to that of Lemma 4.1.6 above.

iii) Is a consequence of identities i) and ii), the coerciveness, growth and periodicity prop-

erties of Qf , and of Lemma 4.1.4.

iv) Let (x, ξ) ∈ Ω×Rd×N . Obviously fhom(x, ξ) > (Qf)hom(x, ξ). Let us prove the converse

inequality. Let n ∈ N and let Tn ∈ N and φn ∈W 1,p0 ((0, Tn)

N ; Rd) be such that

(Qf)hom(x, ξ) +1

n>

1

TNn

∫

(0,Tn)N

Qf(x, y; ξ + ∇φn(y)) dy.

Thus

(Qf)hom(x, ξ) > lim supn→∞

1

TNn

∫

(0,Tn)N

Qf(x, y; ξ + ∇φn(y)) dy. (4.15)

To compare (4.15) with fhom(x, ξ) we apply the Acerbi and Fusco Relaxation Theorem

(Theorem 2.3.30) and the Decomposition Lemma (Lemma 2.2.16). As a consequence of

Theorem 2.3.30, for every n fixed there exists a sequence φn,kk ⊂W 1,p((0, Tn)N ; Rd) such

that φn,k kφn in W 1,p((0, Tn)

N ; Rd) and

1

TNn

∫

(0,Tn)N

Qf(x, y; ξ + ∇φn(y)) dy = limk→∞

1

TNn

∫

(0,Tn)N

f(x, y; ξ + ∇φn,k(y)) dy. (4.16)


By Lemma 2.2.16 we can now find a subsequence (still denoted by φn,kk) and a sequence

ψn,kk ⊂W 1,∞0 (RN ; Rd) such that ψn,k φn in W 1,p((0, Tn)

N ; Rd) with

|∇ψn,k|p equi-integrable (4.17)

and

LNy ∈ (0, Tn)N : ψn,k(y) 6= φn,k(y) −→

k→∞0. (4.18)

As f is nonnegative, by (4.17) and (4.18)

limk→∞

1

TNn

∫

(0,Tn)N

f(x, y; ξ + ∇φn,k(y)) dy

> lim supk→∞

1

TNn

∫

y∈(0,Tn)N : ψn,k(y)=φn,k(y)f(x, y; ξ + ∇ψn,k(y)) dy

= lim supk→∞

1

TNn

∫

(0,Tn)N

f(x, y; ξ + ∇ψn,k(y)) dy. (4.19)

Thus from (4.15), (4.16) and (4.19)

(Qf)hom(x, ξ) > lim supn→∞

lim supk→∞

1

TNn

∫

(0,Tn)N

f(x, y, ξ + ∇ψn,k(y)) dy > fhom(x, ξ).

We are now in a position to prove the continuity property of fhom.

Lemma 4.1.8. The function fhom(x, ·) (or equivalently (Qf)hom(x, ·)) is continuous for all

x ∈ Ω.

Proof. (see also Braides [19]) Fix x ∈ Ω. Let ξ ∈ Rd×N and ξn → ξ in Rd×N . We wish to

show that

fhom(x, ξ) = limn→∞

fhom(x, ξn).

We first establish that (upper semicontinuity)

fhom(x, ξ) > lim supn→∞

fhom(x, ξn). (4.20)

Fixed δ > 0, choose S ∈ N and by density a function ϕ ∈ C∞0 ((0, S)N ; Rd) such that

fhom(x, ξ) + δ >1

SN

∫

(0,S)N

f(x, y, ξ + ∇ϕ(y)) dy

=1

SNlimn→∞

∫

(0,S)N

f(x, y, ξn + ∇ϕ(y)) dy

> lim supn→∞

fhom(x, ξn),


as a consequence of Lemmas 2.1.15 and 4.1.5. Letting δ → 0 we get (4.20).

We show now the converse inequality (lower semicontinuity), i.e.

fhom(x, ξ) 6 lim infn→∞

fhom(x, ξn). (4.21)

We start by remarking that fhom(x, ξn) = (Qf)hom(x, ξn) for all n ∈ N (property iv) in

Lemma 4.1.7). Given n ∈ N, consider Tn ∈ N (Tn ր ∞) and φn ∈ W 1,p0 ((0, Tn)

N ; Rd) such

that

fhom(x, ξn) +1

n>

1

TnN

∫

(0,Tn)N

Qf(x, y, ξn + ∇φn(y)) dy

=

∫

(0,1)N

Qf(x, Tny, ξn + ∇φn(Tny)) dy

=

∫

(0,1)N

Qf(x, Tny, ξn + ∇ψn(y)) dy, (4.22)

where ψn(y) := 1Tnφn(Tny), ψn ∈ W 1,p

0 ((0, 1)N ; Rd). We note that by the p-coervivity con-

dition of (Qf) the sequence ||∇ψn||Lp((0,1)N ;Rd) is bounded. We write∫

(0,1)N

Qf(x, Tny, ξn + ∇ψn(y)) dy

=

∫

(0,1)N

Qf(x, Tny, ξn + ∇ψn(y)) −Qf(x, Tny, ξ + ∇ψn(y)) dy (4.23)

+

∫

(0,1)N

Qf(x, Tny, ξ + ∇ψn(y)) dy.

Our task now is to show that the term (4.23) goes to zero as n goes to infinity. As ξn → ξ

and the sequence ||∇ψn||Lp((0,1)N ;Rd) is bounded, using the p-Lipschitz condition (2.9) and

Holder Inequality we have

lim supn→∞

∫

(0,1)N

|Qf(x, Tny, ξn + ∇ψn(y)) −Qf(x, Tny, ξ + ∇ψn(y)) | dy

6 C lim supn→∞

∫

(0,1)N

(

1 + |ξn + ∇ψn(y)|p−1 + |ξ + ∇ψn(y)|p−1)

|ξn − ξ| dy

6 C lim supn→∞

∫

(0,1)N

(

1 + |∇ψn(y)|p−1)

|ξn − ξ| dy

6 C limn→∞

|ξn − ξ| = 0,

which, together with (4.22), leads to

fhom(x, ξn) +1

n> lim sup

n→∞

∫

(0,1)N

Qf(x, Tny, ξ + ∇ψn(y)) dy > (Qf)hom(x, ξ) = fhom(x, ξ).


Inequality (4.21) holds letting n→ ∞.

By Lemmas 4.1.6 and 4.1.8, we conclude that fhom is a Caratheodory integrand and thus

Ihom is well defined.

Remark 4.1.9. We note that fhom satisfies analogous growth and coercivity conditions to

the ones of f , which, together with the continuity properties of fhom, imply by standard

arguments (approximation ofW 1,p by piecewise affine functions together with the invariance

of the domain of fhom) that this function is quasiconvex with respect to the last variable.

We will show next that in the convex case it is enough to consider one cell period for the

definition of fhom (4.3) (see also Braides [19] or Muller [66]). We define for all (x, ξ) ∈Ω × Rd×N

f⋆hom(x, ξ) = infφ

∫

Qf(x, y, ξ + ∇φ(y)) dy, φ ∈W 1,p

0 ((0, 1)N ; Rd)

and

f⋆⋆hom(x, ξ) = infφ

∫

Qf(x, y, ξ + ∇φ(y)) dy, φ ∈W 1,p

per((0, 1)N ; Rd)

.

Lemma 4.1.10. Assume, in addition to the hypotheses on f , that f(x, y, ·) is convex for

all (x, y) ∈ Ω × RN . Then

f⋆hom = f⋆⋆hom = fhom.

Proof. (see Braides [19] and Muller [66] for an alternative proof). Equality f⋆hom = f⋆⋆hom is

proven by an argument analog to that of the proof of Lemma 4.1.5.

We show that f⋆hom = fhom. Let (x, ξ) ∈ Ω × Rd×N . By definition f⋆hom(x, ξ) > fhom(x, ξ).

To prove the opposite inequality, for each n ∈ N take Tn ∈ N and a function φn ∈W 1,p

0 ((0, Tn)N ; Rd) such that

fhom(x, ξ) + 1n > 1

TNn

∫

(0,Tn)N

f(x, y, ξ + ∇φn(y)) dy

=

∫

Qf(x, Tny, ξ + ∇ψn(y)) dy,

(4.24)

where ψn(y) := 1Tnφn(Tny), ψn ∈ W 1,p

0 (Q; Rd). By the p-growth condition in (H4) the

sequence ||∇ψn||Lp(Q;Rd)n is bounded, and so is ||ψn||W 1,p(Q;Rd)n by Poincare Inequality.


Hence, there exists a subsequence (still denoted by ψnn) such that

ψnW 1,p

ψ

for some ψ = ψ(y) ∈ W 1,p0 (Q; Rd). As a consequence, by Theorem 2.6.7 and up to a

subsequence

ψn2s ψ

and

∇ψn 2s ∇ψ + ∇zψ

for some ψ = ψ(y, z) ∈ Lp(

Q;W 1,pper(Q; Rd)

)

. We divide the rest of the proof in two steps.

Step 1. We follow an argument of Allaire [4] assuming in addition that

(H5)∂f∂η (x, y, η) exists for all (x, y, η) ∈ Ω × RN × Rd×N .

By inequalities (2.5) and (2.6), and by the p-growth condition in (H4) there exists a constant

C > 0 such that for all (x, y, η) ∈ Ω × RN × Rd×N

∣

∣

∣

∂f

∂η(x, y, η)

∣

∣

∣6 C

[

1 + |η|p−1]

. (4.25)

Let ϕ = ϕ(y, z) ∈ C∞c

(

Q;C∞per(Q; Rd×N )

)

. Since f is convex in the last variable then by

inequality (2.5)∫

Qf(

x, Tny, ξ + ∇ψn(y))

dy >

∫

Qf(

x, Tny, ξ + ϕ(y, Tny))

dy

+

∫

Q

∂f

∂η

(

x, Tny, ϕ(y, Tny))

· (∇ψn(y) − ϕ(y, Tny)) dy,

(4.26)

for each n ∈ N. By Lemma 2.6.3, and as ∇ψn 2s ∇ψ +∇zψ (see Definition 2.6.5), we have

limn→∞

∫

Qf(

x, Tny, ξ + ϕ(y, Tny))

dy =

∫

Q

[

∫

Qf(

x, z, ξ + ϕ(y, z))

dz]

dy (4.27)

and

limn→∞

∫

Q

∂f

∂η

(

x, Tny, ϕ(y, Tny))

· (∇ψn(y) − ϕ(y, Tny)) dy

=

∫

Q

[

∫

Q

∂f

∂η

(

x, z, ϕ(y, z))

· (∇ψ(y) + ∇zψ(y, z) − ϕ(y, z)) dz]

dy.

(4.28)

Therefore, from (4.26)-(4.28) we get

lim infn→∞

∫

Qf(


dy >

∫

Q

[ ∫

Qf(

x, z, ξ + ϕ(y, z))

dz

]

dy

+

∫

Q

[ ∫

Q

∂f

∂η

(

x, z, ϕ(y, z))

· (∇ψ(y) + ∇zψ(y, z) − ϕ(y, z)) dz

]

dy,

(4.29)

for all ϕ = ϕ(y, z) ∈ C∞c

(

Q;C∞per(Q; Rd×N )

)

. Let now ϕkk ⊂ C∞c

(

Q;C∞c (Q; Rd×N )

)

be a

convergent sequence to ∇ψ + ∇zψ in Lp(

Q×Q; Rd×N)

. In particular from (4.29) we get

lim infn→∞

∫

Qf(


dy >

∫

Q

[ ∫

Qf(

x, z, ξ + ϕk(y, z))

dz

]

dy

+

∫

Q

[ ∫

Q

∂f

∂η

(

x, z, ϕk(y, z))

· (∇ψ(y) + ∇zψ(y, z) − ϕk(y, z)) dz

]

dy,

(4.30)

for every k ∈ N. Inequality (4.25) and the growth conditions on f and ∂f∂η imply that we

can pass to the limit in (4.30) and get

lim infn→∞

∫

Qf(


dy >

∫

Q

[ ∫

Qf(

x, z, ξ + ∇ψ(y) + ∇zψ(y, z))

dz

]

dy.

(4.31)

By Jensen’s Inequality and Fubini’s Theorem, for each z ∈ Q∫

Qf(

x, z, ξ + ∇ψ(y) + ∇zψ(y, z))

dy > f(

x, z,

∫

Q

[

ξ + ∇ψ(y) + ∇zψ(y, z)]

dy)

= f(

x, z, ξ + ∇z

(

∫

Qψ(y, z) dy

))

. (4.32)

Thus, by (4.24), (4.31), (4.32), and once more Fubini’s Theorem

fhom(ξ) >

∫

Qf(

z, ξ + ∇z

(

∫

Qψ(y, z) dy

))

dz > f⋆⋆hom(ξ) = f⋆hom(ξ).

Step 2. We address now the general case. For each ε > 0 set ζε(η) := 1εN ζ

(

ηε

)

where

ζ ∈ C∞(Rd×N ) denotes the standard mollifier, that is,

ζ(η) :=

C exp(

1|η|2−1

)

if |η| < 1,

0 if |η| > 1,


and the constant C is selected so that

∫

Rd×N

ζ(η) dη = 1. Let

fε(x, y, ξ) :=

∫

B(0,ε)ζε(η)f(x, y, ξ − η) dη

for all ε > 0 and all (x, y, ξ) ∈ Ω × RN × Rd×N . It is straightforward to show that fε

satisfies conditions (H1)-(H5). Fix δ > 0 and, by a density argument, let S ∈ N and

ψ ∈W 1,∞((0, S)N ; Rd) be such that

fhom(x, ξ) + δ >1

SN

∫

(0,S)N

f(x, y, ξ + ∇ψ(y)) dy.

Then


1

SN

∫

(0,S)N

fε(x, y, ξ + ∇ψ(y)) dy,

and thus

fhom(x, ξ) + δ > lim supε→0

(fε)hom(x, ξ)

= lim supε→0

(fε)⋆⋆hom(x, ξ)

> f⋆⋆hom(x, ξ)

since fε is convex and fε > f for all ε > 0. Indeed, it is easy to show that fε is convex and

the last assertion is a consequence of the convexity of f and Jensen’s Inequality.

4.1.2 Proof of the main result

As in the previous subsection, by Remark 4.1.2 we may suppose that f is a positive Borel

function satisfying hypotheses (H1), (H3) and (H4) for every (x, y, ξ) ∈ Ω × RN × Rd×N .

Due to the p-coercivity condition in (H4), to prove Theorem 4.1.1 it suffices to show that


Iε(u) =

∫

Ωfhom(x, u(x),∇u(x)) dx, (4.33)

for all u ∈W 1,p(Ω; Rd), since


Iε(u) = ∞


for all u ∈ Lp(Ω; RN ) \W 1,p(Ω; Rd). To prove identity (4.33) we use the Direct Method of

Γ- convergence. Accordingly, we start by localizing the functionals Iε in order to highlight

their dependence on the domain of integration, that is, we consider a family of functionals

Iε : Lp(Ω; Rd) ×A(Ω) → [0,∞] defined by

Iε(u;A) :=

∫

Af(

x,x

ε,∇u(x)

)

dx if u ∈W 1,p(A; Rd),

∞ otherwise.

Our goal is to show that


Iε(u,A) =

∫

Afhom(x,∇u(x)) dx, (4.34)

for all u ∈W 1,p(Ω; Rd) and A ∈ A(Ω). In particular (4.33) will follow by taking A = Ω.

The next step toward the proof of (4.34) is to establish a compactness property that ensures

the existence of Γ-converging subsequences of Iεε.

Proposition 4.1.11. For every sequence εnn of positive real numbers converging to zero

there exists a further subsequence εnjj ≡ εjj such that

Γ(Lp(A))- limj→∞

Iεj(·;A)(u) =: Iεj(u;A) (4.35)

exists for all u ∈W 1,p(Ω,Rd) and all A ∈ A(Ω).

The proof of this proposition follows an argument analog to the one used in Braides, Fonseca

and Francfort [20], but for completeness we present it here.

Let R be a countable collection of subsets of Ω such that, for any δ > 0 and any A ∈ A(Ω),

there exists a finite union CA of disjoint elements of R satisfying

CA ⊂ A,

LN (A) 6 LN (CA) + δ.

We may take R as the set of open squares with faces parallel to the axes, centered at points

x ∈ Ω∩QN and with rational edge lengths. We denote by R the countable collection of all

finite unions of elements of R, i.e.

R =

k⋃

i=1

Ci : k ∈ N, Ci ∈ R

.

The next lemma is the starting point for the proof of Proposition 4.1.11.

Lemma 4.1.12. For every sequence εnn of positive real numbers converging to zero there

exists a further subsequence εnjj (depending on R) such that the Γ-limit

Γ(Lp(C))- limj→∞

Iεnj(·;C)(u) =: Iεnj

(u;C) (4.36)

exists for all u ∈W 1,p(Ω,Rd) and all C ∈ R.

Proof. Let C ∈ R. From Proposition 2.5.8 and as Lp(Ω; Rd) is separable there exist a

subsequence εnjj (depending on C) such that the Γ(Lp(C))-limit of Iεnj

(·;C) exists for

all u ∈ W 1,p(Ω,Rd). But then by a diagonalization procedure we can find a subsequence

εnjj (depending on R) such that (4.36) holds.

Let now εnn be a fixed sequence of positive real numbers converging to zero and εjj a

subsequence for which (4.36) holds.

Proof of Proposition 4.1.11. We wish to show that for all A ∈ A(Ω) and u ∈W 1,p(Ω,Rd)

inf

lim infj→∞

∫

Af(

x,x

εj,∇uj(x)

)

dx : uj ∈W 1,p(A,Rd), ujLp(A;Rd)−→ u

(4.37)

= inf

lim supj→∞

∫

Af(

x,x

εj,∇uj(x)

)


.

Let A ∈ A(Ω) and u ∈ W 1,p(Ω,Rd). To prove (4.37) it suffices to find a sequence vjj ⊂W 1,p(A,Rd) with vj

Lp(A;Rd)−→ u and such that

inf

lim infj→∞

∫

Af

(

x,x

εj,∇uj(x)

)


> lim supj→∞

∫

Af

(

x,x

εj,∇vj(x)

)

dx.

(4.38)

Fix δ > 0 and choose Cδ ∈ R with Cδ ⊂ A and LN (A \ Cδ) << 1, so that∫

A\Cδ

(1 + |∇u|p) dx 6δ

β, (4.39)


where β is the constant in (H4). By Proposition 2.5.13 consider a sequence wδj ∈W 1,p(Cδ,R

d) such that

wδjLp(Cδ;Rd)−→ u,

Γ(Lp(Cδ))- limj→∞

Iεj(· ;Cδ)(u) = lim

j→∞

∫

Cδ

f

(

x,x

εj, Dwδj (x)

)

dx, (4.40)

and wδj = u on ∂Cδ. Extending wδj by u outside Cδ (still denoted by wδj ) it follows that

wδjLp(A;Rd)−→ u, and in view of (H3), (4.39) and (4.40) we obtain that

lim supδ→0

lim supj→∞

∫

Af

(

x,x

εj,∇wδj (x)

)

dx (4.41)

6 lim supδ→0

limj→∞

∫

Cδ

f

(

x,x

εj,∇wδj (x)

)

dx+ β lim supδ→0

∫

A\Cδ

C(1 + |∇u|p) dx

= lim supδ→0

Γ(Lp(Cδ))- limj→∞

Iεj(·;Cδ)(u)

= lim supδ→0

inf

lim infj→∞

∫

Cδ

f

(

x,x

εj,∇uj(x)

)

dx : uj ∈W 1,p(Cδ,Rd), uj

Lp(Cδ;Rd)−→ u

= inf

lim infj→∞

∫

Af

(

x,x

εj,∇uj(x)

)


(by 4.41)

6 lim infδ→0

lim infj→∞

∫

Af

(

x,x

εj,∇wδj (x)

)

dx.

By Lemma A.2 in the Appendix there exists a decreasing sequence δ(εj) ↓ 0 such that

limj→∞

∫

Af

(

x,x

εj,∇wδ(εj)

j (x)

)

dx

= inf

lim infj→∞

∫

Af

(

x,x

εj,∇uj(x)

)


,

for vj := wδ(εj)j , and this implies (4.38).

We now seek to ensure that Iεj, regarded both as a functional on W 1,p(Ω,Rd) and as a

set function, admits an integral representation of the form

Iεj(u;A) =

∫

Afεj(x,∇u(x)) dx.

We will verify that the hypotheses of Theorem 2.4.1 hold. Using Lemma 2.1.20 and the

conditions imposed on f , it is possible to show that Iεj(u; .) is a measure, more precisely

we prove the following result.

Lemma 4.1.13. For each u ∈W 1,p(Ω; Rd), Iεj(u; .) is the trace of a finite, positive Radon

measure restricted to A(Ω).

Proof. Let u ∈W 1,p(Ω; Rd). In view of Lemma 4.1.11, let uj ⊂W 1,p(Ω; Rd) be a sequence

such that

Iεj(u; Ω) = limj→∞

∫

Ωf

(

x,x

εj,∇uj(x)

)

dx,

and consider µj := f(·, ·εj,∇uj)χΩ(·)LN . By (H4), and up to a subsequence (still denoted

by µj), there exists a finite positive Radon measure on RN such that

µj⋆ µ

(see Proposition 2.1.31). We claim that Iεj(u; .)⌊A(Ω) = µ, i.e. Iεj(u;A) = µ(A) for all

A ∈ A(Ω). We apply Lemma 2.1.20 with Π(·) = Iεj(u; .)

We start by proving condition i) in Lemma 2.1.20, i.e. Iεj(u; .) is nested-subadditive.

Given A, B, C ∈ A(Ω) with C ⊂⊂ B ⊂ A we have to show that

Iεj(u;A) 6 Iεj(u;B) + Iεj(u;A \ C).

Choose Cδ, Dδ ∈ R with Cδ ⊂ C and Dδ ⊂ A \ C such that∫

C\Cδ

(1 + |∇u|p) dx < δ and

∫

(A\C)\Dδ

(1 + |∇u|p) dx < δ.

By Proposition 2.5.13 there exist two sequences vCδ ⊂W 1,p(Cδ; Rd) and vDδ

j ⊂W 1,p(Dδ; Rd)

satisfying

||vDδ

j − u||Lp(Dδ ;Rd) → 0, ||vCδ

j − u||Lp(Cδ;Rd) → 0,

Iεj(u;Dδ) = lim Iεj

(vDδ

j ;Dδ), Iεj(u;Cδ) = lim Iεj(vCδ

j ;Cδ),

vDδ

j = u on ∂Dδ and vCδ

j = u on ∂Cδ.

Extend vCδ

j and vDδ

j by u to all A and set

wδj :=

vDδ

j if x ∈ A \ CvCδ

j if x ∈ C.

Clearly ||wδj − u||Lp(A;Rd) → 0 and we have

Iεj(u;A) 6 lim infδ→0

lim infj→∞

Iεj(wδj ;A)

6 Iεj(u;Dδ) + Iεj(u;Cδ) + lim

δ→0

∫

(A\C)\Dδ∪(C\Cδ)(1 + |∇u|p) dx

= Iεj(u;Dδ) + Iεj(u;Cδ).

To establish condition ii) in Lemma 2.1.20: Given A ∈ A(Ω) and ε > 0, consider Aε ∈ A(Ω)

such that Aε ⊂ A and

β(1 + |∇u(·)|p)χΩ(·)LN (A\Aε) < ε.

Due to the growth conditions (H4)

Iεj(u;A\Aε) 6 lim infj→∞

∫

A\Aε

f

(

x,x

εj,∇u(x)

)

dx

6 β

∫

A\Aε

(1 + |∇u(x)|p) dx

6 ε.

To show iv) fix A ∈ A(Ω). By Proposition 2.1.32

Iεj(u;A) 6 lim infj→∞

∫

Af

(

x,x

εj,∇uj(x)

)

dx

= lim infj→∞

µj(A)

6 µ(A).

Finally, to establish iii) take Ω′ ⊂ RN such that Ω ⊂⊂ Ω′. As µj⋆ µ


µ(Ω′) 6 limj→∞

µj(Ω′) = lim

j→∞

∫

Ωf(

x,x

εj,∇uj(x)

)

dx = Iεj(u; Ω).

Therefore µ(Ω′) 6 Iεj(u; Ω) for all such Ω′. Hence Iεj(u; Ω) > µ(RN ), and as a conse-

quence of Lemma 2.1.20 we conclude that Iεj(u;A) = µ(A) for all A ∈ A(Ω).

By Proposition 2.5.5 the functional Iεj(., A) is lower semicontinuous with respect to the

Lp- topology for all A ∈ A(Ω), hence it is sequentially lower semicontinuous with respect to

the weak topology in W 1,p. As a consequence of the integral representation Theorem 2.4.1

and Remark 2.4.2 we derive the following result.

Lemma 4.1.14. There exist a Caratheodory function

fεj : Ω × Rd×N → [0,∞)

quasiconvex with respect to its second variable for a.e. x ∈ Ω satisfying the same growth

conditions than f does, and such that

Iεj(u,A) =

∫

Afεj(x,∇u(x)) dx

for all u ∈W 1,p(Ω; Rd) and A ∈ A(Ω).

To conclude that (4.34) holds, by Remark 2.5.9 it suffices to prove that the function fεj

is independent on this particular (sub)sequence, so that each Γ-convergent (sub)sequence

has the same limit. The remaining of this section is devoted to showing that

fεj(x, ξ) = fhom(x, ξ) (4.42)

for a.e. x ∈ Ω and for all ξ ∈ Rd×N. To start, let T ∈ N and let ST denote a countable set of

C∞c ((0, T )N ; Rd)-functions dense in W 1,p

0 ((0, T )N ; Rd). Let L be the set of Lebesgue points

x0 for all functions

fεj(·, η) (4.43)

and

x→∫

Qf(x, Ty, η + ∇φ(Ty)) dy, (4.44)


with η ∈ Qd×N , φ ∈ ST and T ∈ N. Since the function f : Ω × RN × Rd×N → [0,∞) is of

Caratheodory-type, by Scorza-Dragoni Theorem there exists a non decreasing sequence of

compact subsets Kmm∈N ⊂ Ω with |Ω\Km| 6 1m such that f : Km×RN ×Rd×N → [0,∞)

is continuous for all m ∈ N. Let K⋆m be the set of Lebesgue points for χKm with m ∈ N,

and define

W :=∞⋃

m=0

(Km ∩K⋆m) and E := L ∩W.

We have |Ω \L| = 0 and |Ω\W | 6 |Ω\Km| 6 1m for each m ∈ N. Consequently |Ω \W | = 0

and |Ω \ E| = 0. In a first step to prove identity (4.42) we derive the following equality.

Proposition 4.1.15. fεj(x0, ξ) = fhom(x0, ξ) for all x0 ∈ E and ξ ∈ Qd×N.

Proof. Consider x0 ∈ E and ξ ∈ Qd×N . By (4.43) we have

fεj(x0, ξ) = limδ→0

1δN

∫

Q(x0,δ)fεj(x, ξ) dx

= limδ→0

Iεj(ξ · ;Q(x0,δ))

δN .

(4.45)

Step 1. We first establish the upper bound inequality for the Γ-limit of Iεjj , i.e.

fεj(x0, ξ) 6 fhom(x0, ξ).

Given n ∈ N, let Tn ∈ N and φn ∈W 1,p0 ((0, Tn)

N ; Rd) such that

fhom(x0, ξ) +1

2n>

1

TnN

∫

(0,Tn)N

f(x0, y, ξ + ∇φn(y)) dy.

By conditions (H1) and (H4), and by the density of STn in W 1,p0 ((0, Tn)

N ; Rd) we may take

φn ∈ STn with

fhom(x0, ξ) +1

n>

1

TnN

∫

(0,Tn)N

f(x0, y, ξ + ∇φn(y)) dy.

Extend φn periodically with period Tn to RN (still denoted by φn). For x ∈ RN define

unj (x) := ξ · x+ εjφn

( x

εj

)


and let δ > 0 be small enough so that Q(x0, δ) ∈ A(Ω). As φnn is bounded in W 1,p, for

fixed n we have limj→∞

unj = v in Lp(Q(x0; δ); Rd) where v(x) = ξ · x. Hence by equality

(4.45)

fεj(x0, ξ) 6 lim infδ→0

lim infj→∞

1

δN

∫

Q(x0;δ)f

(

x,x

εj, ξ + ∇φn

(

x

εj

))

dx. (4.46)

Define now hn(x, y) := f(x, Tny, ξ+∇φn(Tny)) for all x ∈ Ω and y ∈ RN, and for all n ∈ N.

Clearly hn ∈ L1(Q(x0; δ);Cper(Q; Rd)) for n ∈ N - recall Definition 2.6.1 and Lemma 2.6.4

- and then by Lemma 2.6.3

limj→∞

∫

Q(x0;δ)f

(

x,x

εj, ξ + ∇φn

(

x

εj

))

dx

= limj→∞

∫

Q(x0;δ)f

(

x,Tnx

Tnεj, ξ + ∇φn

(

Tnx

Tnεj

))

dx

= limj→∞

∫

Q(x0;δ)hn

(

x,x

Tnεj

)

dx

=

∫

Q(x0;δ)

∫

Qhn(x, y) dy dx

=

∫

Q(x0;δ)

∫

Qf(x, Tny, ξ + ∇φn(Tny)) dy dx (4.47)

(Allaire [3] shows with counterexamples that this convergence does not hold if the continuity

in one of the variables of f is not assumed). Therefore by identities (4.44), (4.46) and (4.47)

fεj(x0, ξ) 6 lim infδ→0

1

δN

∫

Q(x0;δ)

∫

Qf(x, Tny, ξ + ∇φn(Tny)) dy dx

=

∫

Qf(x0, Tny, ξ + ∇φn(Tny)) dy

=1

TnN

∫

(0,Tn)N

f(x0, y, ξ + ∇φn(y)) dy

6 fhom(x0, ξ) +1

n.

Letting n→ ∞ we get


Step 2. We now show that the converse inequality holds, a.e.

fεj(x0, ξ) > fhom(x0, ξ).


Fix δ > 0 small enough so that Q(x0; δ) ∈ A(Ω), and consider uδjj ⊂ W 1,p(Q(x0; δ); Rd)

with limj→∞

uδj = 0 in Lp(Q(x0; δ); Rd) and

Iεj(ξ · ;Q(x0; δ)) = limj→∞

∫

Q(x0;δ)f

(

x,x

εj, ξ + ∇uδj(x)

)

dx.

By identity (4.45)


limj→∞

1δN

∫

Q(x0;δ)f

(

x,x

εj, ξ + ∇uδj(x)

)

dx

= limδ→0

1δN

∫

Q(x0;δ)f

(

x,x

εj(δ), ξ + ∇uδj(δ)(x)

)

dx

= limδ→0

∫

Qf

(

x0 + δy,x0 + δy

εj(δ), ξ + ∇vδj(δ)(y)

)

dy,

where vδj(δ)(y) := 1δu

δj(δ)(x0 + δy) ∈ W 1,p(Q; Rd). Using a diagonalization argument, we

choose the sequence j(δ) in such a way that

δ

εj(δ)>

1

δand lim

δ→0||vδj(δ)||Lp(Q;Rd) = 0. (4.48)

For simplicity denote j(δ) ≡ δ, vδj(δ) ≡ vδ and uδj(δ) ≡ uδ. By Theorem 2.2.16 there exists

a subsequence of vδ (still denoted by vδ) and a sequence wδ ⊂ W 1,∞(RN ; Rd) such

that

wδ 0 in W 1,p, wδ = 0 in a neighborhood of ∂Q,

| ∇wδ |p is equi-integrable (4.49)

and

| y ∈ Q : vδ(y) 6= wδ(y) |→ 0. (4.50)

As, by assumption, f is nonnegative

fεj(x0, ξ) > lim infδ→0

∫

y∈Q: vδ(y)=wδ(y)f

(

x0 + δy,x0 + δy

εδ, ξ + ∇wδ(y)

)

dy

= lim infδ→0

∫

Qf

(

x0 + δy,x0 + δy


)

dy


from (4.49), (4.50) and (H4). Since x0 ∈W , there exists m0 ∈ N such that x0 ∈ Km0 ∩K⋆m0

and then we can write

fεj(x0, ξ) > lim infm→∞

lim infδ→δ

∫

Qm,δ

f

(

x0 + δy,x0 + δy


)

dy, (4.51)

where the set

Qm,δ := y ∈ Q : x0 + δy ∈ Km0 and |∇wδ(y)| 6 m

is such that

limm→∞

limδ→0

|Q \Qm,δ| = 0. (4.52)

Indeed, this set has measure zero because on the one hand, as

|y ∈ Q : x0 + δy 6∈ Km0| 6 1 − 1δN

∫

Q(x0;δ)χKm0

(y) dy

and as x0 ∈ K⋆m0

, we have

limm→∞

limδ→0

|y ∈ Q : x0 + δy 6∈ Km0| = 0,

and on the other hand, by Chebyshev Inequality (Lemma 2.1.10) and the fact that ∇wδδis bounded in Lp(Q; Rd), we have

lim supm→∞

lim supδ→0

|y ∈ Q : |∇wδ(y)| > m| = 0.

Writex0

εδ= mδ + sδ

with mδ ∈ ZN and sδ ∈ [0, 1)N , and define

xδ :=−εδδsδ.

Note that by (4.48) xδ → 0 as δ → 0. After changing variables once more,∫

Qm,δ

f

(

x0 + δy,x0 + δy


)

dy

=

∫

Qm,δ−xδ

f

(

x0 + δ(y + xδ),x0 + δ(y + xδ)

εδ, ξ + ∇wδ(y + xδ)

)

dy

=

∫

Qm,δ−xδ

f

(

x0 + δ(y + xδ),δ

εδy, ξ + ∇wδ(y + xδ)

)

dy,

by the periodicity hypothesis (H3) and the fact that x0+δxδ

εδ= mδ ∈ ZN. Then by inequality

(4.51) we have

fεj(x0, ξ) > lim infm→0

lim infδ→0

∫

Qm,δ−xδ

f

(

x0 + δ(y + xδ),δ


)

dy.

(4.53)

We now write∫

Qm,δ−xδ

f

(

x0 + δ(y + xδ),δ


)

dy

=

∫

Qm,δ−xδ

[

f

(

x0 + δ(y + xδ),δ


)

− f

(

x0,δ


)]

dy

+

∫

Qm,δ−xδ

f

(

x0,δ


)

dy. (4.54)

By hypothesis (H3) and the continuity of f with respect to y, given m ∈ N the restriction

of f to the set Km0 ×RN ×B(ξ,m), where B(ξ,m) := x ∈ RN : |ξ−x| < m, is uniformly

continuous - recall that if g : RN × RN → R is a continuous function kQ-periodic with

respect to the first variable, then for each compact K ⊂ RN , g : RN ×K → R is uniformly

continuous. Hence there exist ρm ∈ (0, 1) such that

|f(x, y, φ) − f(x, y, φ)| < 1

m

for all x, x ∈ Km0 , y, y ∈ RN , and φ, φ ∈ B(ξ,m) satisfying |x− x|+ |y− y|+ |φ− φ| < ρm.

As a result

limm→∞

limδ→0

∫

Qm,δ−xδ

∣

∣

∣f(

x0+δ(y+xδ),δ

εδy, ξ+∇wδ(y+xδ)

)

−f(

x0,δ

εδy, ξ+∇wδ(y+xδ)

)∣

∣

∣ dy = 0,

and then by inequality (4.53) and identity (4.54) we get

fεj(x0, ξ) > lim infm→∞

lim infδ→0

∫

Qm,δ−xδ

f

(

x0,δ


)

dy.

Consequently,


∫

Q−xδ

f

(

x0,δ


)

dy. (4.55)

Indeed, first we note that∫

(Q−xδ)\(Qm,δ−xδ)f

(

x0,δ


)

dy

6 β

∫

(Q−xδ)\(Qm,δ−xδ)(1 + |ξ + ∇wδ(y + xδ)|p) dy


= β

∫

Q\Qm,δ

(1 + |ξ + ∇wδ(y)|p) dy. (4.56)

As |∇wδ|p is equi-integrable, by inequality (4.56) and condition (4.52) we have

lim supm→∞

lim supδ→0

∫

(Q−xδ)\(Qm,δ−xδ)f

(

x0,δ


)

dy = 0,

which, in turn, implies inequality (4.55). In addition, since

∫

Q\Q−xδ

f

(

x0,δ


)

dy 6 β

∫

Q\Q−xδ

(1 + |ξ + ∇wδ(y + xδ)|p)dy

= β

∫

Q+xδ\Q(1 + |ξ + ∇wδ(z)|p)dz → 0

as δ → 0 (once again because |∇wδ|p is equi-integrable and |(Q+ xδ) \Q| → 0 as δ → 0),

we get from inequality (4.55)


∫

Qf

(

x0,δ


)

dy. (4.57)

In order to compare fεj(x0, ξ) with fhom(x0, ξ) we need to modify wδ(· + xδ) close to the

boundary of Q so that it become admissible for fhom. For this purpose, define the sets

Lδ := y ∈ Q : dist(y, ∂Q) 6 |xδ|, Mδ = y ∈ Q : dist(y, ∂Q) > 2|xδ|,

and

Sδ = y ∈ Q : dist(y, ∂Q) ∈ (|xδ|, 2|xδ|).

Consider a function φδ ∈ C∞c (Q; R) with ||∇φδ||L∞ 6 C

|xδ|such that

φδ(y) =

1 if y ∈ Mδ,

0 if y ∈ Lδ


and finally set vδ(y) := φδ(y)wδ(y + xδ) + (1 − φδ(y))wδ(y) ∈W 1,∞0 (Q; Rd). We claim that


∫

Qf(

x0,δ

εδy, ξ + ∇vδ(y)

)

dy, (4.58)

which implies


(εδδ

)N∫

δεδQf(

x0, y, ξ + ∇φδ(y))

dy

with φδ(y) := δεδvδ(

εδ

δ y) ∈W 1,∞0 ( δεδ

Q; Rd), and consequently,

fεj(x0, ξ) > fhom(x0, ξ).

To prove inequality (4.58) we note that

lim infδ→0

∫

Qf

(

x0,δ

εδy, ξ + ∇vδ(y)

)

dy

6 lim infδ→0

∫

Qf

(

x0,δ


)

dy

+ lim supδ→0

β

∫

Lδ

(1 + |∇wδ(x)|p) dx

+ lim supδ→0

β

∫

Sδ

(|∇wδ(x+ xδ)|p + |∇wδ(x)|p) dx

+ lim supδ→0

β||∇φδ||pL∞

∫

Sδ

|wδ(x+ xδ) − wδ(x)|p dx. (4.59)

Due to the integrability property of |∇wδ|pδ

lim supδ→0

∫

Lδ

(1 + |∇wδ(x)|p) dx = 0 = lim supδ→0

∫

Sδ

(|∇wδ(x+ xδ)|p + |∇wδ(x)|p) dx. (4.60)

This property also implies that

lim supδ→0

||∇φδ||pL∞

∫

Sδ

|wδ(x+ xδ) − wδ(x)|p dx = 0, (4.61)


because

||∇φδ||pL∞

∫

Sδ

|wδ(x+ xδ) − wδ(x)|p dx 6C

|xδ|p∫

Sδ

∣

∣

∣

∣

∫ 1

0

dwδ(x+ txδ)

dtdt

∣

∣

∣

∣

p

dx

6C

|xδ|p∫

Sδ

∫ 1

0|∇wδ(x+ txδ)|p . |xδ|p dt dx

6 C

∫

Sδ

∫ 1

0|∇wδ(x+ txδ)|p dt dx

= C

∫ 1

0

∫

Sδ−txδ

|∇wδ(y)|p dy dt

6 C

∫

Nδ

|∇wδ(y)|p dy,

where

Nδ = x ∈ Q : dist(x, ∂Q) 6 3|xδ|.

Hence, inequality (4.58) holds by (4.57), (4.59), (4.60) and (4.61).

As a consequence of this proposition equality (4.42) holds.

Corollary 4.1.16. fεj(x, ξ) = fhom(x, ξ) for a.e. x ∈ Ω and for all ξ ∈ Rd×N.

Proof. By Proposition 4.1.15 we have that for a.e. x ∈ Ω and for all ξ ∈ Qd×N , fεj(x; ξ) =

fhom(x; ξ). By the continuity properties of fεj and fhom with respect to their second

variable, the equality fεj(x; ξ) = fhom(x; ξ) holds true for a.e. x ∈ Ω and for all ξ ∈ Rd×N.

The proof of Theorem 4.1.1 is now straightforward.

Proof of Theorem 4.1.1 As a consequence of Corollary 4.1.16 the Γ(Lp(A))-limit of Iεj( · ;A)

is equal to Ihom( · , A). In particular, since it does not depend upon the extracted sub-

sequence, in view of Remark 2.5.9, the whole sequence Iε( · ;A) Γ(Lp(A))-converges to

Ihom( · ;A). Taking A = Ω we conclude the proof of Theorem 4.1.1.


4.2 Multiple scale functionals

In this section we allow our functionals to depend on one more scale of periodicity, namely

our goal is to prove the following result.

Theorem 4.2.1. Let f : Ω × RN × RN × Rd×N → R be a function satisfying

(H1) f(x, · , · , · ) is continuous for a.e. x ∈ Ω;

(H2) f( · , y, z, ξ) is measurable for all (y, z, ξ) ∈ RN × RN × Rd×N ;

(H3) f(x, · , z, ξ) is Q-periodic for all (z, ξ) ∈ RN × Rd×N and for a.e. x ∈ Ω; f(x, y, · , ξ)is Q-periodic for all (y, ξ) ∈ RN × Rd×N and for a.e. x ∈ Ω;

(H4) there exists β > 0 and a real number p > 1 such that for all (y, z, ξ) ∈ RN×RN×Rd×N

and for a.e. x ∈ Ω1

β|ξ|p − β 6 f(x, y, z, ξ) 6 β(1 + |ξ|p).

For each ε > 0 define Iε : Lp(Ω,Rd) → R by

Iε(u) :=

∫

Ωf(

x,x

ε,x

ε2,∇u(x)

)


∞ otherwise.

Then the Γ(Lp(Ω))-limit of the family Iεε is given by the functional

Ihom(u) :=

∫


∞ otherwise,

where fhom is defined for all ξ ∈ Rd×N and a.e. x ∈ Ω by


infφ

1

TN

∫

(0,T )N

fhom(x, y, ξ + ∇φ(y)) dy : φ ∈W 1,p0 ((0, T )N ; Rd)

,

(4.62)

and

fhom(x, y, ξ) := limT→∞

infφ

1

TN

∫

(0,T )N

f(x, y, z, ξ + ∇φ(z)) dz : φ ∈W 1,p0 ((0, T )N ; Rd)

(4.63)


for a.e. x ∈ Ω and all (y, ξ) ∈ RN × Rd×N .

Theorem 4.2.1 was obtained in collaboration with J.F. Babadjian [10]. As previously re-

marked, formula (4.62) is obtained by homogenizing first with respect to z, considering y

as a parameter, and then homogenizing with respect to y. That is, fhom is the density

obtained by iterating twice the homogenization formula (4.3). The generalization of this

result to any number of scales k > 2 follows by an iterated argument similar to the one

used in Braides and Defranceschi (see Remark 22.8 in [19]).

As in Section 4.1 (see Remark 4.1.2), without loss of generality we may assume that f is a

positive Borel function such that hypothesis (H1), (H3) and (H4) hold for every (x, y, z, ξ) ∈Ω × RN × RN × Rd×N .

We note that most of the proofs presented here follow along the lines of the ones in Braides

and Defranceschi [19] (Theorem 22.1 and Remark 22.8), and that our main contribution

is to weaken the strong uniform continuity hypothesis (1.6). Let us briefly describe how

we proceed: The idea consists in proving the result for integrands which do not depend

explicitly on x (see Theorem 4.2.2), and then to treat the general case by freezing this

macroscopic variable and proceeding as in the proof of Theorem 4.1.1 by using the “uni-

formly continuous structure” of the integrand f up to sets of measure zero.

We divide this section as follows. In Subsubsection 4.2.1 we state the main properties of

fhom and fhom that are basic for our analysis. In Subsubsection 4.2.2 we present some

auxiliary results for the proof of the homogeneous counterpart of Theorem 4.2.1, Theorem

4.2.2, in which we assume that f does not depend explicitly on x. The proof of Theorem

4.2.1 in its full generality is presented in Subsection 4.2.3. Finally, in Subsection 4.2.4

we remark an alternative proof for convex integrands relying on arguments of multiscale

convergence.


Repeating the argument used in Section 4.1, we can see that the function fhom given in

(4.63) is well defined and is a Caratheodory function:

fhom(·, · , ξ) is LN ⊗ LN -measurable for all ξ ∈ Rd×N , (4.64)

fhom(x, y, ·) is continuous for all (x, y) ∈ Ω × RN . (4.65)


By condition (H3) it follows that

fhom(x, · , ξ) is Q-periodic for all x ∈ Ω and all ξ ∈ Rd×N . (4.66)

Moreover, fhom is quasiconvex in the variable ξ and satisfies the same p-growth and p-

coercivity condition as f :

1

β|ξ|p − β 6 fhom(x, y, ξ) 6 β(1 + |ξ|p) (4.67)

for all x ∈ Ω and all (y, ξ) ∈ RN ×Rd×N , where β is the constant in (H4). As a consequence

of (4.64)-(4.67), the function fhom given in (4.62) is also well defined, and is a Caratheodory

function, which implies the functional Ihom is well defined on W 1,p(Ω; Rd). Finally, fhom is

also quasiconvex in the variable ξ and satisfies the same p-growth and p-coercivity condition

as f and fhom:1

β|ξ|p − β 6 fhom(x, ξ) 6 β(1 + |ξ|p) (4.68)

for all x ∈ Ω and all ξ ∈ Rd×N , where, as before, β is the constant in (H4).

In what follows limk,m,n

:= limk

limm

limn

with obvious generalizations.

4.2.2 Main result when the integrands do not depend on the

macroscopic variable

We assume that f does not depend explicitly on x, namely f : RN ×RN ×Rd×N → [0,∞),

and that it satisfies hypotheses (H3)-(H4). In addition, according to (H1)-(H2), and unless

we specify the contrary, we assume f to be continuous.

For each ε > 0 consider the functional Iε : Lp(Ω; Rd) → [0,∞] defined by

Iε(u) :=

∫

Ωf(x

ε,x

ε2,∇u(x)

)


∞ otherwise.

(4.69)

Our objective is to prove the following result.

Theorem 4.2.2. Under the above assumptions on f the Γ(Lp(Ω))-limit of the family Iεεis given by


Ihom(u) =

∫

Ωfhom(∇u(x)) dx if u ∈W 1,p(Ω; Rd),

∞ otherwise,

where fhom is defined by

fhom(ξ) := limT→∞

infφ

1

TN

∫

(0,T )N

fhom(y, ξ + ∇φ(y)) dy : φ ∈W 1,p0

(

(0, T )N ; Rd)

(4.70)

for all ξ ∈ Rd×N , and where

fhom(y, ξ) := limT→∞

infφ

1

TN

∫

(0,T )N

f(y, z, ξ + ∇φ(z)) dz : φ ∈W 1,p0

(

(0, T )N ; Rd)

(4.71)

for all (y, ξ) ∈ RN × Rd×N .

This result can be seen as a generalization of Theorem 1.9 in Fonseca and Zappale [51]

(for first derivatives), in which, as it is usual for the convex case, it is enough to consider

variations that are periodic in the cell Q. Their multiscale argument (see Subsection 4.2.4

below) does not apply here since, as it is expected in the non convex case, the variations

should be considered to be periodic over an infinite ensemble of cells, as it is seen from

(4.70) and (4.71).

We divide the proof of Theorem 4.2.2 in four steps.

STEP 1. Localization of our functionals (4.69).

We highlight their dependence on the class of bounded, open subsets of RN , denoted by

A0(RN ). As it will be clear from the proofs of Lemmas 4.2.7 and 4.2.8 below, it would

not be sufficient to localize, as in Section 4.1, on any open subset of Ω. Indeed, formulas

(4.70) and (4.71) suggest working in cubes of the type (0, T )N , with T arbitrarily large, not

necessarily contained in Ω.

For each ε > 0 consider Iε : Lp(RN ; Rd) ×A0(RN ) → [0,∞] defined by

Iε(u;A) :=

∫

Af(x

ε,x

ε2,∇u(x)

)


∞ otherwise.

(4.72)


We will prove that the family of functionals Iε(· ;A)ε, with A ∈ A0(Ω), Γ-converges

with respect to the strong Lp(A; Rd)-topology to the functional Ihom(· ;A), where Ihom :

Lp(RN ; Rd) ×A0(RN ) → [0,∞] is given by

Ihom(u;A) =

∫

Afhom(∇u(x)) dx if u ∈W 1,p(A; Rd),

∞ otherwise.

As a consequence, taking A = Ω yields Theorem 4.2.2.

STEP 2. Existence of Γ-convergent subsequences.

Let εn ↓ 0. For every A ∈ A0(RN ) consider the Γ-lower limit of Iεn(· ;A)n for the

Lp(A; Rd)-topology defined for u ∈ Lp(RN ; Rd) by

Iεn(u;A) := infun

lim infn→∞

Iεn(un;A) : un → u in Lp(A; Rd)

.

In view of the p-coercivity condition (H4) it follows that Iεn(u;A) is infinite whenever

u ∈ Lp(RN ; Rd) \W 1,p(A; Rd) for each A ∈ A0(RN ), so it suffices to study the case where

u ∈W 1,p(A; Rd).

By a similar argument to the one used in Section 4.1 it can be seen that there exists a

subsequence εnjj ≡ εjj such that Iεj(· ;A) is the Γ-limit of Iεj

(· ;A)j for each

A ∈ A0(RN ).

STEP 3. Integral representation of the Γ-limit.

Our goal is to study the behavior of Iεj(u; ·) in A(A) for each u ∈ W 1,p(A; Rd) and

A ∈ A0(RN ). Following the proof of Lemma 4.1.13, it is possible to show that Iεj(u; ·) is

a measure on A(A) for all A ∈ A0(RN ). Namely, the following result holds.

Lemma 4.2.3. For each A ∈ A0(RN ) and all u ∈W 1,p(A; Rd), the restriction of Iεj(u; ·)

to A(A) is a Radon measure, absolutely continuous with respect to the N -dimensional

Lebesgue measure.

For the moment, we are not in position to apply Buttazzo-Dal Maso Integral Representation

Theorem (Theorem 2.4.1) because, a priori, the integrand would depend on the open set


A ∈ A0(RN ). The following result prevents this dependence from holding since it leads to

an homogeneous integrand as it will be seen in Lemma 4.2.5 below.

Lemma 4.2.4. For all ξ ∈ Rd×N , y0 and z0 ∈ RN , and δ > 0

Iεj(ξ · ;Q(y0, δ)) = Iεj(ξ · ;Q(z0, δ)).

Proof. Clearly, it suffices to establish the inequality

Iεj(ξ · ;Q(y0, δ)) > Iεj(ξ · ;Q(z0, δ)).

Let wjj ⊂W 1,p0 (Q(y0, δ); R

d), with wj → 0 in Lp(Q(y0, δ); Rd), be such that

Iεj(ξ· ;Q(y0, δ)) = limj→∞

∫

Q(y0,δ)f( x

εj,x

ε2j, ξ + ∇wj(x)

)

dx

(see Proposition 2.5.13). By hypothesis (H4) and the Poincare Inequality, we can suppose

that the sequence wjj is uniformly bounded in W 1,p(Q(y0, δ); Rd). Thus by the Decompo-

sition Lemma, there exists a subsequence of wjj (still denoted by wjj) and a sequence

ujj ⊂W 1,∞0 (Q(y0, δ); R

d) such that uj 0 in W 1,p(Q(y0, δ); Rd),

|∇uj |p is equi-integrable (4.73)

and

LN (y ∈ Q(y0, δ) : uj(y) 6= wj(y)) → 0. (4.74)

Then, in view of (4.73), (4.74) and the p-growth condition (H4),

Iεj(ξ· ;Q(y0, δ)) > lim supj→∞

∫

Q(y0,δ)∩uj=wjf( x

εj,x

ε2j, ξ + ∇uj(x)

)

dx

> lim supj→∞

∫

Q(y0,δ)f( x

εj,x

ε2j, ξ + ∇uj(x)

)

dx (4.75)

For all j ∈ N we writey0 − z0εj

= mεj+ sεj

with mεj∈ ZN and sεj

∈ [0, 1)N ,mεj

εj= θεj

+ lεj(4.76)


with θεj∈ ZN and lεj

∈ [0, 1)N , and we define

xεj:= mεj

εj − ε2j lεj. (4.77)

Note that xεj= y0−z0−εjsεj

−ε2j lεj→ y0−z0 as j → ∞. For all j ∈ N, extend uj by zero to

the whole RN and set vj(x) = uj(x+xεj) for x ∈ Q(z0, δ). Then vjj ⊂W 1,p(Q(z0, δ); R

d)

and vj → 0 in Lp(Q(z0, δ); Rd) because

∫

Q(z0,δ)|vj(x)|pdx =

∫

Q(z0,δ)|uj(x+ xεj

)|pdx

=

∫

Q(z0+xεj,δ)

|uj(x)|pdx

6

∫

Q(y0,δ)|uj(x)|pdx, (4.78)

since uj ≡ 0 outside Q(y0, δ). We also remark that by the translation invariance of the

Lebesgue measure, the sequence |∇vj |pj is equi-integrable. In view of (4.75), (4.76),

(4.77) and (H3),


∫

Q(y0−xεj,δ)f(x+ xεj

εj,x+ xεj

ε2j, ξ + ∇uj(x+ xεj

))

dx

= lim supj→∞

∫

Q(y0−xεj,δ)f( x

εj− εjlεj

,x

ε2j, ξ + ∇vj(x)

)

dx

> lim supj→∞

∫

Q(z0,δ)f( x

εj− εjlεj

,x

ε2j, ξ + ∇vj(x)

)

dx

− lim supj→∞

∫

Q(z0,δ)\Q(y0−xεj,δ)f( x

εj− εjlεj

,x

ε2j, ξ + ∇vj(x)

)

dx.

Since vj ≡ 0 outsideQ(y0−xεj, δ), the p-growth condition (H4) and the fact that LN (Q(z0, δ)\

Q(y0 − xεj, δ)) → 0 yield

lim supj→∞

∫

Q(z0,δ)\Q(y0−xεj,δ)f( x

εj− εjlεj

,x

ε2j, ξ + ∇vj(x)

)

dx

6 lim supj→∞

β(1 + |ξ|p)LN (Q(z0, δ) \Q(y0 − xεj, δ)) = 0,

and therefore


∫

Q(z0,δ)f( x

εj− εjlεj

,x

ε2j, ξ + ∇vj(x)

)

dx. (4.79)


To eliminate the term εj lεjin (4.79), and thus to recover Iεj(ξ· ;Q(z0, δ)), we would like

to apply a uniform continuity argument. Since f is continuous on RN × RN × Rd×N and

separately Q-periodic with respect to its two first variables, by hypothesis (H3), then f is

uniformly continuous on RN × RN ×B(0, λ) for any λ > 0. We define

Rλj := x ∈ Q(z0, δ) : |ξ + ∇vj(x)| 6 λ,

and we note that by Chebyshev’s inequality

LN (Q(z0, δ) \Rλj ) 6 C/λp, (4.80)

for some constant C > 0 independent of λ or j. Thus, in view of (4.79) and the fact that f

is nonnegative,

Iεj(ξ· ;Q(y0, δ)) > lim supλ→∞

lim supj→∞

∫

Rλj

f( x

εj− εjlεj

,x

ε2j, ξ + ∇vj(x)

)

dx.

Denoting by ωλ : [0,∞) → [0,∞) the modulus of continuity of f on RN ×RN ×B(0, λ), we

get that for any x ∈ Rλj∣

∣

∣

∣

∣

f( x

εj,x

ε2j, ξ + ∇vj(x)

)

− f( x

εj− εjlεj

,x

ε2j, ξ + ∇vj(x)

)

∣

∣

∣

∣

∣

6 ωλ(εjlεj).

Then, the continuity of ωλ and the fact that ωλ(0) = 0 yield

Iεj(ξ· ;Q(y0, δ)) > lim supλ→∞

lim supj→∞

∫

Rλj

f( x

εj,x

ε2j, ξ + ∇vj(x)

)

dx− δNωλ(εjlεj)

= lim supλ→∞

lim supj→∞

∫

Rλj

f( x

εj,x

ε2j, ξ + ∇vj(x)

)

dx.

The equi-integrability of |∇vj |p, the p-growth condition (H4) and (4.80), imply that

lim supλ→∞

lim supj→∞

∫

Q(z0,δ)\Rλj

f( x

εj,x

ε2j, ξ + ∇vj(x)

)

dx

6 β lim supλ→∞

supj∈N

∫

Q(z0,δ)\Rλj

(1 + |∇vj(x)|p)dx = 0,

and since vj → 0 in Lp(Q(z0, δ); Rd),


∫

Q(z0,δ)f( x

εj,x

ε2j, ξ + ∇vj(x)

)

dx

> Iεj(ξ· ;Q(z0, δ)).


As a consequence of this lemma, we derive the following result.

Lemma 4.2.5. There exists a continuous function fεj : Rd×N → [0,∞) such that for all

A ∈ A0(RN ) and all u ∈W 1,p(A; Rd),

Iεj(u;A) =

∫

Afεj(∇u(x)) dx.

Proof. Let A ∈ A0(RN ). By Theorem 2.4.1, there exists a Caratheodory function fAεj

:

A× Rd×N → [0,∞) satisfying

Iεj(u;U) =

∫

UfAεj

(x,∇u(x)) dx

for all U ∈ A(A) and all u ∈W 1,p(U ; Rd). Furthermore, for a.e. x ∈ A and all ξ ∈ Rd×N

fAεj(x, ξ) = lim

δ→0

Iεj(ξ· ;Q(x, δ))

δN.

Define fεj : Rd×N → [0,∞) by

fεj(ξ) = limδ→0

Iεj(ξ· ;Q(0, δ))

δN.

As a consequence of Lemma 4.2.4, fAεj(x, ξ) = fεj(ξ) for a.e. x ∈ A and for all ξ ∈ Rd×N ,

and we conclude that

Iεj(u;A) =

∫

Afεj(∇u(x)) dx

holds for all u ∈W 1,p(A; Rd).

STEP 4. Characterization of the Γ-limit.

Our next objective is to show that Iεj(u;A) = Ihom(u;A) for any A ∈ A0(RN ) and all

u ∈ W 1,p(A; Rd). In view of Lemma 4.2.5, we only need to prove that fhom(ξ) = fεj(ξ)

for all ξ ∈ Rd×N , thus it suffices to work with affine functions instead of general Sobolev

functions. In order to estimate fεj from below in terms of fhom, we will need the following

result, close in spirit to Proposition 22.4 in Braides and Defranceschi [19].

Proposition 4.2.6. Let f : RN × RN × Rd×N → [0,∞) be a (not necessarily continuous)

function such that f(x, · , ·) is continuous, f( ·, y; ξ) is measurable, and (H3) and (H4) hold.

Let A be an open, bounded, connected and Lipschitz subset of RN . Given M and η two

positive numbers, and ϕ : [0,∞) → [0,∞] a continuous and increasing function satisfying

ϕ(t)/t → ∞ as t → ∞, there exists ε0 ≡ ε0(ϕ,M, η) > 0 such that for every 0 < ε < ε0,

every a ∈ RN and every u ∈W 1,p(a+A; Rd) with

∫

a+Aϕ(|∇u|p) dx 6 M,

there exists v ∈W 1,p0 (a+A; Rd) with ‖v‖Lp(a+A;Rd) 6 η satisfying

∫

a+Af(

x,x

ε,∇u

)

dx >

∫

a+Afhom(x,∇u+ ∇v) dx− η.

Proof. The proof is divided into two steps. First, we prove this proposition under the

additional hypothesis that a belongs to a compact set of RN . Then, we conclude the result

in its full generality replacing a by its fractional part a− [[a]] and using the periodicity of

the integrands f and fhom.

Step 1. For a ∈ [ − 1, 1]N , the claim of Proposition 4.2.6 holds. Indeed, if not then we

may find ϕ, M and η as above, and sequences εj ↓ 0, ajj ⊂ [−1, 1]N and ujj ⊂W 1,p(aj +A; Rd) with

∫

aj+Aϕ(|∇uj |p) dx 6 M (4.81)

such that, for every j ∈ N

∫

aj+Af

(

x,x

εj,∇uj

)

dx

< infv∈W 1,p

0 (aj+A;Rd)

∫

aj+Afhom(x,∇uj + ∇v) dx : ‖v‖Lp(aj+A;Rd) 6 η

− η. (4.82)

From (4.81) and the Poincare Inequality, up to a translation argument, we can suppose that

the sequence ‖uj‖W 1,p(aj+A;Rd) is uniformly bounded. From this fact and since the set

aj+A is an extension domain (see Theorem 2.2.2), there is no loss of generality in assuming

that ujj is bounded in W 1,p(RN ; Rd) and that, due to (4.81),

supj∈N

∫

RN

ϕ(|∇uj |p) dx 6 M1 (4.83)

for some constant M1 > 0 depending only on M (see the proof of the Extension Theorem

for Sobolev functions, Theorem 1, Section 4.4 in Evans and Gariepy [46]). Passing to a

subsequence, we can also assume that uj u in W 1,p(RN ; Rd). Let B a ball of sufficiently

large radius so that aj+A ⊂ B for all j ∈ N. De La Vallee Poussin criterion (see Proposition

2.2.10) and (4.83) guarantee that the sequence |∇uj |pj is equi-integrable on B. This

implies that there exists δ = δ(η) such that

supj∈N

β

∫

E(1 + |∇u|p + |∇uj |p) dx 6

η

2(4.84)

whenever E is a measurable subset of B satisfying LN (E) 6 δ, and where β is the constant

given in (H4). As aj ⊂ [−1, 1]N we may suppose, without loss of generality, that this

sequence aj → a ∈ [−1, 1]N , and that for fixed 0 < ρ < 1, with ρN ≪ δ, the following holds

for j large enough:

a+ (1 − ρ)A ⊂ aj +A ⊂ a+ (1 + ρ)A,

LN (Sj) 6 δ, where Sj := [aj +A] \ [a+ (1 − ρ)A] ⊂ B,

and ‖uj − u‖Lp(a+(1+ρ)A;Rd) 6 η.

(4.85)

Take now a sequence of cut-off functions ϕj ∈ C∞c (RN ; [0, 1]) such that

ϕj =

1 on a+ (1 − ρ)A,

0 outside aj +A,

and ‖∇ϕj‖L∞(RN ) 6 C/ρ for some constant C > 0. Let wj = ϕju + (1 − ϕj)uj . Then

wj − uj ∈W 1,p0 (aj +A; Rd) and

∫

aj+A|wj − uj |p dx 6

∫

aj+Aϕj |u− uj |p dx 6

∫

a+(1+ρ)A|u− uj |p dx 6 ηp.

Then, taking v := wj−uj as test function in (4.82), it follows from (4.67), (4.84), and (4.85)

that

∫

aj+Af

(

x,x

εj,∇uj

)

dx <

∫

aj+Afhom(x,∇wj) dx− η

6

∫

a+(1−ρ)Afhom(x,∇u) dx

+β

∫

Sj

(

1 + |∇u|p + |∇uj |p +C

ρ|u− uj |p

)

dx− η

6

∫

a+(1−ρ)Afhom(x,∇u) dx− η

2+β C

ρ

∫

Sj

|u− uj |p dx. (4.86)

Since uj → u in Lp(RN ; Rd), by (4.85) and (4.86) we have

lim supj→∞

∫

aj+Af

(

x,x

εj,∇uj

)

dx 6

∫


2, (4.87)

and as uj u in W 1,p(a+ (1 − ρ)A; Rd), by Theorem 1.1 and (4.87) we get∫

a+(1−ρ)Afhom(x,∇u) dx 6 lim inf

j→∞

∫

a+(1−ρ)Af

(

x,x

εj,∇uj

)

dx

6 lim infj→∞

∫

aj+Af

(

x,x

εj,∇uj

)

dx

6

∫


2

which is a contradiction.

Step 2. (General case) Let a ∈ RN . Then a− [[a]] ∈ [−1, 1]N . Given u ∈ W 1,p(a+ A; Rd),

set u(x) := u(x + [[a]]) and thus u ∈ W 1,p(a − [[a]] + A; Rd). Applying Step 1 with η/3,

we get the existence of 0 < ε′0 ≡ ε′0(M,ϕ, η) such that, for all 0 < ε < ε′0, there exist

v ∈W 1,p0 (a− [[a]] +A; Rd) satisfying ‖v‖Lp(a−[[a]]+A;Rd) 6 η/3 and∫

a−[[a]]+Af(

x,x

ε,∇u(x)

)

dx >

∫

a−[[a]]+Afhom(x,∇u(x) + ∇v(x)) dx− η

3.

Setting v(x) := v(x− [[a]]), then v ∈W 1,p0 (a+A; Rd) and ‖v‖Lp(a+A;Rd) 6

η3 6 η. Therefore,

by a change of variables∫

a+Af

(

x,x− [[a]]

ε,∇u(x)

)

dx >

∫

a+Afhom(x,∇u(x) + ∇v(x)) dx− η

3, (4.88)

where we have used condition (H3) and (4.66). Writing

[[a]]

ε=: mε + rε, with mε ∈ ZN and |rε| <

√Nε,

by (H3) the inequality (4.88) reduces to∫

a+Af(

x,x

ε− rε,∇u(x)

)

dx >

∫

a+Afhom(x,∇u(x) + ∇v(x)) dx− η

3. (4.89)

Choose λ > 0 large enough (depending on η) so that

β

∫

|∇u|>λ∩[a+A](1 + |∇u|p) dx 6

η

6. (4.90)

Fixed ρ > 0, by Scorza-Dragoni’s Theorem there exists a compact set Kρ ⊂ a + A with

LN ([a + A] \Kρ) 6 ρ such that f : Kρ × RN × Rd×N → R is continuous. Take ρ ≡ ρ(η)

small enough so that

β

∫

[a+A]\Kρ

(1 + |∇u|p) dx 6η

6. (4.91)

Then, from (4.90), (4.91) and the p-growth condition (H4)∫

a+Af(

x,x

ε− rε,∇u(x)

)

dx 6

∫

|∇u|6λ∩Kρ

f(

x,x

ε− rε,∇u(x)

)

dx+η

3. (4.92)

But since f is Q-periodic in its second variable, then f is uniformly continuous on Kρ ×RN × B(0, λ). Thus, as rε → 0, for any η > 0 there exists ε′′0 ≡ ε′′0(η) > 0 such that for all

ε < ε′′0 and all x ∈ |∇u| 6 λ ∩Kρ,∣

∣

∣f(

x,x

ε− rε,∇u(x)

)

− f(

x,x

ε,∇u(x)

)∣

∣

∣<

η

3LN (A).

Hence∫

|∇u|6λ∩Kρ

f(

x,x

ε− rε,∇u(x)

)

dx 6

∫

|∇u|6λ∩Kρ

f(x,x

ε,∇u) dx+

η

3(4.93)

and, consequently, by (4.92) and (4.93) we have∫

a+Af(

x,x

ε− rε,∇u(x)

)

dx 6

∫

|∇u|6λ∩Kρ

f(

x,x

ε,∇u(x)

)

dx+2η

3. (4.94)

Thus, for all ε < ε0 := minε′0, ε′′0, and as a result of (4.89), (4.94) and the fact that f is

nonnegative,∫

a+Af(

x,x

ε,∇u(x)

)

dx >

∫

a+Afhom(x,∇u(x) + ∇v(x)) dx− η.

We are now in position to prove that fεj = fhom.


Lemma 4.2.7. For all ξ ∈ Rd×N , fhom(ξ) 6 fεj(ξ).

Proof. By Lemma 4.2.5, given ξ ∈ Rd×N

fεj(ξ) =

∫

Qfεj(ξ) dx = Iεj(ξ· ;Q). (4.95)

Let wj ⊂W 1,p0 (Q; Rd) be a sequence such that wj → 0 in Lp(Q; Rd) and

Iεj(ξ · ;Q) = limj→∞

∫

Qf

(

x

εj,x

εj2; ξ + ∇wj(x)

)

dx

(Proposition 2.5.13). Following the same argument as in Lemma 4.2.4, by the Decomposition

Lemma, there is no loss of generality in assuming that |∇wj |p is equi-integrable. Thus,

from De La Vallee Poussin criterion (see Proposition 2.2.10) there exists an increasing

continuous function ϕ : [0,∞) → [0,∞] satisfying ϕ(t)/t→ ∞ as t→ ∞ and such that

supj∈N

∫

Qϕ(|∇wj |p) dx 6 1.

Changing variables

fεj(ξ) = limj→∞

1

TNj

∫

(0,Tj)N

f

(

x,x

εj, ξ + ∇zj(x)

)

dx

and

supj∈N

1

TNj

∫

(0,Tj)N

ϕ(|∇zj |p) dx 6 1, (4.96)

where we set Tj := 1/εj and zj(x) := Tjwj(x/Tj) with zj ∈ W 1,p0 ((0, Tj)

N ; Rd). For any

j ∈ N define Ij :=

1, ..., [[Tj ]]N

, and for i ∈ Ij take aji ∈ ZN such that

⋃

i∈Ij

(aji +Q) ⊆ (0, Tj)N . (4.97)

Thus

fεj(ξ) > lim supj→∞

1

TNj

∑

i∈Ij

∫

aji +Q

f

(

x,x

εj, ξ + ∇zj(x)

)

dx. (4.98)

Let M > 2 and η > 0. For j ∈ N define

IMj :=

i ∈ Ij :

∫

aji +Q

ϕ(|∇zj |p) dx 6 M

.

We note that for any M > 2, there exists j(M) ∈ N such that for all j > n(M) sufficiently

large so that Tj > M , IMj 6= ∅. In fact, otherwise we could find M > 2 and a subsequence

jk ∈ N satisfying∫

ajki +Q

ϕ(|∇zjk |p)dx > M,

for all i ∈ Ijk . Summation in i and (4.97) would yield

∫

(0,Tjk)N

ϕ(|∇zjk |p)dx > M [[Tjk ]]N

which is in contradiction with (4.96). We also note that in view of (4.96)

Card(Ij \ IMj )M 6∑

i∈Ij\IMj

∫

aji +Q

ϕ(|∇zj |p) dx 6

∫

(0,Tj)N

ϕ(|∇zj |p) dx 6 TNj ,

and so

Card(Ij \ IMj ) 6TNjM

. (4.99)

By Lemma 4.2.6 there exists ε0 ≡ ε0(M,η) such that, for any j large enough satisfying 0 6

εj < ε0 and for any i ∈ IMj , we can find vj,M,ηi ∈W 1,p

0 (aji+Q; Rd) with ‖vj,M,ηi ‖

Lp(aji +Q;Rd)

6

η and

∫

aji +Q

f

(

x,x

εj, ξ + ∇zj(x)

)

dx >

∫

aji +Q

fhom

(

x; ξ + ∇zj + ∇vj,M,ηi

)

dx− η.

Consequently, for j large enough

∑

i∈Ij

∫

aji +Q

f

(

x,x

εj, ξ + ∇zj(x)

)

dx >∑

i∈IMj

∫

aji +Q

fhom(x, ξ + ∇zj + ∇vj,M,ηi ) dx

−η card(IMj ).

As Card(IMj ) 6 [[Tj ]]N , dividing by TNj and passing to the limit when j → ∞ we obtain

lim supj→∞

1

TNj

∑

i∈Ij

∫

aji +Q

f

(

x,x

εj, ξ + ∇zj(x)

)

dx

> lim supj→∞

1

TNj

∑

i∈IMj

∫

aji +Q

fhom(x, ξ + ∇zj + ∇vj,M,ηi ) dx− η. (4.100)


Hence, from (4.98) and (4.100),

fεj(ξ) > lim supM,η,j

1

TNj

∑

i∈IMj

∫

aji +Q

fhom(x, ξ + ∇φj,M,η) dx (4.101)

where φj,M,η ∈W 1,p0 ((0;Tj)

N ; Rd) is defined by

φj,M,η(x) :=

zj(x) + vj,M,ηi (x) if x ∈ aji +Q and i ∈ IMj ,

zj(x) otherwise.

Now, in view of the definition of φj,M,η, the p-growth condition (4.67) and (4.99),

1

TNj

∑

i∈Ij\IMj

∫

aji +Q

fhom(x; ξ + ∇φj,M,η) dx

=1

TNj

∑

i∈Ij\IMj

∫

aji +Q

fhom(x; ξ + ∇zj) dx

6β

TNj

∑

i∈Ij\IMj

∫

aji +Q

(1 + |∇zj |p) dx

6β

M+

β

TNj

∑

i∈Ij\IMj

∫

aji +Q

|∇zj |p dx

6β

M+ β

∫

⋃

i∈Ij\IMj

1Tj

(aji +Q)

|∇wj |p dx. (4.102)

By (4.99)

LN

⋃

i∈Ij\IMj

1

Tj(aji +Q)

6

1

M.

Consequently, in view of the equi-integrability of |∇wj |p and (4.102), we get

lim supM,η,j

1

TNj

∑

i∈Ij\IMj

∫

aji +Q

fhom(x; ξ + ∇φj,M,η) dx = 0. (4.103)

Therefore, (4.101) and (4.103) imply


fεj(ξ) > lim supM,η,j

1

TNj

∑

i∈Ij

∫

aji +Q

fhom(x, ξ + ∇φj,M,η) dx

= lim supM,η,j

1

TNj

∫

(0,Tj)N

fhom(x, ξ + ∇φj,M,η) dx, (4.104)

because by definition of φj,M,η and the p-growth property of fhom, (4.67),

1

TNj

∫

(

0,TNj

)

\

⋃

i∈Ij

(aji +Q)

fhom(x, ξ + ∇φj,M,η) dx

=1

TNj

∫

(

0,TNj

)

\

⋃

i∈Ij

(aji +Q)

fhom(x, ξ + ∇zj) dx

6β

TNj

∫

(

0,TNj

)

\

⋃

i∈Ij

(aji +Q)

(1 + |∇zj |p) dx

= β

∫

Q\

⋃

i∈Ij

1Tj

(aji +Q)

(1 + |∇wj |p) dx.

and consequently, the equi-integrability of |∇wj |p and the fact that

LN

Q \

⋃

i∈Ij

1

Tj(aji +Q)

→ 0

as j → ∞ yield

lim supM,η,j

1

TNj

∫

(

0,TNj

)

\

⋃

i∈Ij

(aji +Q)

fhom(x, ξ + ∇φj,M,η) dx = 0.

Hence by (4.104) and (4.70) we get that

fεj(ξ) > fhom(ξ).

Let us now prove the converse inequality.

Lemma 4.2.8. For all ξ ∈ Rd×N , fhom(ξ) > fεj(ξ).


Proof. In view of (4.70), for δ > 0 fixed take T ≡ Tδ ∈ N, with Tδ → ∞ as δ → 0, and let

φ ≡ φδ ∈W 1,p0 ((0, T )N ; Rd) be such that

fhom(ξ) + δ >1

TN

∫

(0,T )N

fhom(x; ξ + ∇φ(x)) dx. (4.105)

By Theorem 1.1 and Proposition 2.5.13, there exists a sequence φj ⊂ W 1,p0 ((0, T )N ; Rd)

with φj → φ in Lp((0, T )N ; Rd) such that

∫

(0,T )N

fhom(x; ξ + ∇φ(x)) dx = limj→∞

∫

(0,T )N

f

(

x,x

εj, ξ + ∇φj(x)

)

dx. (4.106)

Further, in view of the Decomposition Lemma, we can assume – upon extracting a subse-

quence still denoted by φj – |∇φj |p to be equi-integrable. Fix j ∈ N such that εj ≪ 1.

For all i ∈ ZN let aji ∈ εjZN ∩ [i(T + 1), εj)

N (uniquely defined).

T=1 N=2, ,i=(0,1)

jε

jε

aji

1 2

In particular, the cubes Q(aji , T ) are not overlapping because if i, k ∈ ZN with i 6= k, then

|i− k| > 1 and thus |aji − akj | > T . Set

φj(x) :=

φj(x− aji ) if x ∈ Q(aji , T ) and i ∈ ZN ,

0 otherwise.

Then φj ∈W 1,p(RN ; Rd). Let Ij := i ∈ ZN : (0, T/εj)N ∩Q(aji , T ) 6= ∅. Note that

Card(Ij) 6

([[

1

εj

]]

+ 1

)N

. (4.107)

If ψj(x) := εjφj(x/εj), then ψj → 0 in Lp((0, T )N ; Rd), as j → ∞, because

∫

(0,T )N

|ψj(x)|pdx = εpj

∫

(0,T )N

∣

∣

∣

∣

φj

(

x

εj

)∣

∣

∣

∣

p

dx

= εp+Nj

∫

(0,T/εj)N

|φj(x)|pdx

6 εp+Nj

∑

i∈Ij

∫

Q(aji ,T )

|φj(x− aji )|pdx

= εp+Nj Card(Ij)

∫

(0,T )N

|φj(x)|pdx

6 εp+Nj

([[

1

εj

]]

+ 1

)N ∫

(0,T )N

|φj(x)|pdx→ 0,

where we have used the fact that φj ≡ 0 on (0, T/εj)N \⋃i∈Ij

Q(aji , T ) and that

supj∈N

‖φj‖Lp((0,T )N ;Rd) <∞

by the Poincare Inequality and (4.106). Consequently,

Iεj(ξ· ; (0, T )N ) 6 lim infj→∞

∫

(0,T )N

f

(

x

εj,x

ε2j, ξ + ∇ψj(x)

)

dx

= lim infj→∞

∫

(0,T )N

f

(

x

εj,x

ε2j, ξ + ∇φj

(

x

εj

)

)

dx

= lim infj→∞

εNj

∫

(0,T/εj)N

f

(

x,x

εj; ξ + ∇φj(x)

)

dx. (4.108)

Note that

εNj LN

(

0,T

εj

)N

\⋃

i∈Ij

Q(aji , T )

= LN

(0, T )N \⋃

i∈Ij

Q(aji , εjT )

Thus


εNj LN

(

0,T

εj

)N

\⋃

i∈Ij

Q(aji , T )

6 LN

(0, T )N \

⋃

i∈Ij :Q(aji ,εjT )⊂(0,T )N

Q(aji , εjT )

= TN − Card(

i ∈ Ij : Q(aji , εjT ) ⊂ (0, T )N)

εNj TN

6 TN

(

1 − εNj

[[

T

εj(T + 1)

]]N)

.

Hence, from inequality (4.108) and the p-growth condition (H4) it follows that

Iεj(ξ· ; (0, T )N ) 6 lim infj→∞

εNj

∑

i∈Ij

∫

Q(aji ,T )

f

(

x,x

εj, ξ + ∇φj(x)

)

dx

+β(1 + |ξ|p)LN

(

0,T

εj

)N

\⋃

i∈Ij

Q(aji , T )

6 lim infj→∞

εNj∑

i∈Ij

∫

Q(aji ,T )

f

(

x,x

εj, ξ + ∇φj(x)

)

dx

+β(1 + |ξ|p)TN(

1 −(

T

T + 1

)N)

. (4.109)

By a change of variables, for all i ∈ Ij

∫

Q(aji ,T )

f

(

x,x

εj, ξ + ∇φj(x)

)

dx

=

∫

(0,T )N

f

(

x+ aji ,x+ ajiεj

, ξ + ∇φj(x))

dx

=

∫

(0,T )N

f

(

x+ aji − i(T + 1),x

εj, ξ + ∇φj(x)

)

dx, (4.110)

where we have used (H3), the fact that T ∈ N and aji/εj ∈ ZN . In order to apply a uniform

continuity argument and recover (4.106) we define

Rλj := x ∈ (0, T )N : |ξ + ∇φj(x)| 6 λ


and we observe that, according to Chebyshev’s inequality,

LN ((0, T )N \Rλj ) 6 C/λp,

for some constant C > 0 which does not depend on j and λ. Then by (4.109) and (4.110)

Iεj(ξ· ; (0, T )N )

6 lim infλ→∞

lim infj→∞

∑

i∈Ij

εNj

∫

Rλj

f

(

x+ aji − i(T + 1),x

εj, ξ + ∇φj(x)

)

dx

+β(1 + |ξ|p)TN(

1 −(

T

T + 1

)N)

. (4.111)

Indeed, the p-growth condition (H4) and the equi-integrability of |∇φj |pj imply that

lim supλ→∞

lim supj→∞

∑

i∈Ij

εNj

∫

(0,T )N\Rλj

f

(

x+ aji − i(T + 1),x

εj, ξ + ∇φj(x)

)

dx

6 lim supλ→∞

lim supj→∞

β εNj Card(Ij)

∫

(0,T )N\Rλj

(1 + |ξ + ∇φj(x)|p)dx

6 lim supλ→∞

C1LN ((0, T )N \Rλj ) + C2 supj∈N

∫

(0,T )N\Rλj

|∇φj(x)|pdx

= 0.

As f is continuous and separately periodic in its two first variables, f is uniformly continuous

on RN × RN × B(0, λ) for any λ > 0. Let ωλ : [0,∞) → [0,∞) the modulus of continuity

of f on RN × RN ×B(0, λ). Then, for all x ∈ Rλj ,

∣

∣

∣

∣

f

(

x,x

εj, ξ + ∇φj(x)

)

− f

(

x+ aji − i(T + 1),x

εj, ξ + ∇φj(x)

)∣

∣

∣

∣

6 ωλ(|aji − i(T + 1)|) 6 ωλ(εj). (4.112)

In view of (4.112), (4.107), and since ωλ is continuous and satisfies ωλ(0) = 0, we get

lim infλ→∞

lim infj→∞

εNj Card(Ij)

∫

Rλj

f

(

x,x

εj, ξ + ∇φj(x)

)

dx+ TNωλ(εj)

6 lim infλ→∞

lim infj→∞

(1 + εNj )

∫

Rλj

f

(

x,x

εj, ξ + ∇φj(x)

)

dx+ TNωλ(εj)

6 lim infλ→∞

lim infj→∞

∫

Rλj

f

(

x,x

εj, ξ + ∇φj(x)

)

dx

6 lim infj→∞

∫

(0,T )N

f

(

x,x

εj, ξ + ∇φj(x)

)

dx.


Consequently by (4.105), (4.106), (4.111) and Lemma 4.2.5,

fεj(ξ) 6 fhom(ξ) + δ + β(1 + |ξ|p)(

1 −(

T

T + 1

)N)

.

The result follows by letting δ tend to zero.

Proof of Theorem 4.2.2. From Lemma 4.2.7 and Lemma 4.2.8, we conclude that fhom(ξ) =

fεj(ξ) for all ξ ∈ Rd×N . As a consequence, Iεj(u;A) = Ihom(u;A) for all A ∈ A0(RN )

and all u ∈W 1,p(A; Rd). Since the Γ-limit does not depend upon the extracted subsequence,

Remark 2.5.9 implies that the whole sequence Iε(· ;A) Γ(Lp(A))-converges to Ihom(· ;A).

In particular,

Γ(Lp(A))- limε→0

Iε(u;A) = Ihom(u;A) :=

∫

Afhom(∇u) dx.

4.2.3 The general case

Our objective now is to prove Theorem 4.2.1.

STEP 1. Existence and integral representation of the Γ-limit.

The idea in this case is to freeze the macroscopic variable and to use Theorem 4.2.2 through

a blow up argument. This leads us to work on small cubes centered at convenient Lebesgue

points of Ω which, contrary to the homogeneous case, allow us to localize our functionals

on A(Ω).

We define Iε : Lp(Ω; Rd) ×A(Ω) → [0,∞] by

Iε(u;A) :=

∫

Af(

x,x

ε,x

ε2,∇u(x)

)


∞ otherwise,

and we introduce the functional Ihom : Lp(Ω; Rd) ×A(Ω) → [0,∞]

Ihom(u;A) :=

∫

Afhom(x,∇u(x)) dx if u ∈W 1,p(A; Rd),

∞ otherwise.


Given εn ↓ 0 and A ∈ A(Ω), consider the Γ-lower limit of Iεn(· ;A)n for the Lp(A; Rd)-

topology defined, for u ∈W 1,p(Ω; Rd), by

Iεn(u;A) := inf

lim infj→∞

Iεn(uj ;A) : un → u in Lp(A; Rd)

.

Due to the p-coercivity condition in (H4), to prove Theorem 4.2.1 it suffices to show that

for all u ∈W 1,p(Ω; Rd)


Iε(u) =

∫

Ωfhom(x,∇u(x)) dx.

As in Section 4.1 there exists a subsequence εjj ≡ εnjj such that for any A ∈ A(Ω),

Iεj(· ;A) is the Γ(Lp(A))-limit of Iεj(· ;A) and, for all u ∈ W 1,p(Ω; Rd), the set function

Iεj(u; ·) is the restriction of a Radon measure to A(Ω). Furthermore, from the integral

representation Theorem 2.4.1, we have

Lemma 4.2.9. There exists a Caratheodory function fεj : Ω × Rd×N → R, quasiconvex

in its second variable, satisfying the same coercivity and growth conditions as f , such that

Iεj(u;A) =

∫

Afεj(x,∇u(x)) dx

for every A ∈ A(Ω) and u ∈W 1,p(Ω; Rd). Moreover, for all ξ ∈ Rd×N and a.e. x ∈ Ω,

fεj(x, ξ) = limδ→0

Iεj(ξ· ;Q(x, δ))

δN.


As before we only need to prove that fεj(x, ξ) = fhom(x, ξ) for a.e. x and all ξ. For this

purpose let L be the set of Lebesgue points x0 for all functions fεj(· , ξ) and fhom(· , ξ), for

all ξ ∈ Qd×N . We have LN (Ω \ L) = 0 and we will first show in Lemma 4.2.10 and 4.2.11

below that the equality fεj(x, ξ) = fhom(x, ξ) holds for all x ∈ L and all ξ ∈ Qd×N . By

definition of the set L it is enough to show that

∫

Q(x0,δ)fεj(x, ξ) dx =

∫

Q(x0,δ)fhom(x, ξ) dx, (4.113)

for every x0 ∈ L and each δ > 0 small enough so that Q(x0, δ) ∈ A(Ω).

Lemma 4.2.10. For all ξ ∈ Qd×N and all x0 ∈ L,∫

Q(x0,δ)fεj(x, ξ) dx >

∫

Q(x0,δ)fhom(x, ξ) dx.

Proof. Let ξ ∈ Qd×N and x0 ∈ L. From Lemma 4.2.9, we have∫

Q(x0,δ)fεj(x, ξ) dx = Iεj(ξ· ;Q(x0; δ)).

Let uj ⊂ W 1,p(Q(x0, δ); Rd) be a recovery sequence for Iεj(ξ· ;Q(x0; δ)), that is, a

sequence uj such that uj → 0 in Lp(Q(x0, δ); Rd) and

Iεj(ξ· ;Q(x0; δ)) = limj→∞

∫

Q(x0,δ)f

(

x,x

εj,x

ε2j, ξ + ∇uj(x)

)

dx.

As before, the Decomposition Lemma lets us assume that |∇uj |p is equi-integrable. We

split Q(x0, δ) into hN small disjoint cubes Qi,h such that

Q(x0, δ) =hN⋃

i=1

Qi,h and LN (Qi,h) = (δ/h)N . (4.114)

Then

∫

Q(x0,δ)fεj(x, ξ) dx = lim

h→∞limj→∞

hN∑

i=1

∫

Qi,h

f

(

x,x

εj,x

ε2j, ξ + ∇uj(x)

)

dx.

Let η > 0. By Scorza-Dragoni’s Theorem, there exists a compact set Kη ⊂ Ω such that

LN (Ω \Kη) < η, (4.115)

and the restriction of f to Kη × RN × RN × Rd×N is a continuous function. Given λ > 0,

we introduce

Rλj := x ∈ Ω : |ξ + ∇uj(x)| 6 λ,

for all j ∈ N and we note that due to Chebyshev’s inequality, we have

LN (Ω \Rλj ) 6C

λp, (4.116)

for some constant C > 0 independent of j and λ. Then

∫

Q(x0,δ)fεj(x, ξ) dx > lim sup

λ,η,h,j

hN∑

i=1

∫

Qi,h∩Kη∩Rλj

f

(

x,x

εj,x

ε2j, ξ + ∇uj(x)

)

dx.


In view of condition (H3), f is uniformly continuous on Kη×RN ×RN ×B(0, λ). Denoting

by ωη,λ : [0,∞) → [0,∞) the modulus of continuity of f on Kη × RN × RN × B(0, λ), for

every (x, x′) ∈ [Qi,h ∩Kη ∩Rλj ] × [Qi,h ∩Kη],∣

∣

∣

∣

∣

f

(

x,x

εj,x

ε2j, ξ + ∇uj(x)

)

− f

(

x′,x

εj,x

ε2j, ξ + ∇uj(x)

)∣

∣

∣

∣

∣

6 ωη,λ(|x− x′|)

6 ωη,λ

(√Nδ

h

)

. (4.117)

From (4.114) and (4.117), after integrating in (x, x′) over [Qi,h ∩Kη ∩Rλj ]× [Qi,h ∩Kη], we

get, since ωη,λ is continuous and satisfies ωη,λ(0) = 0,

hN∑

i=1

hN

δN

∫

Qi,h∩Kη

∫

Qi,h∩Kη∩Rλj

∣

∣

∣

∣

∣

f

(

x,x

εj,x

ε2j, ξ + ∇uj(x)

)

−f(

x′,x

εj,x

ε2j, ξ + ∇uj(x)

)∣

∣

∣

∣

∣

dx

dx′

6 δNωη,λ

(√Nδ

h

)

−−−→h→∞

0,

uniformly in j ∈ N, for all η > 0 and λ > 0. Hence, by Fubini’s Theorem∫


> lim supλ,η,h,j

hN

δN

hN∑

i=1

∫

Qi,h∩Kη∩Rλj

∫

Qi,h∩Kη

f

(

x′,x

εj,x

ε2j, ξ + ∇uj(x)

)

dx′

dx. (4.118)

However, as a consequence of (H4) and (4.115) we have that for all λ > 0,

hN

δN

hN∑

i=1

∫

Qi,h∩Kη∩Rλj

∫

Qi,h\Kη

f

(

x′,x

εj,x

ε2j, ξ + ∇uj(x)

)

dx′

dx

6 βhN

δN

hN∑

i=1

δN

hN(1 + λp)LN (Qi,h \Kη)

6 β(1 + λp)LN (Ω \Kη)

6 β(1 + λp)η −−−→η→0

0 (4.119)

uniformly in j ∈ N and h ∈ N, and similarly

hN

δN

hN∑

i=1

∫

Qi,h∩Rλj \Kη

∫

Qi,h

f

(

x′,x

εj,x

ε2j, ξ + ∇uj(x)

)

dx′

dx 6 β(1 + λp)η −−−→η→0

0,

(4.120)


uniformly in j ∈ N and h ∈ N. Moreover, (4.114) and (4.116), together with the equi-

integrability of |∇uj |p, yield

supj,h,η

hN

δN

hN∑

i=1

∫

Qi,h\Rλj

∫

Qi,h

f

(

x′,x

εj,x

ε2j, ξ + ∇uj(x)

)

dx′

dx

6 supj∈N

β

∫

Q(x0,δ)\Rλj

(1 + |∇uj(x)|p) dx −−−→λ→∞

0. (4.121)

Finally, (4.118)-(4.121) and Fubini’s Theorem lead to

∫


> lim suph→∞

lim supj→∞

hN

δN

hN∑

i=1

∫

Qi,h

∫

Qi,h

f

(

x′,x

εj,x

ε2j, ξ + ∇uj(x)

)

dx

dx′

> lim suph→∞

hN

δN

hN∑

i=1

∫

Qi,h

lim infj→∞

∫

Qi,h

f

(

x′,x

εj,x

ε2j, ξ + ∇uj(x)

)

dx

dx′, (4.122)

where we have used Fatou’s Lemma. Fix x′ ∈ Qi,h such that fhom(x′, ξ) is well defined and

apply Theorem 4.2.2 to the continuous function (y, z, ξ) 7→ f(x′, y, z, ξ). Since uj → 0 in

Lp(Q(x0, δ); Rd), we can use the Γ-lim inf inequality to get

lim infj→∞

∫

Qi,h

f

(

x′,x

εj,x

ε2j, ξ + ∇uj(x)

)

dx >δN

hNfhom(x′, ξ).

Then, in view of (4.122) we conclude (4.113), that is,

∫

Q(x0,δ)fεj(x, ξ) dx >

∫


Lemma 4.2.11. For all ξ ∈ Qd×N and all x0 ∈ L,

∫

Q(x0,δ)fεj(x, ξ) dx 6

∫


Proof. As in Lemma 4.2.10, we decompose Q(x0, δ) into hN small disjoints cubes Qi,h such

that

Q(x0, δ) =hN⋃

i=1

Qi,h and LN (Qi,h) = (δ/h)N .

Since f and fhom are Caratheodory functions, by Scorza-Dragoni’s Theorem for each η > 0,

we can find a compact set Kη ⊂ Q(x0, δ) such that

LN (Q(x0, δ) \Kη) < η, (4.123)

f is continuous on Kη × RN × RN × Rd×N and fhom is continuous on Kη × Rd×N . Let

Ih,η :=

i ∈ 1, · · · , hN : Kη ∩Qi,h 6= ∅

.

For i ∈ Ih,η, choose xh,ηi ∈ Kη ∩ Qi,h. Theorem 4.2.2, together with e.g. Theorem 21.1

in Dal Maso [19], implies the existence of a sequence uj,h,ηi ⊂ W 1,p0 (Qi,h; R

d) such that

uj,h,ηi → 0 in Lp(Qi,h; Rd) as j → ∞ and

∫

Qi,h

fhom(xh,ηi , ξ) dx =( δ

h

)Nfhom(xh,ηi , ξ) = lim

j→∞

∫

Qi,h

f

(

xh,ηi ,x

εj,x

ε2j, ξ + ∇uj,h,ηi

)

dx.

Set

uηj (x) :=

uj,h,ηi (x) if x ∈ Qi,h and i ∈ Ih,η,

0 otherwise.

(4.124)

Then uηj ⊂W 1,p0 (Q(x0, δ); R

d), uηj → 0 in Lp(Q(x0, δ); Rd) as j → ∞ and

lim infh→∞

∑

i∈Ih,η

∫

Qi,h

fhom(xh,ηi , ξ) dx

= lim infh→∞

limj→∞

∑

i∈Ih,η

∫

Qi,h

f

(

xh,ηi ,x

εj,x

ε2j, ξ + ∇uηj

)

dx. (4.125)

In view of (4.68) and (4.123) we have

suph∈N

∑

i∈Ih,η

∫

Qi,h\Kη

fhom(xh,ηi , ξ) dx 6 β(1 + |ξ|p)LN (Q(x0, δ) \Kη) −−−→η→0

0, (4.126)

thus from (4.125) and (4.126),

lim infη→0

lim infh→∞

∑

i∈Ih,η

∫

Qi,h∩Kη


> lim infη,h,j

∑

i∈Ih,η

∫

Qi,h∩Kη

f

(

xh,ηi ,x

εj,x

ε2j, ξ + ∇uηj

)

dx. (4.127)

Since fhom(· , ξ) is continuous on Kη, it is uniformly continuous. Thus, denoting by ωη its

modulus of continuity on Kη, we have for all x ∈ Qi,h ∩Kη,

|fhom(x, ξ) − fhom(xh,ηi , ξ)| 6 ωη(|x− xh,ηi |) 6 ωη

(√Nδ

h

)

−−−→h→∞

0. (4.128)

Since Qi,h ∩Kη = ∅ for i 6∈ Ih,η, then by (4.123), (4.128) and (4.127) we get

∫

Q(x0,δ)fhom(x, ξ) dx

= limη→0

∫

Kη

fhom(x, ξ) dx

= lim infη→0

lim infh→∞

∑

i∈Ih,η

∫

Qi,h∩Kη

fhom(x, ξ) dx

= lim infη→0

lim infh→∞

∑

i∈Ih,η

∫

Qi,h∩Kη


> lim infλ,η,h,j

∑

i∈Ih,η

∫

Qi,h∩Kη∩Rλj,η

f

(

xh,ηi ,x

εj,x

ε2j, ξ + ∇uηj

)

dx, (4.129)

where Rλj,η :=

x ∈ Q(x0, δ) : |ξ + ∇uηj (x)| 6 λ

. From (4.125) and the fact that uηj ≡ 0 on

Q(x0, δ) \⋃

i∈Ih,ηQi,h, we get

supj∈N,η>0

∫

Q(x0,δ)|∇uηj |pdx <∞. (4.130)

In particular, according to Chebyshev’s inequality, we have

LN (Q(x0, δ) \Rλj,η) 6C

λp, (4.131)

for some constant C > 0 independent of j, η and λ. Since f is continuous on Kη × RN ×RN × Rd×N and separately Q-periodic in its second and third variable (see assumption

(H3)), it is uniformly continuous on Kη × RN × RN × B(0, λ). Thus, denoting by ωη,λ its

modulus of continuity on Kη × RN × RN ×B(0, λ), we have for all x ∈ Qi,h ∩Kη ∩Rλj,η,∣

∣

∣

∣

∣

f

(

x,x

εj,x

ε2j, ξ + ∇uηj (x)

)

− f

(

xh,ηi ,x

εj,x

ε2j, ξ + ∇uηj (x)

)∣

∣

∣

∣

∣

6 ωη,λ(|x− xh,ηi |)

6 ωη,λ

(√Nδ

h

)

−−−→h→∞

0.


Then, according to (4.129) and the fact that Qi,h ∩Kη = ∅ for i 6∈ Ih,η,∫

Q(x0,δ)fhom(x, ξ) dx

> lim infλ,η,h,j

∑

i∈Ih,η

∫

Qi,h∩Kη∩Rλj,η

f

(

x,x

εj,x

ε2j, ξ + ∇uηj

)

dx,

= lim infλ,η,j

∫

Kη∩Rλj,η

f

(

x,x

εj,x

ε2j, ξ + ∇uηj

)

dx.

In view of the p-growth condition (H4), (4.123) and the definition of Rλj,η,

supj∈N

∫

Rλj,η\Kη

f

(

x,x

εj,x

ε2j, ξ + ∇uηj

)

dx 6 β(1 + λp)η −−−→η→0

0,

so∫

Q(x0,δ)fhom(x, ξ) dx > lim inf

λ,η,j

∫

Rλj,η

f

(

x,x

εj,x

ε2j, ξ + ∇uηj

)

dx.

Let λk ր ∞ and ηk ↓ 0. By a diagonalization procedure, it is possible to find a subsequence

jk of j such that, upon setting vk := uηk

jkand Rk := Rλk

jk,ηk, then vk ∈W 1,p

0 (Q(x0, δ); Rd),

vk → 0 in Lp(Q(x0, δ); Rd) and

∫


k→∞

∫

Rk

f

(

x,x

εjk,x

ε2jk, ξ + ∇vk

)

dx.

By (4.130) and the Poincare Inequality, the sequence vk is bounded in W 1,p(Q(x0, δ); Rd)

uniformly with respect to k ∈ N so that, according to the Decomposition Lemma, it is no

loss of generality to assume that |∇vk|p is equi-integrable. It turns out, in view of the

p-growth condition (H4) and (4.131) that

∫

Q(x0,δ)\Rk

f

(

x,x

εjk,x

ε2jk, ξ + ∇vk

)

dx 6 β supl∈N

∫

Q(x0,δ)\Rk

(1 + |∇vl|p) dx −−−→k→∞

0.

Thus, using the Γ-lim inf inequality,

∫


k→∞

∫

Q(x0,δ)f

(

x,x

εjk,x

ε2jk, ξ + ∇vk

)

dx

>

∫

Q(x0,δ)fεj(x, ξ) dx.


Proof of Theorem 4.2.1. As a consequence of Lemma 4.2.10 and 4.2.11, we have fhom(x, ξ) =

fεj(x, ξ) for all x ∈ L and all ξ ∈ Qd×N . By Lemma 4.2.9 and the fact that fhom is a

Caratheodory function we obtain fhom(x, ξ) = fεj(x, ξ) for all a.e. x ∈ Ω and all ξ ∈ Rd×N .

Since the result does not depend upon the specific choice of the subsequence, we get by

Remark 2.5.9 that the whole sequence Iε(· ;A) Γ(Lp(A))-converges to Ihom(· ;A). Taking

A = Ω we conclude the proof of Theorem 4.2.1.

4.2.4 Some remarks in the convex case

As in Section 4.1, we note that under the additional hypothesis that f(x, y, z, ·) is convex

for all x and all (y, z) equality (4.62) and (4.63) simplify to read

fhom(x, ξ) = infφ

∫

Qfhom(x, y, ξ + ∇φ(y)) dy, φ ∈W 1,p

0 (Q; Rd)

for all ξ ∈ Rd×N and all x ∈ Ω, and

fhom(x, y, ξ) = infφ

∫

Qf(x, y, z, ξ + ∇φ(z)) dz, φ ∈W 1,p

0 (Q; Rd)

for all x ∈ Ω and all (y, ξ) ∈ RN × Rd×N (see Muller [66] and Braides and Defranceschi

[19]).

Our objective here is to present an alternative proof of Lemma 4.2.11 in the convex case.

Namely, we would like to show that fεj(x0, ξ) 6 fhom(x0, ξ) a.e. x0 ∈ Ω and all ξ ∈ Rd×N ,

without appealing to Theorem 4.2.2. For this purpose, let us denote by S (resp. S) a

countable set of functions in C∞c (Q; Rd) (resp. C∞

c (Q×Q; Rd)) dense in W 1,p0 (Q; Rd) (resp.

Lp(Q;W 1,p0 (Q; Rd))). Define L to be the set of Lebesgue points x0 for all functions

fεj(· , ξ), fhom(· , ξ)

and

x→∫

Q

∫

Qf(x, y, z, ξ + ∇yφ(y) + ∇zψ(y, z)) dy dz,


with φ ∈ S, ψ ∈ S. Note that LN (Ω \ L) = 0. If x0 ∈ L and ξ ∈ Qd×N , then


1

δN

∫


= limδ→0

Iεj(ξ · ;Q(x0, δ))

δN.

(4.132)

Given m ∈ N consider φm ∈ S such that

fhom(x0, ξ) +1

m>

∫

Qfhom(x0, y, ξ + ∇φm(y)) dy. (4.133)

Then by Theorem 2.1.9, and following a similar argument as in Lemma 4.6 in Fonseca and

Zappale [51], there exist Φm ∈ Lp(

Q;W 1,p0 (Q; Rd)

)

such that

fhom(x0, y, ξ + ∇φm(y)) +1

m>

∫

Qf(x0, y, z, ξ + ∇yφm(y) + ∇zΦm(y, z)) dz.

We now choose Φm,k ∈ S such that

‖Φm,k − Φm‖Lp(Q;W 1,p0 (Q;Rd))

−−−→k→∞

0, (4.134)

and we extend φm and Φm,k periodically to RN and RN×RN , respectively. For each x ∈ RN

define

ujm,k(x) := ξ · x+ εjφm

( x

εj

)

+ ε2jΦm,k

( x

εj,x

ε2j

)

and consider δ > 0 small enough so that Q(x0, δ) ∈ A(Ω). For fixed m and k we have

ujm,k → v in Lp(Q(x0; δ); Rd) as j → ∞, where v(x) = ξ · x. Hence by (4.132) and the

p-Lipschitz property of f(x, y, ·) (see 2.9)

fεj(x0, ξ) 6 lim infk,δ,j

1

δN

∫

Q(x0;δ)f

(

x,x

εj,x

ε2j, ξ + ∇yφm

(

x

εj

)

+εj∇yΦm,k

( x

εj,x

ε2j

)

+ ∇zΦm,k

( x

εj,x

ε2j

)

)

dx

6 lim infk,δ,j

1

δN

∫

Q(x0;δ)f

(

x,x

εj,x

ε2j, ξ + ∇yφm

(

x

εj

)

+∇zΦm,k

( x

εj,x

ε2j

)

)

dx. (4.135)


Arguing as in Proposition 4.1.15 we define

hm,k(x, y, z) := f(

x, y, z; ξ + ∇yφm(y) + ∇zΦm,k(y, z))

.

Then by Lemma 2.6.3 since hm,k ∈ Lp(Q(x0, δ);Cper(Q×Q)), we get

lim infj→∞

∫

Q(x0;δ)f

(

x,x

εj,x

ε2j, ξ + ∇yφm

(

x

εj

)

+ ∇zΦm,k

( x

εj,x

ε2j

)

)

dx

= limj→∞

∫

Q(x0;δ)hm,k

(

x,x

εj,x

ε2j

)

dx

=

∫

Q(x0;δ)

∫

Q

∫

Qhm,k(x, y, z) dz dy dx

=

∫

Q(x0;δ)

∫

Q

∫

Qf(

x, y, z; ξ + ∇yφm (y) + ∇zΦm,k(y, z))

dz dy dx.

Therefore, by (4.44)

lim infδ→0

lim infj→∞

1

δN

∫

Q(x0;δ)f

(

x,x

εj,x

ε2j, ξ + ∇yφm

(

x

εj

)

+ ∇zΦm,k

( x

εj,x

ε2j

)

)

dx

=

∫

Q

∫

Qf(x0, y, z; ξ + ∇yφm(y) + ∇zΦm,k(y, z)) dz dy,

and thus, by (4.133)-(4.135), (H4), and Fubini’s Theorem, we obtain

fεj(x0, ξ) 6

∫

Q

∫

Qf(x0, y, z, ξ + ∇yφm(y) + ∇zΦm(y, z)) dz dy

=

∫

Q

[∫

Qf(x0, y, z, ξ + ∇yφm(y) + ∇zΦm(y, z)) dz

]

dy

6

∫

Qfhom(x0, y, ξ + ∇φm(y)) dy +

1

m

6 fhom(x0, ξ) +2

m.

Letting m→ ∞ we deduce that


5. APPLICATION TO THIN FILMS

This part is devoted to studying a reiterated homogenization problem in thin domains of

elastic type with multiple scale and periodic microstructure. The results presented here

were obtained in collaboration with J. F. Babadjian [10, 9].

Throughout this chapter ω stands for an open bounded set in R2 and Ω := ω × I, with

I := (−1, 1). We will identify W 1,p(ω; R3) with the set of functions u ∈ W 1,p(Ω; R3) such

that D3u(x) = 0 for a.e. x ∈ Ω, and we set Q′(a, δ) := a+ δQ′ for a ∈ R2 and δ > 0, where

Q′ = (0, 1)2.

5.1 Thin films with periodic microstructure in the in-plane

direction

We start with the case where heterogeneities are allowed in the in-plane direction of the

film and they scale as the thickness of this body.

For each ε > 0 we define Wε : Lp(Ω; R3) → R by

Wε(u) :=

∫

ΩW

(

xα, x3,xαε,∇αu(x)

∣

∣

∣

1

ε∇3u(x)

)


∞ otherwise,

(5.1)

with 1 < p < ∞, where we assume that W : Ω × R2 × R3×3 → R satisfies the following

hypotheses:

(A1) W (x, · , · ) is continuous for a.e. x ∈ Ω;

(A2) W ( · , · , ξ) is L3 ⊗ L2-measurable for all ξ ∈ R3×3;

5. Application to thin films 134

(A3) there exists 0 < β <∞ such that

1

β|ξ|p−β 6 W (x, yα, ξ) 6 β(1+|ξ|p), for a.e. x ∈ Ω and for all (yα, ξ) ∈ R2×R3×3;

(A4) W (x, · , ξ) is Q′-periodic for a.e. x ∈ Ω and all ξ ∈ R3×3, where we denote by Q′ =

(0, 1)2 the unit cube of R2.

The goal of this section is to prove the following result.

Theorem 5.1.1. If W satisfies (A1)-(A4), then the family Wεε Γ(Lp(Ω))-converges to

the functional Whom : Lp(Ω; R3) → R defined by

Whom(u) :=

2

∫

ωWhom(xα,∇αu(xα)) dxα if u ∈W 1,p(ω; R3),

∞ otherwise,

(5.2)

where Whom is given by

Whom(xα, ξ) := limT→∞

infφ

1

2T 2

∫

(0,T )2×IW(

xα, y3, yα, ξ + ∇αφ(y)|∇3φ(y))

dy :

φ ∈W 1,p((0, T )2 × I; R3), φ = 0 on ∂(0, T )2 × I

(5.3)

for a.e. xα ∈ ω and all ξ ∈ R3×2.

Remark 5.1.2. We remark that, due to hypotheses (A1) and (A2), the function W is a

Caratheodory integrand as W (x, ·; ·) is continuous a.e. x ∈ Ω and W (·, yα; ξ) is measurable

for all yα ∈ R2 and ξ ∈ R3×3. This implies (see Proposition 2.3.27) that W is equivalent to

a Borel function, that is there exist a Borel function W such that W (x, · ; · ) = W (x, · ; · )for a.e. x ∈ Ω. As a consequence the integral in (5.1) is well defined. As in the results of

Chapter 4 to prove Theorem 5.1.1 we may assume, without loss of generality, that W is

non negative. Indeed, in view of (A3) it suffices to replace W by W + β.

As a consequence of Theorem 5.1.1, we deduce the usual convergence of (almost) minimizers.

Corollary 5.1.3. Let f ∈ Lp′(Ω; R3) and g ∈ Lp

′(Σ; R3), where Σ := ω × −1, 1 (1/p +

1/p′ = 1). Then every sequence uεε ⊂ Vε := u ∈W 1,p(Ω; R3) : u(x) = (xα, εx3) on ∂ω×I of (almost) minimizers of


Wf,gε (u) =

∫

ΩW

(

x,xαε,∇αu(x)

∣

∣

∣

1

ε∇3u(x)

)

dx−∫

Ωf · u dx−

∫

Σg · u dS

is weakly relatively compact in W 1,p(Ω; R3). Furthermore, any limit point u of this sequence

is a solution of the minimization problem

minv−(xα,0)∈W

1,p0 (ω;R3)

2

∫

ωWhom(xα,∇αv(xα)) dxα −

∫

ω(f + g+ + g−)(xα) · v(xα) dxα

,

where f := 12

∫

If(·, x3) dx3 and g± := g(·,±1).

Corollary 5.1.3 departs from Corollary 4.1.3 on the type of boundary condition that has

been considered. This difficulty is overcome thanks to Remark 5.1.11, which says that we

can prescribe the lateral boundary condition of the recovery sequence. We do not include

the proof of this corollary because it is similar to that of Corollary 1.3 in Bouchitte, Fonseca

and Mascarenhas [25].

The plan of this section is as follows. In Subsection 5.1.1 we will discuss some properties

of Whom, namely that it is well defined and that Whom(xα; · ) is continuous for a.e. xα ∈ ω.

Section 5.1.2 is devoted to the proof of Theorem 5.1.1. The starting point of our analysis

is the Γ-limit integral representation result Theorem 3.2.1. Our objective is to identify the

integrand, showing that it coincides (almost everywhere) with Whom. As in Proposition

4.1.15, we will use a two-scale convergence argument to derive an upper bound for the limit

integral (Lemma 5.1.12). However, we cannot use the same argument as in Proposition

4.1.15 to derive a lower bound. The argument in this case (see Lemma 5.1.13) is more

technical and the difficulty comes from the fact that the problem, at fixed ε, and the

asymptotic problem, when ε → 0, are of different nature (one is a full three-dimensional

problem, the other a two-dimensional one). We will need to use a decoupling argument to

take into account the different nature of the two variables xα and yα that appear in the

structure of the limit functional. For this purpose it will be convenient to extend W to a

function which is (separately) continuous everywhere. This is the aim of Lemma B.1 which

provides conditions under which a Caratheodory function such as W can be extended to a

separately continuous function in the macroscopic variable xα and the microscopic variable

xα/ε.1

1 We could also have used the Scorza Dragoni Theorem and the Tietze Extension Theorem (see Theorem

B.3 in the Appendix). This argument will be used in Subsection 5.2.



In this section we follow very closely the arguments of Section 4.1 to derive some properties

of the stored energy Whom that will be of use in the proof of Theorem 5.1.1.

We begin by showing that in the definition (5.3) of Whom the limit as T → ∞ exists. We

introduce the following new condition :

(A′1) W (x, yα; ·) is continuous for a.e. x ∈ Ω and all yα ∈ R2.

Remark 5.1.4. Note that (A1) implies (A′1). Furthermore, if W satisfies (A′

1) and (A2), then

W is a Caratheodory function in the following sense : W (·, ·; ξ) is L3⊗L2-measurable for all

ξ ∈ R3×3 and W (x, yα; ·) is continuous for L3 ⊗L2-a.e. (x, yα) ∈ Ω×R2. As a consequence,

there exists a Borel function W ′ on Ω × R2 × R3×3 such that W (x, yα; ·) = W ′(x, yα; ·) for

L3 ⊗ L2-a.e. (x, yα) ∈ Ω × R2. Thus the integral in (5.3) is well defined. We insist on the

fact that, in principle, W ′ and W (see Remark 5.1.2) need not to be equal.

Lemma 5.1.5. If W satisfies (A′1), (A2)-(A4), then

Whom(xα, ξ) = limT→∞

infφ

1

2T 2

∫

(0,T )2×IW(

xα, y3, yα, ξ + ∇αφ(y)|∇3φ(y))

dy :

φ ∈W 1,p((0, T )2 × I; R3), φ = 0 on ∂(0, T )2 × I

exists for a.e. xα ∈ ω and all ξ ∈ R3×2.

Proof. Let xα ∈ ω be such that (A′1), (A3) and (A4) hold and let ξ ∈ R3×2. Define

µ : A(R2) → R+ by

µ(A) := infφ

1

2

∫

A×IW (xα, y3, yα, ξ + ∇αφ(y)|∇3φ(y)) dy :

φ ∈W 1,p(A× I; R3), φ = 0 on ∂A× I

.

In view of Remark 5.1.4, µ is well defined and, thanks to (A3), it is a finite function.

Moreover this set function satisfies the assumptions of Lemma A.1. Indeed firstly, by

(A3), µ(A) 6 β(1 + |ξ|p)L2(A) for all A ∈ A(R2). Secondly, µ is subadditive, that is


µ(C) 6 µ(A) + µ(B) for all A, B, C ∈ A(R2) with A ∩B 6= ∅ and C = A ∪B. Finally, by

(A4), for any i ∈ Z2, µ(A+ i) = µ(A) for all A ∈ A(R2). As a consequence the limit

limT→∞

µ((0, T )2)

T 2= Whom(xα, ξ)

exists.

Remark 5.1.6. As in Section 4.1, the limit as T → ∞ in (5.3) can be replaced by an infimum

taken for every T > 0.

Now that Whom is well defined, we will show that Whom(xα; ·) is continuous for a.e. xα ∈ ω,

for later use in Theorem 5.1.1. To prove this property directly it seems that we would need a

little bit more than only the continuity condition imposed on W (x, yα; ·) (e.g. a p-Lipschitz

condition). We remark that if W (x, yα; ·) were quasiconvex, then by the p-growth condition

(A3), W (x, yα; ·) would satisfy a p-Lipschitz condition (see Lemma 5.1.9 below). Since we

do not want to a priori restrict too much the stored energy density, in order to compensate

for this lack of regularity we prove first in Lemma 5.1.8 that the value of Whom does not

change if we replace W by its quasiconvexification QW (see Remark 5.1.7 below).

Remark 5.1.7. For a.e. x ∈ Ω, all yα ∈ R2 and all ξ ∈ R3×3 the functions QW (x, yα; · )(usual quasiconvexification with respect to the last variable ξ) are quasiconvex. By Remark

5.1.4 if W satisfies (A1)-(A4), then so does QW , except that QW (x, ·; ξ) may only be upper

semicontinuous (as the infimum of continuous functions) for a.e. x ∈ Ω, and all ξ ∈ R3×3.

In particular, since QW satisfies (A′1), (A2)-(A4), by Lemma 5.1.5 it follows that

(QW )hom(xα, ξ) = limT→∞

infφ

1

2T 2

∫

(0,T )2×IQW

(

xα, y3, yα, ξ + ∇αφ(y)|∇3φ(y))

dy :

φ ∈W 1,p((0, T )2 × I; R3), φ = 0 on ∂(0, T )2 × I

exists for a.e. xα ∈ ω and all ξ ∈ R3×2.

Lemma 5.1.8. If W satisfies (A1)-(A4), then (QW )hom(xα, ξ) = Whom(xα, ξ) for a.e.

xα ∈ ω and all ξ ∈ R3×2.

Proof. Let xα ∈ ω be such that both (QW )hom(xα; · ) and Whom(xα; · ) are well defined.

Since W > QW , we have Whom(xα, ξ) > (QW )hom(xα, ξ) for all ξ ∈ R3×2. Let us prove now

the opposite inequality. Let ξ ∈ R3×2. For each n > 0, let Tn ∈ N and φn ∈W 1,∞((0, Tn)2×


I; R3), satisfying φn = 0 on ∂(0, Tn)2 × I, be such that

(QW )hom(xα, ξ) +1

n>

1

2T 2n

∫

(0,Tn)2×IQW (xα, y3, yα, ξ + ∇αφn(y)|∇3φn(y)) dy.

The Lipschitz regularity of φn may be ensured due to the density of W 1,∞((0, Tn)2 × I; R3)

in W 1,p((0, Tn)2 × I; R3) together with the p-growth condition (A3). Thus

(QW )hom(xα, ξ) > lim supn→∞

1

2T 2n

∫

(0,Tn)2×IQW (xα, y3, yα, ξ + ∇αφn(y)|∇3φn(y)) dy. (5.4)

For each n ∈ N fixed, by the Acerbi-Fusco Relaxation Theorem 2.3.30 and Remark 5.1.2,

there exists a sequence φn,kk ⊂W 1,∞((0, Tn)2×I; R3) satisfying φn,k = φn on ∂[(0, Tn)

2×I] with φn,k

k→∞φn and such that

1

2T 2n

∫

(0,Tn)2×IQW (xα, y3, yα, ξ + ∇αφn(y)|∇3φn(y)) dy

= limk→∞

1

2T 2n

∫

(0,Tn)2×IW (xα, y3, yα, ξ + ∇αφn,k(y)|∇3φn,k(y)) dy.

From (5.4) we have

(QW )hom(xα, ξ) > lim supn→∞

lim supk→∞

1

2T 2n

∫

(0,Tn)2×IW (xα, y3, yα, ξ + ∇αφn,k(y)|∇3φn,k(y)) dy

> lim supn→∞

infφ

1

2T 2n

∫

(0,Tn)2×IW (xα, y3, yα, ξ + ∇αφ(y)|∇3φ(y)) dy :

φ ∈W 1,p((0, Tn)2 × I; R3), φ = 0 on ∂(0, Tn)

2 × I

= Whom(xα, ξ).

We are now in position to prove the continuity of Whom in its second variable :

Lemma 5.1.9. Let W satisfying (A1)-(A4), then Whom(xα; · ) is continuous on R3×2 for

a.e. xα ∈ ω.

Proof. We observe that by the p-growth condition in (A3) and Remark 5.1.7, QW satisfies

a p-Lipschitz condition: There exists β > 0 such that for all yα ∈ R2 and a.e. x ∈ Ω,

|QW (x, yα; ξ1) −QW (x, yα; ξ2)| 6 β(1 + |ξ1|p−1 + |ξ2|p−1)|ξ1 − ξ2|, ξ1, ξ2 ∈ R3×3. (5.5)


Take xα ∈ ω such that both (QW )hom(xα; · ) and Whom(xα; · ) are well defined. By Lemma

5.1.8 we have (QW )hom(xα; · ) = Whom(xα; · ). Given ξ ∈ R3×2 let ξn → ξ in R3×2. From the

definition of Whom(xα, ξ), for fixed δ > 0 choose T ∈ N and φ ∈ W 1,p((0, T )2 × I; R3), φ =

0 on ∂(0, T )2 × I, such that

Whom(xα, ξ) + δ >1

2T 2

∫

(0,T )2×IW (xα, y3, yα, ξ + ∇αφ(y)|∇3φ(y)) dy. (5.6)

Therefore, Remark 5.1.6 yields

lim supn→∞

Whom(xα, ξn) 6 lim supn→∞

1

2T 2

∫

(0,T )2×IW (xα, y3, yα; ξn + ∇αφ(y)|∇3φ(y)) dy

=1

2T 2

∫

(0,T )2×IW (xα, y3, yα, ξ + ∇αφ(y)|∇3φ(y)) dy

due to hypothesis (A1), the p-growth condition in (A3) and Lebesgue’s Dominated Conver-

gence Theorem. So by (5.6) and letting δ → 0 we conclude that

lim supn→∞

Whom(xα, ξn) 6 Whom(xα, ξ). (5.7)

Similarly, for each n ∈ N consider Tn ∈ N (Tn ր ∞) and φn ∈W 1,p((0, Tn)2 × I; R3), φn =

0 on ∂(0, Tn)2 × I, such that

Whom(xα, ξn) +1

n>

1

2T 2n

∫

(0,Tn)2×IQW (xα, y3, yα, ξn + ∇αφn(y)|∇3φn(y)) dy

=1

2

∫

Q′×IQW (xα, y3, Tnyα, ξn + ∇αφn(Tnyα, y3)|∇3φn(Tnyα, y3)) dy

=1

2

∫

Q′×IQW (xα, y3, Tnyα, ξn + ∇αψn(y)|Tn∇3ψn(y)) dy,

after a change of variables and where ψn(y) := 1Tnφn(Tnyα, y3). Clearly the function ψn

belongs to W 1,p(Q′ × I; R3) and ψn = 0 on ∂Q′ × I. By the p-coercivity hypothesis in (A3)

and (5.7), the sequence (∇αψn|Tn∇3ψn) is bounded in Lp(Q′ × I; R3×3) uniformly in n.

We can write that

lim infn→∞

∫

Q′×IQW (xα, y3, Tnyα, ξn + ∇αψn(y)|Tn∇3ψn(y)) dy

> lim infn→∞

∫

Q′×I

[

QW (xα, y3, Tnyα, ξn + ∇αψn(y)|Tn∇3ψn(y))

−QW (xα, y3, Tnyα, ξ + ∇αψn(y)|Tn∇3ψn(y))]

dy

+ lim infn→∞

∫

Q′×IQW (xα, y3, Tnyα, ξ + ∇αψn(y)|Tn∇3ψn(y)) dy.


Using (5.5), Holder inequality, the fact that ‖(∇αψn|Tn∇3ψn)‖Lp(Q′×I;R3×3) is bounded

and ξn → ξ, we obtain

lim infn→∞

∫

Q′×I

[

QW (xα, y3, Tnyα, ξn + ∇αψn(y)|Tn∇3ψn(y))

−QW (xα, y3, Tnyα, ξ + ∇αψn(y)|Tn∇3ψn(y))]

dy = 0,

and consequently

lim infn→∞

Whom(xα, ξn) > lim infn→∞

1

2

∫

Q′×IQW (xα, y3, Tnyα, ξ + ∇αψn(y)|Tn∇3ψn(y)) dy

= lim infn→∞

1

2T 2n

∫

(0,Tn)2×IQW (xα, y3, yα, ξ + ∇αφn(y)|∇3φn(y)) dy

> (QW )hom(xα, ξ)

= Whom(xα, ξ). (5.8)

From (5.7) and (5.8), we conclude that Whom(xα; ·) is continuous at ξ.

5.1.2 Main result

We start by localizing our functionals. Define Wε : Lp(Ω; R3) ×A(ω) → R by

Wε(u;A) :=

∫

A×IW

(

xα, x3,xαε,∇αu(x)

∣

∣

∣

1

ε∇3u(x)

)


∞ otherwise.

We will prove that the family of functionals Wε(·;A)ε Γ-converges with respect to the

Lp(A× I; R3)-topology to the functional Whom(·;A) : Lp(Ω; R3) → R,

Whom(u;A) :=

2

∫

AWhom(xα,∇αu(xα)) dxα if u ∈W 1,p(A; R3),

∞ otherwise,

(5.9)

for all A ∈ A(ω). As a consequence, taking A = ω yields Theorem 5.1.1.

For any A ∈ A(ω) and any sequence εn ↓ 0, consider Wεn(·;A) : Lp(Ω; R3) → R the

Γ-lower limit of Wεn(·;A)n,

Wεn(u;A) := infun

lim infn→∞

Wεj(un;A) : un → u in Lp(A× I; R3)

. (5.10)


Remark 5.1.10. In view of the coercivity condition (A4), for all A ∈ A(ω) we have that

Wεn(u;A) = ∞whenever u ∈ Lp(Ω; R3) \W 1,p(A; R3), hence our objective is to charac-

terize Wεn(u;A) for u ∈W 1,p(A; R3).

By virtue of Remark 5.1.10, together with Theorem 3.2.1, it follows that every sequence

εnn admits a subsequence εnjj ≡ εjj such that Wεj( · ;A), defined in (5.10), is the

Γ(Lp(A× I))-limit of Wεn( · ;A)j for all A ∈ A(ω). Further, there exists a Caratheodory

function Wεj : ω × R3×2 → R such that

Wεj(u;A) = 2

∫

AWεj(xα,∇αu(xα)) dxα, (5.11)

for all A ∈ A(ω) and all u ∈W 1,p(A; R3).

Our aim is to show that Wεj(·;A) = Whom(·;A) on W 1,p(A; R3) for all A ∈ A(ω). Given

A ∈ A(ω), in view of the integral representation (5.11) and (5.9), it is enough to show

that Wεj(xα, ξ) = Whom(xα, ξ) for a.e. xα ∈ A and all ξ ∈ R3×2. We will prove that

Wεj(xα, ξ) = Whom(xα, ξ) for a.e. xα ∈ ω and all ξ ∈ R3×2.

Remark 5.1.11. Lemma 3.2.2 implies that Wεj(u;A) is unchanged if the approximating

sequences uj are constrained to match the lateral boundary condition of their target, i.e.

uj ≡ u on ∂A× I.

From now on, εjj will denote a subsequence of εj for which the Γ(Lp(A× I))-limit of

Wεj(·;A)j∈N exists and coincides with Wεj(·;A) for all A ∈ A(ω).

For each T > 0 consider ST a countable set of functions in C∞([0, T ]2 × [−1, 1]; R3) that is

dense in

WT = φ ∈W 1,p((0, T )2 × I; R3) : φ = 0 on ∂(0, T )2 × I.

Let L be the set of Lebesgue points x0α for all functions

Wεj(·, ξ), Whom(·, ξ) (5.12)

and

xα 7→∫

Q′×IW (xα, y3, T yα, ξ + ∇αφ(Tyα, y3)|∇3φ(Tyα, y3)) dyα dy3, (5.13)

with T ∈ N, φ ∈ ST and ξ ∈ Q3×2, and for which Whom(x0α; · ) is well defined.


To prove that Wεj(xα, ξ) = Whom(xα, ξ) for a.e. xα ∈ ω and all ξ ∈ R3×2 we first show

in Lemmas 5.1.12 and 5.1.13 below that both functions coincide on L× Q3×2. The general

case will only be treated at the end of that section using the Caratheodory property of both

integrands.

Fix ξ ∈ Q3×2 and set v(x) := ξ · xα. By (5.11) and (5.12)

Wεj(x0α, ξ) = lim

δ→0

1

δ2

∫

Q′(x0α,δ)

Wεj(xα; ξ) dxα

= limδ→0

Iεj(v;Q′(x0

α, δ))

2δ2. (5.14)

Lemma 5.1.12. Wεj(x0α, ξ) 6 Whom(x0

α, ξ) for all x0α ∈ L and all ξ ∈ Q3×2.

Proof. Given k ∈ N, let Tk ∈ N and φk ∈ STkwith φk = 0 on ∂(0, Tk)

2 × I, be such that

Whom(x0α, ξ) +

1

k>

1

2Tk2

∫

(0,Tk)2×IW (x0

α, y3, yα; ξ + ∇αφk(y)|∇3φk(y)) dy.

This is possible because of the continuity properties (A1) of W , the growth conditions

(A3) and the density of STkin WTk

. Extend φk periodically with period Tk to R2 × I. For

x ∈ R2×I, define ukj (x) := ξ ·xα+εjφk(xα

εj, x3). Let δ small enough so that Q′(x0

α, δ) ∈ A(ω).

For fixed k, ukj → v in Lp(Q′(x0α, δ) × I; R3) as j → ∞, hence, by (5.14)

Wεj(x0α, ξ) 6 lim inf

δ→0lim infj→∞

1

2δ2

∫

Q′(x0α,δ)×I

W

(

xα, x3,xαεj,∇αu

kj

∣

∣

∣

1

εj∇3u

kj

)

dx

= lim infδ→0

lim infj→∞

1

2δ2

∫

Q′(x0α,δ)×I

W

(

xα, x3,xαεj, ξ + ∇αφk

(

xαεj, x3

)

∣

∣

∣∇3φk

(

xαεj, x3

))

dx.

Define

hk(xα, yα) :=

∫ 1

−1W (xα, x3, Tkyα, ξ + ∇αφk(Tkyα, x3)|∇3φk(Tkyα, x3))dx3,

for xα ∈ ω and yα ∈ R2.

The continuity of W with respect to yα, its measurability and periodicity properties, and

the fact that Tk ∈ N lead us to conclude that the function hk ∈ L1(Q′(x0α, δ); Cper(Q

′)) for


fixed δ > 0. Lemma 2.6.3 together with Fubini’s Theorem yields

limj→∞

∫

Q′(x0α,δ)×I

W

(

xα, x3,xαεj, ξ + ∇αφk

(

xαεj, x3

)

∣

∣

∣∇3φk

(

xαεj, x3

))

dx

= limj→∞

∫

Q′(x0α,δ)

hk

(

xα,xαTkεj

)

dxα

=

∫

Q′(x0α,δ)

∫

Q′

hk(xα, yα) dyα dxα

=

∫

Q′(x0α,δ)

∫

Q′×IW (xα, x3, Tkyα; ξ + ∇αφk(Tkyα, x3)|∇3φk(Tkyα, x3))dyα dx3 dxα.

Using (5.13) we have

Wεj(x0α, ξ)

6 lim infδ→0

1

2δ2

∫

Q′(x0α,δ)

∫

Q′×IW (xα, x3, Tkyα; ξ +∇αφk(Tkyα, x3)|∇3φk(Tkyα, x3))dyα dx3dxα

=1

2

∫

Q′×IW (x0

α, x3, Tkyα; ξ + ∇αφk(Tkyα, x3)|∇3φk(Tkyα, x3))dyα dx3

6 Whom(x0α, ξ) +

1

k.

Letting k → ∞, we assert the claim.

Note that the same argument could be used to prove Lemma 2.5 in Babadjian and Francfort

[11].

Lemma 5.1.13. Wεj(x0α, ξ) > Whom(x0

α, ξ) for all x0α ∈ L and all ξ ∈ Q3×2.

Proof. Let vj ⊂W 1,p(Q′(x0α, δ) × I; R3) be a recovery sequence for the Γ-limit, i.e.

vj → 0 in Lp(Q′(x0α, δ) × I; R3)

and

Wεj(v;Q′(x0

α, δ)) = limj→∞

∫

Q′(x0α,δ)×I

W

(

xα, x3,xαεj, ξ + ∇αvj

∣

∣

∣

1

εj∇3vj

)

dx.

According to the Decomposition Lemma result for a sequence of scaled gradients, The-

orem 2.2.17, there exists a subsequence of εj (not relabelled) and a sequence uj ⊂

W 1,p(Q′(x0α, δ) × I; R3) such that, upon setting Ej := x ∈ Q′(x0

α, δ) × I : uj(x) = vj(x),then

uj → 0 in Lp(Q′(x0α, δ) × I; R3),

∣

∣

∣

(

∇αuj∣

∣

1εj∇3uj

)∣

∣

∣

p

is equi-integrable,

limj→∞

L3([Q′(x0α, δ) × I] \ Ej) = 0.

(5.15)

Thus, in view of the p-growth condition (A3) together with (5.15) and Remark 5.1.2 it

follows that

Wεj(v;Q′(x0

α, δ)) > lim supj→∞

∫

Ej

W

(

xα, x3,xαεj, ξ + ∇αuj

∣

∣

∣

1

εj∇3uj

)

dx

= lim supj→∞

∫

Q′(x0α,δ)×I

W

(


∣

∣

∣

1

εj∇3uj

)

dx

− lim supj→∞

∫

[Q′(x0α,δ)×I]\Ej

W

(


∣

∣

∣

1

εj∇3uj

)

dx

> lim supj→∞

∫

Q′(x0α,δ)×I

W

(


∣

∣

∣

1

εj∇3uj

)

dx.

For any h ∈ N, we split Q′(x0α, δ) into h2 disjoints cubes Q′

i,h of side length δ/h so that

Q′(x0α, δ) =

⋃h2

i=1Q′i,h and

Wεj(v;Q′(x0

α, δ)) > lim suph→∞

lim supj→∞

h2∑

i=1

∫

Q′i,h

×IW

(


∣

∣

∣

1

εj∇3uj

)

dx.

(5.16)

For every η > 0 and λ > 0, let Kη ⊂ Ω and W η,λ be given by Lemma B.1 below (with

N = d = 3, m = 2 and f = W ). Then

L3(Ω \Kη) < η. (5.17)

On the other hand, define

Rλj :=

x ∈ Q′(x0α, δ) × I :

∣

∣

∣

∣

(

ξ + ∇αuj(x)∣

∣

∣

1

εj∇3uj(x)

)∣

∣

∣

∣

6 λ

.

Chebyshev’s inequality implies that there exists a constant C > 0 – which does not depend

on j or λ – such that

L3([Q′(x0α, δ) × I] \Rλj ) <

C

λp. (5.18)


Since W and W η,λ coincide on Kη × R2 ×B(0, λ), we get in view of (5.16)

Wεj(v;Q′(x0

α, δ)) >

lim supλ,η,h,j

h2∑

i=1

∫

[Q′i,h

×I]∩Rλj ∩Kη

W η,λ

(


∣

∣

∣

1

εj∇3uj

)

dx.

By virtue of inequality (0.1) below (Appendix) and (5.17),

h2∑

i=1

∫

([Q′i,h

×I]∩Rλj )\Kη

W η,λ

(


∣

∣

∣

1

εj∇3uj

)

dx 6 β(1 + λp)η −−−→η→0

0,

uniformly in (j, h), so that

Wεj(v;Q′(x0

α, δ)) >

lim supλ,η,h,j

h2∑

i=1

∫

[Q′i,h

×I]∩Rλj

W η,λ

(


∣

∣

∣

1

εj∇3uj

)

dx.

Fix yα ∈ Q′. Since W η,λ(·, yα; · ) is continuous, it is uniformly continuous on Ω × B(0, λ),

and we define the modulus of continuity ωη,λ : Q′ × R+ → R+ by

ωη,λ(yα, t) := sup(x,ξ), (x′,ξ′)∈Ω×B(0,λ)

|W η,λ(x, yα; ξ) −W η,λ(x′, yα; ξ′)| : |(x; ξ) − (x′; ξ′)| 6 t.

Then

ωη,λ(·, t) is lower semicontinuous for all t ∈ R+,

ωη,λ(yα, ·) is continuous and increasing for all yα ∈ Q′,

ωη,λ(yα, 0) = 0 for all yα ∈ Q′,

and

|W η,λ(x, yα; ξ)−W η,λ(x′, yα; ξ′)| 6 ωη,λ(yα, |x−x′|+|ξ−ξ′|) for all (x, ξ), (x′, ξ′) ∈ Ω×B(0, λ).

(5.19)

The first property is a consequence of the fact that the supremum of continuous functions is

lower semicontinuous, while the other ones are classical properties of moduli of continuity.

For all t ∈ R+, we extend ωη,λ(·, t) to R2 byQ′-periodicity. SinceW η,λ(x, · ; ξ) isQ′-periodic,

inequality (5.19) holds for all yα ∈ R2. Consequently, for every (xα, x3) ∈ [Q′i,h × I] ∩ Rλj


and every x′α ∈ Q′i,h,

∣

∣

∣

∣

W η,λ

(

xα, x3,xαεj, ξ + ∇αuj(xα, x3)

∣

∣

∣

1

εj∇3uj(xα, x3)

)

−W η,λ

(

x′α, x3,xαεj, ξ + ∇αuj(xα, x3)

∣

∣

∣

1

εj∇3uj(xα, x3)

)∣

∣

∣

∣

6 ωη,λ

(

xαεj, |xα − x′α|

)

6 ωη,λ

(

xαεj,

√2δ

h

)

.

We get, after integration in (xα, x3, x′α) and summation,

h2∑

i=1

h2

δ2

∫

Q′i,h

∫

Rλj ∩[Q′

i,h×I]

∣

∣

∣

∣

W η,λ

(

xα, x3,xαεj, ξ + ∇αuj(xα, x3)

∣

∣

∣

1

εj∇3uj(xα, x3)

)

−W η,λ

(


∣

∣

∣

1

εj∇3uj(xα, x3)

)∣

∣

∣

∣

dx

dx′α

6 2

∫

Q′(x0α,δ)

ωη,λ

(

xαεj,

√2δ

h

)

dxα.

Riemann-Lebesgue’s Lemma applied to the Q′-periodic function ωη,λ( · ,√

2δ/h) yields,

limj→∞

2

∫

Q′(x0α,δ)

ωη,λ

(

xαεj,

√2δ

h

)

dxα = 2δ2∫

Q′

ωη,λ

(

xα,

√2δ

h

)

dxα,

and by Levi’s Monotone Convergence Theorem

limh→∞

2δ2∫

Q′

ωη,λ

(

xα,

√2δ

h

)

dxα = 0.

Hence

Wεj(v;Q′(x0

α, δ)) >

lim supλ,η,h,j

h2∑

i=1

h2

δ2

∫

Q′i,h

∫

[Q′i,h

×I]∩Rλj

W η,λ

(

x′α, x3,xαεj

; ξ + ∇αuj(xα, x3)∣

∣

∣

1

εj∇3uj(xα, x3)

)

dx

dx′α.

We define the following sets which depend on all parameters (η, λ, i, h, n)

T := (x′α, xα, x3) ∈ Q′i,h ×Q′

i,h × I : (x′α, x3) ∈ Kη and (xα, x3) ∈ Rλj ,

T1 := (x′α, xα, x3) ∈ Q′i,h ×Q′

i,h × I : (x′α, x3) 6∈ Kη and (xα, x3) ∈ Rλj ,

T2 := (x′α, xα, x3) ∈ Q′i,h ×Q′

i,h × I : (xα, x3) 6∈ Rλj ,

and note that Q′i,h × Q′

i,h × I = T ∪ T1 ∪ T2. Since W (·, yα; · ) and W η,λ(·, yα; · ) coincide

on Kη ×B(0, λ), we have

Wεj(v;Q′(x0

α, δ))

> lim supλ,η,h,j

h2∑

i=1

h2

δ2

∫

TW η,λ

(


∣

∣

∣

1

εj∇3uj(xα, x3)

)

dx dx′α

= lim supλ,η,h,j

h2∑

i=1

h2

δ2

∫

TW

(


∣

∣

∣

1

εj∇3uj(xα, x3)

)

dx dx′α.

(5.20)

We will prove that the corresponding integrals over T1 and T2 are zero. Indeed, in view of

(5.17) and the p-growth condition (A3),

h2∑

i=1

h2

δ2

∫

T1

W

(


∣

∣

∣

1

εj∇3uj(xα, x3)

)

dx dx′α

6

h2∑

i=1

h2

δ2L2(Q′

i,h)L3([Q′i,h × I] \Kη)β(1 + λp)

< β(1 + λp)η −−−→η→0

0, (5.21)

uniformly in (j, h). The bound from above in (A3), the equi-integrability of the sequence∣

∣

∣

(

∇αuj∣

∣

1εj∇3uj

)∣

∣

∣

p

and (5.18) imply that

h2∑

i=1

h2

δ2

∫

T2

W

(


∣

∣

∣

1

εj∇3uj(xα, x3)

)

dx dx′α

6

h2∑

i=1

h2

δ2L2(Q′

i,h)β

∫

[Q′i,h

×I]\Rλj

(

1 +

∣

∣

∣

∣

(

∇αuj

∣

∣

∣

1

εj∇3uj

)∣

∣

∣

∣

p)

dx

= β

∫

[Q′(x0α,δ)×I]\R

λj

(

1 +

∣

∣

∣

∣

(

∇αuj

∣

∣

∣

1

εj∇3uj

)∣

∣

∣

∣

p)

dx −−−→λ→∞

0, (5.22)

uniformly in (η, n, h). Thus, in view of (5.20), (5.21), (5.22), Fatou’s Lemma yields

Wεj(v;Q′(x0

α, δ))

> lim suph→∞

lim supj→∞

h2∑

i=1

h2

δ2

∫

Q′i,h

∫

Q′i,h

×IW

(


∣

∣

∣

1

εj∇3uj(xα, x3)

)

dx dx′α

> lim suph→∞

h2∑

i=1

h2

δ2

∫

Q′i,h

lim infj→∞

∫

Q′i,h

×IW

(


∣

∣

∣

1

εj∇3uj(xα, x3)

)

dx dx′α.


Fix x′α ∈ Q′i,h such that Whom(x′α, ξ) is well defined and set Z(x; ξ) := W (x′α, x3, xα; ξ). It

is easy to check that Z is a Caratheodory integrand. Hence, applying (3.10), we get since

uj → 0 in Lp(Q′(x0α, δ) × I; R3),

2δ2

h2Z(ξ) 6 lim inf

j→∞

∫

Q′(x0α,δ)×I

Z

(

xαεj, x3, ξ + ∇αuj(x)

∣

∣

∣

1

ε∇3uj(x)

)

dx,

where

Z(ξ) := infT>0, φ

∫

(0,T )2×IZ(x, ξ + ∇αφ(x)|∇3φ(x)) dx :

φ ∈W 1,p((0, T )2 × I; R3), φ = 0 on ∂(0, T )2 × I

.

In view of the previous formula together with (5.3) and Remark 5.1.6, we have that Z(ξ) =

Whom(x′α, ξ). Then

lim infj→∞

∫

Q′i,h

×IW

(


∣

∣

∣

1

εj∇3uj(xα, x3)

)

dx >2δ2

h2Whom(x′α, ξ),

and so

Wεj(v;Q′(x0

α, δ)) > lim suph→∞

h2∑

i=1

h2

δ2

∫

Q′i,h

2δ2

h2Whom(x′α, ξ)dx

′α = 2

∫

Q′(x0α,δ)

Whom(x′α, ξ)dx′α.

Dividing both sides of the previous inequality by δ2 and passing to the limit when δ ↓ 0, we

obtain by (5.12) and (5.14)

Wεj(x0α, ξ) > Whom(x0

α, ξ).

Proposition 5.1.14. Wεj(xα, ξ) = Whom(xα, ξ) a.e. xα ∈ ω and all ξ ∈ R3×2.

Proof. Let E be the intersection of the set L with the subset of points x0α ∈ ω where

Wεj(x0α; · ) and Whom(x0

α; · ) are continuous (see Lemma 5.1.9). Then L2(ω \ E) = 0 and

in view of Lemma 5.1.12 and 5.1.13, we have that for all x0α ∈ E and for all ξ ∈ Q3×2,

Wεj(x0α, ξ) = Whom(x0

α, ξ). Since Wεj(x0α; · ) and Whom(x0

α; · ) are continuous for all

x0α ∈ E, the equality Wεj(x

0α, ξ) = Whom(x0

α, ξ) holds true for x0α ∈ E and all ξ ∈ R3×2.


Corollary 5.1.15. Γ(Lp(A × I))- limε

Wε( · ;A) = Whom( · ;A) for all A ∈ A(ω), where

Whom(·;A) is the functional defined in (5.9).

Proof. From Proposition 5.1.14 we can conclude that Whom(·;A) is well defined and

Γ(Lp(A× I))- limj

Wεj( · ;A) = Whom( · ;A)

for all A ∈ A(ω) (see Remark 5.1.10). Since this limit does not depend upon the ex-

tracted subsequence, in view of Remark 2.5.9, the whole sequence Wε( · ;A)ε Γ(Lp(A×I))-converges to Whom( · ;A) for each A ∈ A(ω).

The proof of Theorem 5.1.1 is a consequence of Corollary 5.1.15 taking A = ω.

5.2 When heterogeneities are allowed also in the transverse

direction

Following the lines of the previous section and those of Babadjian and Francfort [11] we

assume that

(A1) W (x, · , · ; · ) is continuous for a.e. x ∈ Ω;

(A2) W ( · , · , · ; ξ) is L3 ⊗ L3 ⊗ L2-measurable for all ξ ∈ R3×3;

(A3)

yα 7→W (x, yα, y3, zα; ξ) is Q′-periodic for all (zα, y3, ξ) ∈ R3 × R3×3 and a.e. x ∈ Ω,

(zα, y3) 7→W (x, yα, y3, zα; ξ) is Q -periodic for all (yα, ξ) ∈ R2 × R3×3 and a.e. x ∈ Ω.

(A4) there exists β > 0 such that

1

β|ξ|p−β 6 W (x, y, zα; ξ) 6 β(1+|ξ|p) for all (y, zα, ξ) ∈ R3×R2×R3×3 and a.e x ∈ Ω.

We prove the following theorem.

Theorem 5.2.1. Let W : Ω×R3 ×R2 ×R3×3 → R be a function satisfying (A1)-(A4). For

each ε > 0, consider the functional Wε : Lp(Ω; R3) → R defined by

Wε(u) :=

∫

ΩW

(

x,x

ε,xαε2

;∇αu(x)∣

∣

∣

1

ε∇3u(x)

)


∞ otherwise.

(5.23)


Then the Γ(Lp(Ω))-limit of the family Wεε is given by the functional

Whom(u) :=

2

∫

ωW hom(xα;∇αu(xα)) dxα if u ∈W 1,p(ω; R3),

∞ otherwise,

(5.24)

where W hom is defined, for all ξ ∈ R3×2 and a.e. xα ∈ ω, by

W hom(xα; ξ) := limT→∞

infφ

1

2T 2

∫

(0,T )2×IWhom(xα, y3, yα; ξ + ∇αφ(y)|∇3φ(y)) dy :

φ ∈W 1,p((0, T )2 × I; R3), φ = 0 on ∂(0, T )2 × I

(5.25)

and

Whom(x, yα; ξ) := limT→∞

infφ

1

T 3

∫

(0,T )3W (x, yα, z3, zα; ξ + ∇φ(z)) dz :

φ ∈W 1,p0 ((0, T )3; R3)

, (5.26)

for a.e x ∈ Ω and all (yα, ξ) ∈ R2 × R3×3.

Remark 5.2.2. As before, the limits as T → ∞ in (5.25) and (5.26) can be replaced by an

infimum taken over T > 0.

Let us formally justify the periodicity assumptions (A3): Since the volume of Ωε is of order

ε and ε2 ≪ ε, in a first step, we can think of ε as being a fixed parameter and let ε2

tend to zero. At this point dimension reduction is not occurring and (5.23) can be seen

as a single one-scale homogenization problem as in (4.1), in which it is natural to assume

(zα, y3) 7→ W (x, yα, y3, zα; ξ) to be Q-periodic. The homogenization formula for this case

gives us an homogenized stored energy density Whom(x, yα; ξ) that, in a second step, is

used as the integrand of a problem similar to the one treated in Section 5.1. In particular,

the required Q′- periodicity of Whom(x, · ; ξ) can be obtained from the Q′-periodicity of

yα 7→W (x, yα, y3, zα; ξ).

We would also like to remark the difference between assumptions (H2) in Section 4.2

and (A2): In the 3D-2D case, if we assume only W (·, y, zα; ξ) to be L3-measurable for

all (y, zα, ξ) ∈ R3 × R2 × R3×3, then the functional (5.23) is well defined (at least for

u ∈ W 1,p(Ω; R3)) but this would not be the case for the integrals in (5.25) and (5.26)

because their integrands would be only separately measurable.


We will first study the case where W does not depend on the macroscopic variable x (The-

orem 5.2.4). We observe that the proof of Theorem 5.2.4 is very close to its N -dimensional

analogue Theorem 4.2.2, the main difference being the use of the Scaled Gradients Decom-

position Lemma, Lemma 2.2.17, in place of Lemma 2.2.16. As before (compare Theorems

4.1.1 and 5.1.1), Theorem 5.2.1 and Theorem 4.2.1 cannot be treated similarly. We will

need an argument along the lines of what is done in Section 5.1 where we had to consider a

suitable continuous extension of W ; in this case we will use a corollary of the Scorza-Dragoni

Theorem and Tietze Extension Theorem (see Lemma B.3 in the Appendix).

We organize this section as follows. In Subsection 5.2.1 we discuss the main properties of

Whom and W hom. Then, in Subsection 5.2.2 we address the case where W is independent

of the macroscopic in-plane variable xα (Theorem 5.2.4). Finally, Theorem 5.2.1 is proved

in Subsection 5.2.3.

Remark 5.2.3. As before, without loss of generality we assume that W is non negative upon

replacing W by W + β which is non negative in view of (A4).


As in Section 4.2.1 we can see that the function Whom given in (5.26) is well defined and is

(equivalent to) a Caratheodory function:

Whom(·, · ; ξ) is L3 ⊗ L2-measurable for all ξ ∈ R3×3, (5.27)

Whom(x, yα; ·) is continuous for L3 ⊗ L2-a.e (x, yα) ∈ Ω × R2. (5.28)

By condition (A3) it follows that

Whom(x, · ; ξ) is Q′-periodic for a.e. x ∈ Ω and all ξ ∈ R3×3. (5.29)

Moreover, Whom is quasiconvex in the ξ variable and satisfies the same p-growth and p-

coercivity condition as W :

1

β|ξ|p−β 6 Whom(x, yα; ξ) 6 β(1+|ξ|p) for a.e. x ∈ Ω and all (yα, ξ) ∈ R2×R3×3, (5.30)

where β is the constant in (A4). Just as before, (5.27), (5.28) and (5.30) imply that the

function W hom given in (5.25) is also well defined, and is (equivalent to) a Caratheodory

function, which implies that the definition of Whom makes sense on W 1,p(Ω; R3). Finally,


W hom is also quasiconvex in the ξ variable and satisfies the same p-growth and p-coercivity

condition as W and Whom:

1

β|ξ|p − β 6 W hom(xα; ξ) 6 β(1 + |ξ|p) for a.e. xα ∈ ω and all ξ ∈ R3×2, (5.31)

where, as before, β is the constant in (A4).

5.2.2 Main result when the integrands do not depend on the

macroscopic variable

In this section, we assume that W does not depend explicitly on xα, namely W : I × R3 ×R2 × R3×3 → [0,∞). For each ε > 0, consider the functional Wε : Lp(Ω; R3) → [0,∞]

defined by

Wε(u) :=

∫

ΩW

(

x3,x

ε,xαε2

;∇αu(x)∣

∣

∣

1

ε∇3u(x)

)


∞ otherwise.

(5.32)

Our objective is to prove the following result.

Theorem 5.2.4. Under assumptions (A1)-(A4) the Γ(Lp(Ω))-limit of the family Wεε is

given by

Whom(u) =

2

∫

ωW hom(∇αu(xα)) dxα if u ∈W 1,p(ω; R3),

∞ otherwise,

where W hom is defined, for all ξ ∈ R3×2, by

W hom(ξ) := limT→∞

infφ

1

2T 2

∫

(0,T )2×IWhom(y3, yα; ξ + ∇αφ(y)|∇3φ(y)) dy :

φ ∈W 1,p((0, T )2 × I; R3) and φ = 0 on ∂(0, T )2 × I

(5.33)

and

Whom(y3, yα; ξ) := limT→∞

infφ

1

T 3

∫

(0,T )3W (y3, yα, z3, zα; ξ + ∇φ(z)) dz :

φ ∈W 1,p0 ((0, T )3; R3)

, (5.34)

for all (y, ξ) ∈ R3 × R3×3.


Since the proofs are very similar to those of Section 4.2.2, we just sketch them highlighting

the main differences.

STEP 1. Localization and existence of Γ-convergent subsequences.

For the same reason than in the proof of Theorem 4.2.2 in Section 4.2.2, we localize the

functionals given in (5.32) on the class of bounded open subsets of R2, denoted by A0(ω).

For each ε > 0, consider Wε : Lp(R2 × I; R3) ×A0(ω) → [0,∞] defined by

Wε(u;A) :=

∫

A×IW

(

x3,x

ε,xαε2

;∇αu(x)∣

∣

∣

1

ε∇3u(x)

)


∞ otherwise.

(5.35)

Given εnn ↓ 0 and A ∈ A0(R2), consider the Γ-lower limit of Wεn(· ;A)n for the

Lp(A× I; R3)-topology, defined for u ∈ Lp(R2 × I; R3), by

Wεn(u;A) := infun

lim infn→∞

Wεn(un;A) : un → u in Lp(A× I; R3)

.

In view of the p-coercivity condition (A4), for each A ∈ A0(R2) it follows that Wεn(u;A)

is infinite whenever u ∈ Lp(R2 × I; R3) \ W 1,p(A; R3), so it suffices to study the case

where u ∈ W 1,p(A; R3). Arguing exactly as in Section 5.1, we can prove the existence of a

subsequence εnjj ≡ εjj such that Wεj(· ;A) is the Γ(Lp(A×I)-limit of Wεj

(· ;A)n∈N

for each A ∈ A0(R2).

Our next ojective is to show that for every A ∈ A0(R2) and every u ∈ W 1,p(A; R3), then

Wεj(u;A) = Whom(u;A), where Whom : Lp(R2 × I; R3) ×A0(R2) → [0,∞] is given by

Whom(u;A) =

2

∫

AW hom(∇αu(xα)) dxα if u ∈W 1,p(A; R3),

∞ otherwise.

STEP 2. Integral representation of the Γ-limit.

Following the proof of Lemma 3.2.2, it is possible to show that for each A ∈ A0(R2) and

all u ∈ W 1,p(A; R3), the restriction of Wεj(u; ·) to A(A) is a Radon measure, absolutely


continuous with respect to the two-dimensional Lebesgue measure. But as before, one has

to ensure that the integral representation given by Theorem 2.4.1 is independent of the

open set A ∈ A0(R2). The following result, prevents this dependence from holding since it

leads to an homogeneous integrand, as will be seen in Lemma 5.2.6 below.

Lemma 5.2.5. For all ξ ∈ R3×2, y0α and z0

α ∈ R2, and δ > 0

Wεj(ξ · ;Q′(y0α, δ)) = Wεj(ξ · ;Q′(z0

α, δ)).

Proof. It is obviously enough to show that

Wεj(ξ · ;Q′(y0α, δ)) > Wεj(ξ · ;Q′(z0

α, δ)).

According to Theorem 2.2.17 and Lemma 3.2.2 there exists a sequence uj ⊂W 1,p(Q′(y0α, δ)×

I; R3) such that∣

∣

(

∇αuj | 1εj∇3uj

)∣

∣

pis equi-integrable, uj = 0 on ∂Q′(y0

α, δ) × I, uj → 0

in Lp(Q′(y0α, δ) × I; R3) and

Wεj(ξ· ;Q′(y0α, δ)) = lim

j→∞

∫

Q′(y0α,δ)×IW

(

x3,x

εj,xαε2j

; ξ + ∇αuj(x)∣

∣

∣

1

εj∇3uj(x)

)

dx.

We argue exactly as in the proof of Lemma 4.2.4 with y0α and z0

α in place of y0 and z0. For

all j ∈ N, extend uj by zero to the whole R2 × I and set vj(xα, x3) = uj(xα + xεjα , x3) for

(xα, x3) ∈ Q′(z0α, δ) × I, where x

εjα := mεj

εj − ε2j lεj. Then vj ⊂ W 1,p(Q′(z0

α, δ) × I; R3),

vj → 0 in Lp(Q′(z0α, δ) × I; R3), the sequence

∣

∣

(

∇αvj∣

∣

1εj∇3vj

)∣

∣

pis equi-integrable and

Wεj(ξ· ;Q′(y0α, δ))

= lim supj→∞

∫

Q′(z0α,δ)×IW

(

x3,xαεj

− εjlεj,x3

εj,xαε2j

; ξ + ∇αvj(x)∣

∣

∣

1

εj∇3vj(x)

)

dx,(5.36)

where we have used the p-growth condition (A4) and the fact that L2(Q′(z0α, δ) \ Q′(y0

α −xεjα , δ)) → 0. To eliminate the term εj lεj

in (5.36), we would like to apply a uniform

continuity argument. Since for a.e. x3 ∈ I the function W (x3, ·, · ; ·) is continuous on

R3 ×R2 ×R3×3, then (A3) implies that it is uniformly continuous on R3 ×R2 ×B(0, λ) for

any λ > 0. We define

Rλj :=

x ∈ Q′(z0α, δ) × I :

∣

∣

∣

∣

(

ξ + ∇αvj(x)∣

∣

∣

1

εj∇3vj(x)

)∣

∣

∣

∣

6 λ

,


and we note that by Chebyshev’s inequality

L3([Q′(z0α, δ) × I] \Rλj ) 6 C/λp, (5.37)

for some constant C > 0 independent of λ or j. Thus, in view of (5.36) and the fact that

W is nonnegative,

Wεj(ξ· ;Q′(y0α, δ))

> lim supλ→∞

lim supj→∞

∫

Rλj

W

(

x3,xαεj

− εjlεj,x3

εj,xαε2j

; ξ + ∇αvj(x)∣

∣

∣

1

εj∇3vj(x)

)

dx.

Denoting by ωλ(x3, ·) : [0,∞) → [0,∞) the modulus of continuity of W (x3, ·, · ; ·) on R3 ×R2 × B(0, λ), we can check that for a.e. x3 ∈ I, the function t 7→ ωλ(x3, t) is continuous,

increasing and satisfies ωλ(x3, 0) = 0 while, for all t ∈ [0,∞), the function x3 7→ ωλ(x3, t) is

measurable (as the supremum of measurable functions). We get, for any x ∈ Rλj∣

∣

∣

∣

∣

W

(

x3,x

εj,xαε2j

; ξ + ∇αvj(x)∣

∣

∣

1

εj∇3vj(x)

)

−W(

x3,xαεj

− εjlεj,x3

εj

xαε2j

; ξ + ∇αvj(x)∣

∣

∣

1

εj∇3vj(x)

)∣

∣

∣

∣

∣

6 ωλ(x3, εjlεj).

The properties of ωλ, Levi’s Monotone Convergence Theorem and (5.36) yield

Wεj(ξ· ;Q′(y0α, δ)) > lim sup

λ→∞lim supj→∞

∫

Rλj

W

(

x3,x

εj,xαε2j

; ξ + ∇αvj(x)∣

∣

∣

1

εj∇3vj(x)

)

dx

−δ2∫ 1

−1ωλ(x3, εjlεj

) dx3

= lim infj→∞

∫

Q′(z0α,δ)×IW

(

x3,x

εj,xαε2j

; ξ + ∇αvj(x)∣

∣

∣

1

εj∇3vj(x)

)

dx

> Wεj(ξ· ;Q′(z0α, δ)),

where we have used the equi-integrability of∣

∣

(

∇αvj∣

∣

1εj∇3vj

)∣

∣

p, the p-growth condition

(A4), (5.37) and the fact that vj → 0 in Lp(Q′(z0α, δ) × I; R3).

As a consequence of this lemma, and adapting the argument used in the proof of Lemma

4.2.5, we deduce the following integral representation result.

Lemma 5.2.6. There exists a continuous function Wεj : R3×2 → [0,∞) such that for all

A ∈ A0(R2) and all u ∈W 1,p(A; R3),

Wεj(u;A) = 2

∫

AWεj(∇αu(xα)) dxα.


In view of Lemma 5.2.6, we only need to prove that W hom(ξ) = Wεj(ξ) for all ξ ∈ R3×2,

and thus it suffices to work with affine functions instead of with general Sobolev functions.

We state, without proof, a result equivalent to Proposition 4.2.6 for the dimension reduction

case.

Proposition 5.2.7. Given M > 0, η > 0, and ϕ : [0,∞) → [0,∞] a continuous and

increasing function satisfying ϕ(t)/t → ∞ as t → ∞, there exists ε0 ≡ ε0(M,η) > 0 such

that for every 0 < ε < ε0, every a ∈ R2 and every u ∈W 1,p((a+Q′) × I; R3) with∫

(a+Q′)×Iϕ(|∇u|p) dx 6 M, (5.38)

there exists v ∈W 1,p0 ((a+Q′) × I; R3) with ‖v‖Lp((a+Q′)×I;R3) 6 η satisfying

∫

(a+Q′)×IW(

x3, xα,x3

ε,xαε

;∇u)

dx >

∫

(a+Q′)×IWhom(x3, xα;∇u+ ∇v) dx− η.

Lemma 5.2.8. For all ξ ∈ R3×2, W hom(ξ) 6 Wεj(ξ).

Proof. From Lemma 5.2.6, Theorem 2.2.17 and Lemma 3.2.2 we may find a sequence

wj ⊂ W 1,p(Q′ × I; R3) such that∣

∣

(

∇αwj∣

∣

1εj∇3wj

)∣

∣

p

jis equi-integrable, wj = 0 on

∂Q′ × I, wj → 0 in Lp(Q′ × I; R3) and

2Wεj(ξ) = limj→∞

∫

Q′×IW

(

x3,x

εj,xαεj2

; ξ + ∇αwj(x)∣

∣

∣

1

εj∇3wj(x)

)

dx.

Thus, from De La Vallee Poussin criterion (Proposition 2.2.10) there exists an increasing

continuous function ϕ : [0,∞) → [0,∞] satisfying ϕ(t)/t→ ∞ as t→ ∞ and such that

supj∈N

∫

Q′×Iϕ

(∣

∣

∣

∣

(

∇αwj

∣

∣

∣

1

εj∇3wj

)∣

∣

∣

∣

p)

dx 6 1.

Changing variables yields

Wεj(ξ) = limj→∞

1

2T 2j

∫

(0,Tj)2×IW

(

x3, xα,x3

εj,x3

ε2j; ξ + ∇αzj(x)|∇3zj(x)

)

dx


and

supj∈N

1

T 2j

∫

(0,Tj)2×Iϕ(|∇zj |p) dx 6 1,

where we set Tj := 1/εj and zj(x) := Tjwj(xα/Tj , x3). Note that zj ∈W 1,p((0, Tj)2×I; R3)

and zj = 0 on ∂(0, Tj)2 × I. For all j ∈ N, define Ij :=

1, · · · , [[Tj ]]2

and for any i ∈ Ij ,

take aji ∈ Z2 such that⋃

i∈Ij

(aji +Q′) ⊂ (0, Tj)2.

Moreover, for all M > 0, let

IMj :=

i ∈ Ij :

∫

(aji +Q

′)×Iϕ(|∇zj |p) dx 6 M

.

Applying Proposition 5.2.7, we get for any η > 0 and any i ∈ IMj the existence of vj,M,ηi ∈

W 1,p0 ((aji +Q′) × I; R3) with ‖vj,M,η

i ‖Lp((aj

i +Q′)×I;R3)

6 η and

∫

(aji +Q

′)×IW

(

x3, xα,x3

εj,xαεj

; ξ + ∇αzj |∇3zj

)

dx

>1

T 2j

∫

(aji +Q

′)×IWhom

(

x3, xα; ξ + ∇αzj + ∇αvj,M,ηi |∇3zj + ∇3v

j,M,ηi

)

dx− η.

Hence,

Wεj(ξ) > lim supM,η,j

1

2T 2j

∑

i∈IMj

∫

(aji +Q

′)×IWhom

(

x3, xα; (ξ|0) + ∇φj,M,η)

dx (5.39)

where φj,M,η ∈W 1,p((0;Tj)2 × I; R3) is defined by

φj,M,η(x) :=

zj(x) + vj,M,ηi (x) if x ∈ (aji +Q′) × I and i ∈ IMj ,

zj(x) otherwise

and satisfies φj,M,η = 0 on ∂(0, Tj)2 × I. In view of the definition of φj,M,η, the p-growth

condition (5.30) and the equi-integrability of∣

∣

(

∇αwj∣

∣

1εj∇3wj

)∣

∣

p, arguing exactly as in

Lemma 4.2.7, we get

Wεj(ξ) > lim supM,η,j

1

2T 2j

∫

(0,Tj)2×IWhom(x3, xα; ξ + ∇αφ

j,M,η|∇3φj,M,η) dx

> W hom(ξ).


Let us now prove the opposite inequality.

Lemma 5.2.9. For all ξ ∈ R3×2, W hom(ξ) > Wεj(ξ).

Proof. In view of (5.33), for δ > 0 fixed take Tδ ≡ T ∈ N, with Tδ → ∞ as δ → 0, and let

φδ ≡ φ ∈W 1,p((0, T )2 × I; R3) be such that φ = 0 on ∂(0, T )2 × I and

W hom(ξ) + δ >1

2T 2

∫

(0,T )2×IWhom(x3, xα; ξ + ∇αφ(x)|∇αφ(x)) dx. (5.40)

From Theorem 4.1.1, Theorem 2.5.13 and the Decomposition Lemma 2.2.16 there exists

φj ⊂ W 1,p0 ((0, T )2 × I; R3) such that |∇φj |p is equi-integrable, φj → φ in Lp((0, T )2 ×

I; R3) and

∫

(0,T )2×IWhom(x3, xα; ξ + ∇αφ(x)|∇3φ(x)) dx

= limj→∞

∫

(0,T )2×IW

(

x3, xα,x3

εj,x3

εj; ξ + ∇αφj(x)|∇3φj(x)

)

dx. (5.41)

Fix j ∈ N such that εj ≪ 1. For all i ∈ Z2 let aji ∈ εjZ2 ∩ [i(T + 1), εj)

2 (uniquely defined).

Set

φj(x) :=

φj(xα − aji , x3) if x ∈ Q′(aji , T ) × I and i ∈ Z2,

0 otherwise,

then φj ∈ W 1,p(R2 × I; R3). Let Ij := i ∈ Z2 : (0, T/εj)2 ∩ Q′(aji , T ) 6= ∅. If ψj(x) :=

εjφj(xα/εj , x3) then ψj → 0 in Lp((0, T )2 × I; R3), as j → ∞. Consequently, the p-growth


condition (A4) implies that

Wεj(ξ· ; (0, T )2)

6 lim infj→∞

∫

(0,T )2×IW

(

x3,x

εj,xαε2j

; ξ + ∇αψj(x)∣

∣

∣

1

εj∇3ψj(x)

)

dx

6 lim infj→∞

ε2j

∑

i∈Ij

∫

Q′(aji ,T )×I

W

(

x3, xα,x3

εj,xαεj

; ξ + ∇αφj(x)|∇3φj(x)

)

dx

+2β(1 + |ξ|p)LN

(

0,T

εj

)2\⋃

i∈Ij

Q′(aji , T )

= lim infj→∞

ε2j∑

i∈Ij

∫

(0,T )2×IW

(

x3, xα + aji − i(T + 1),x3

εj,xαεj

; ξ + ∇αφj(x)|∇3φj(x)

)

dx

+2β(1 + |ξ|p)T 2

(

1 −(

T

T + 1

)2)

(5.42)

where we have used (A3), the fact that T ∈ N and aji/εj ∈ Z2. We now use the same

uniform continuity argument than in the proof of Lemmas 5.2.5 and 4.2.8. We get

Wεj(ξ) 6 W hom(ξ) + δ + 2β(1 + |ξ|p)(

1 −(

T

T + 1

)2)

.

The result follows by letting δ tend to zero.

Proof of Theorem 5.2.4. From Lemma 5.2.8 and Lemma 5.2.9, we conclude that W hom(ξ) =

Wεj(ξ) for all ξ ∈ R3×2. As a consequence, Wεj(u;A) = Whom(u;A) for all A ∈ A0(R2)

and all u ∈ W 1,p(A; R3). Since the Γ-limit does not depend upon the extracted subse-

quence, Proposition 2.5.8 implies that the whole sequence Wε(· ;A) Γ(Lp(A× I))-converges

to Whom(· ;A).

5.2.3 The general case

Our aim here is to study the case where the function W depends also on the in-plane

variable.

STEP 1. Localization of the functionals.


As in Subsection 4.2.3, to prove Theorem 5.2.1 it is convenient to localize the functionals

Wε in (5.23) on the class of all bounded open subsets of ω, denoted by A(ω). For each

ε > 0 we consider the family of functionals Wε : Lp(Ω; R3) ×A(ω) → [0,∞] defined by

Wε(u;A) :=

∫

A×IW

(

x,x

ε,xαε2

;∇αu(x)∣

∣

∣

1

ε∇3u(x)

)


∞ otherwise.

(5.43)

Given εn ↓ 0 and A ∈ A(ω) we define the Γ-lower limit of Wεn(· ;A)n with respect to the

Lp(A× I; R3)-topology by

Wεn(u;A) := infun

lim infn→∞

Wεn(un;A) : un → u in Lp(A× I; R3)

for all u ∈ Lp(Ω; R3). Our main objective is to show that

Wεn = Whom (5.44)

where Whom : Lp(Ω; R3) ×A(ω) → [0,∞] is given by

Whom(u;A) =

2

∫

AW hom(xα;∇αu(xα)) dxα if u ∈W 1,p(A; R3),

∞ otherwise.

The conclusion of Theorem 5.2.1 would follow taking A = ω.

By hypotheses (A4) it follows that Wεn(u;A) = ∞ for each A ∈ A(ω) whenever u ∈Lp(Ω; R3) \W 1,p(A; R3).

As a consequence of Theorem 3.2.1, given εn ↓ 0 there exists a subsequence εnjj ≡ εjj

for which the functional Wεj(· ;A) is the Γ(Lp(A × I))-limit of Wεj(· ;A)j for each

A ∈ A(ω). Moreover given u ∈W 1,p(A; R3)

Wεj(u;A) = 2

∫

AWεj(xα;∇α(xα)) dxα,

for some Caratheodory function Wεj : ω × R3×2 → R.

Accordingly, to prove equality (5.44) it suffices to show that Wεj(xα; ξ) = W hom(xα; ξ)

for a.e. xα ∈ ω and all ξ ∈ R3×2.



For each T > 0 consider ST a countable set of functions in C∞([0, T ]2 × [−1, 1]; R3) that is

dense in

φ ∈W 1,p((0, T )2 × I; R3) : φ = 0 on ∂(0, T )2 × I

.

Let L be the set of Lebesgue points x0α for all functions

Wεj(·; ξ), W hom(·; ξ)

and

xα 7→∫

(0,T )2×IWhom(xα, y3, yα; ξ + ∇αφ(y)|∇3φ(y)) dy,

with T ∈ N, φ ∈ ST and ξ ∈ Q3×2, and for which W hom(x0α; · ) is well defined. Note that

L2(ω \ L) = 0.

We start by proving the following inequality.

Lemma 5.2.10. For all x0α ∈ L and all ξ ∈ Q3×2, we have Wεj(x

0α; ξ) > W hom(x0

α; ξ).

Proof. Let δ > 0 small enough so that Q′(x0α, δ) ∈ A(ω). By Theorem 2.2.17 we can find

a sequence ujj ⊂W 1,p(Q′(x0α, δ)× I; R3) with uj → 0 in Lp(Q′(x0

α, δ)× I; R3), such that

the sequence of scaled gradients(


)

is p-equi-integrable and

Wεj(ξ· ;Q′(x0α, δ)) = 2

∫

Q′(x0α,δ)

Wεj(xα; ξ) dxα

= limj→∞

∫

Q′(x0α,δ)×I

W

(

x,x

εj,xαε2j

; ξ + ∇αuj(x)∣

∣

∣

1

εj∇3uj(x)

)

dx.

Given m ∈ N let Cm and Wm be given by Lemma B.3. Then, since W > 0 and W = Wm

on Cm × R3 × R2 × R3×3, we get

Wεj(ξ· ;Q′(x0α, δ)) > lim sup

m→∞lim supj→∞

∫

[Q′(x0α,δ)×I]∩Cm

Wm

(

x,x

εj,xαε2j

; ξ + ∇αuj(x)∣

∣

∣

1

εj∇3uj(x)

)

dx.

By the p-growth condition (0.5), the equi-integrability of∣

∣

(


)∣

∣

pand relation

(0.4), we obtain

∫

[Q′(x0α,δ)×I]\Cm

Wm

(

x,x

εj,xαε2j

; ξ + ∇αuj(x)∣

∣

∣

1

εj∇3uj(x)

)

dx

6 β

∫

[Q′(x0α,δ)×I]\Cm

(

1 +

∣

∣

∣

∣

(

ξ + ∇αuj(x)∣

∣

∣

1

εj∇3uj(x)

)∣

∣

∣

∣

p)

dx −−−−→m→∞

0,


uniformly with respect to j ∈ N. Then, we get


m→∞lim supj→∞

∫

Q′(x0α,δ)×I

Wm

(

x,x

εj,xαε2j

; ξ + ∇αuj(x)∣

∣

∣

1

εj∇3uj(x)

)

dx.

For any h ∈ N, we split Q′(x0α, δ) into h2 disjoint cubes Q′

i,h of side length δ/h so that

Q′(x0α, δ) =

h2⋃

i=1

Q′i,h

and


m,h,j

h2∑

i=1

∫

Q′i,h

×IWm

(

x,x

εj,xαε2j

; ξ + ∇αuj(x)∣

∣

∣

1

εj∇3uj(x)

)

dx

> lim supm,λ,h,j

h2∑

i=1

∫

[Q′i,h

×I]∩Rλj

Wm

(

x,x

εj,xαε2j

; ξ + ∇αuj(x)∣

∣

∣

1

εj∇3uj(x)

)

dx

where, given λ > 0, we define

Rλj :=

x ∈ Q′(x0α, δ) × I :

∣

∣

∣

∣

(

ξ + ∇αuj(x)∣

∣

∣

1

εj∇3uj(x)

)∣

∣

∣

∣

6 λ

.

Since Wm is continuous and separately periodic it is in particular uniformly continuous on

Ω × R3 × R2 ×B(0, λ). With similar arguments to that used in the proof of Lemma 5.1.13

(with Wm in place of Wm,λ), we obtain

Wεj(ξ· ;Q′(x0α, δ))

> lim suph→∞

lim supj→∞

h2

δ2

h2∑

i=1

∫

Q′i,h

∫

Q′i,h

×IW

(

x′α, x3,x

εj,xαε2j

; ξ + ∇αuj(x)∣

∣

∣

1

εj∇3uj(x)

)

dx dx′α

>lim suph→∞

h2

δ2

h2∑

i=1

∫

Q′i,h

lim infj→∞

∫

Q′i,h

×IW

(

x′α, x3,x

εj,xαε2j

; ξ + ∇αuj(x)∣

∣

∣

1

εj∇3uj(x)

)

dx dx′α, (5.45)

where we have used Fatou’s Lemma. We now fix x′α ∈ Q′i,h such that W hom(x′α, x3, y, zα; ξ)

is well defined. Then by Theorem 5.2.4 we get that

lim infj→∞

∫

Q′i,h

×IW

(

x′α, x3,x

εj,xαε2j

; ξ + ∇αuj(x)∣

∣

∣

1

εj∇3uj(x)

)

dx > 2δ2

h2W hom(x′α; ξ). (5.46)


Gathering (5.45) and (5.46), it turns out that∫

Q′(x0α,δ)

Wεj(xα; ξ) dxα >

∫

Q′(x0α,δ)

W hom(x′α; ξ) dx′α.

As a consequence the claim follows by the choice of x0α, after dividing the previous inequality

by δ2 and letting δ → 0.

We now prove the converse inequality.

Lemma 5.2.11. For all ξ ∈ Q3×2 and all x0α ∈ L, Wεj(x

0α; ξ) 6 W hom(xα; ξ).

Proof. For every m ∈ N, consider the set Cm and the function Wm given by Lemma B.3,

and define (Wm)hom and (Wm)hom as (5.25) and (5.26), with Wm in place of W . For fixed

η > 0 and any m ∈ N let Kmη be a compact subset of ω given by Scorza-Dragoni’s Theorem

with L2(ω \Kmη ) 6 η and such that (Wm)hom : Km

η × R3×2 → R is continuous .

Step 1. There exists a subsequence mk ր ∞ such that

2 lim infk→∞

∫

Q′(x0α,δ)

(Wmk)hom(xα; ξ) dxα > 2

∫

Q′(x0α,δ)

Wεj

(

xα, ξ)

dxα. (5.47)

To show this inequality we follow an argument similar to that of Lemma 4.2.11. We first

decompose Q′(x0α, δ) into h2 small disjoint cubes Q′

i,h such that

Q′(x0α, δ) =

h2⋃

i=1

Q′i,h and L2(Q′

i,h) = (δ/h)2.

Let

Imh,η :=

i ∈ 1, · · · , h2 : Kmη ∩Q′

i,h 6= ∅

.

For i ∈ Imh,η choose xh,η,mi ∈ Kmη ∩Q′

i,h. By Theorem 5.2.4 and Lemma 3.2.2 there exists a

sequence uj,h,η,mi ⊂W 1,p(Q′i,h × I; R3) with uj,h,η,mi = 0 on ∂Q′

i,h × I, uj,h,η,mi −−−→n→∞

0 in

Lp(Q′i,h × I; R3), and such that

2

∫

Q′i,h

(Wm)hom(xh,η,mi ; ξ) dxα

= limj→∞

∫

Q′i,h

×IWm

(

xh,η,mi , x3,x

εj,xαε2j

; ξ + ∇αuj,h,η,mi

∣

∣

∣

1

εj∇3u

j,h,η,mi

)

dx.


Setting

uη,mj (x) :=

uj,h,η,mi (x) if xα ∈ Q′i,h and i ∈ Imh,η,

0 otherwise,

it follows that uη,mj ⊂W 1,p(Q′(x0α, δ) × I; R3) and uη,mj −−−→

n→∞0 in Lp(Q′(x0

α, δ) × I; R3).

Thus,

2 lim infη,h

∑

i∈Imh,η

∫

Q′i,h

(Wm)hom(xh,η,mi ; ξ) dxα

> lim infη,h,j

∑

i∈Imh,η

∫

Q′i,h

×IWm

(

xh,η,mi , x3,x

εj,xαε2j

; ξ + ∇αuη,mj

∣

∣

∣

1

εj∇3u

η,mj

)

dx.

As in Lemma 4.2.11 we obtain

2

∫

Q′(x0α,δ)

(Wm)hom(xα; ξ) dxα

> lim infλ,η,h,j

∑

i∈Imh,η

∫

(Q′i,h

×I)∩Rλj,η,m

Wm

(

xh,η,mi , x3,x

εj,xαε2j

; ξ + ∇αuη,mj

∣

∣

∣

1

εj∇3u

η,mj

)

dx,

where

Rλj,η,m :=

x ∈ Q′(x0α, δ) × I :

∣

∣

∣

∣

(

ξ + ∇αuη,mj (x)

∣

∣

∣

1

εj∇3u

η,mj (x)

)∣

∣

∣

∣

6 λ

with

L3(

[

Q′(x0α, δ) × I

]

\Rλj,η,m)

6C

λp, (5.48)

for some constant C > 0 independent of j, η, m and λ. Taking into account that Wm is

continuous we get

2

∫

Q′(x0α,δ)

(Wm)hom(xα; ξ) dxα > lim infλ,η,j

∫

Rλj,η,m

Wm

(

x,x

εj,xαε2j

; ξ + ∇αuη,mj

∣

∣

∣

1

εj∇3u

η,mj

)

dx.

By a diagonalization argument, given mk ր ∞, λk ր ∞, and ηk ↓ 0 there exists jk ր ∞such that

2 lim infk→∞

∫

Q′(x0α,δ)

(Wmk)hom(xα; ξ) dxα > lim infk→∞

∫

Rk

Wmk

(

x,x

εjk,xαε2jk

; ξ + ∇αvk

∣

∣

∣

1

εjk∇3vk

)

dx,

where vk := uηk,mk

jk∈ W 1,p(Q′(x0

α, δ) × I; R3) with vk → 0 in Lp(Q′(x0α, δ) × I; R3), and

where Rk := Rλk

jk,ηk,mk. Using Theorem 2.2.17 we can assume, without loss of generality,


that the sequence∣

∣

(

∇αvk| 1εjk

∇3vk)∣

∣

pis equi-integrable. Then, since Wmk = W on

Cmk× R3 × R2 × R3×3,

2 lim infk→∞

∫

Q′(x0α,δ)

(Wmk)hom(xα; ξ) dxα

> lim infk→∞

∫

Q′(x0α,δ)×I

Wmk

(

x,x

εjk,xαε2jk

; ξ + ∇αvk

∣

∣

∣

1

εjk∇3vk

)

dx

> lim infk→∞

∫

[Q′(x0α,δ)×I]∩Cmk

W

(

x,x

εjk,xαε2jk

; ξ + ∇αvk

∣

∣

∣

1

εjk∇3vk

)

dx

= lim infk→∞

∫

Q′(x0α,δ)×I

W

(

x,x

εjk,xαε2jk

; ξ + ∇αvk

∣

∣

∣

1

εjk∇3vk

)

dx

by the growth conditions on W , the p-equi-integrability of the above sequence of scaled

gradients, (0.4) (see the Appendix) and (5.48). As a result we get inequality (5.47).

Step 2. Fixed ρ > 0, let T ∈ N and φ ∈ ST be such that

W hom(x0α; ξ) + ρ >

1

2T 2

∫

(0,T )2×IWhom(x0

α, y3, yα; ξ + ∇αφ(y)|∇3φ(y)) dy. (5.49)

Taking (T, φ) in the definition of (Wmk)hom and recalling Remark 5.2.2 (with Wmk in place

of W ) it follows that

∫

Q′(x0α,δ)


61

2T 2

∫

Q′(x0α,δ)

∫

(0,T )2×I(Wmk)hom(xα, y3, yα; ξ + ∇αφ(y)|∇3φ(y)) dy dxα. (5.50)

Define

Ek := (xα, yα, y3) ∈ Q′(x0α, δ) × (0, T )2 × I : (xα, y3) ∈ Cmk

.

From (0.4) it follows that

L2 ⊗ L3(

[Q′(x0α, δ) × (0, T )2 × I] \ Ek

)

= L3 ⊗ L2((

[Q′(x0α, δ) × I] \ Cmk

)

× (0, T )2)

= T 2L3(

[Q′(x0α, δ) × I] \ Cmk

)

6 T 2/mk. (5.51)


Since (Wmk)hom = Whom on Cmk× R2 × R3×3,

∫

Q′(x0α,δ)

∫

(0,T )2×I(Wmk)hom(xα, y3, yα; ξ + ∇αφ(y)|∇3φ(y)) dy dxα

=

∫

Ek

Whom(xα, y3, yα; ξ + ∇αφ(y)|∇3φ(y)) dy dxα

+

∫

[Q′(x0α,δ)×(0,T )2×I]\Ek

(Wmk)hom(xα, y3, yα; ξ + ∇αφ(y)|∇3φ(y)) dy dxα

6

∫

Q′(x0α,δ)×(0,T )2×I

Whom(xα, y3, yα; ξ + ∇αφ(y)|∇3φ(y)) dy dxα

+C

∫

[Q′(x0α,δ)×(0,T )2×I]\Ek

(1 + |∇φ(y)|p) dy dxα (5.52)

by property (5.30) with Wmk in place of W . Passing to the limit as k → ∞, relations

(5.50), (5.51) and (5.52) yield

lim supk→∞

∫

Q′(x0α,δ)


61

2T 2

∫

Q′(x0α,δ)

∫

(0,T )2×IWhom(xα, y3, yα; ξ + ∇αφ(y)|∇3φ(y)) dy dxα.

Hence by (5.47) we obtain∫

Q′(x0α,δ)

Wεj

(

xα, ξ)

dxα 61

2T 2

∫

Q′(x0α,δ)

∫

(0,T )2×IWhom(xα, y3, yα; ξ + ∇αφ(y)|∇3φ(y)) dy dxα.

As a consequence, by the choice of x0α together with (5.49) we finally get, after dividing the

previous inequality by δ2 and letting δ → 0, that

Wεj

(

x0α, ξ)

61

2T 2

∫

(0,T )2×IWhom(x0

α, y3, yα; ξ + ∇αφ(y)|∇3φ(y)) dy

6 W hom(x0α; ξ) + ρ

and the result follows by letting ρ→ 0.

Proof of Theorem 5.2.1. As a consequence of Lemmas 5.2.11 and 5.2.10, we haveW hom(xα; ξ) =

Wεj(xα; ξ) for all xα ∈ L and all ξ ∈ Q3×2. Since W hom and Wεj are Caratheodory func-

tions, denoting by E the intersection between L and the set of points xα ∈ ω such that both

W hom(xα; ·) and Wεj(xα; ·) are well defined and continuous, we obtain W hom(xα; ξ) =

Wεj(xα; ξ) for all xα ∈ E and all ξ ∈ R3×2. But since L2(ω \ E) = 0, it follows that


the equality holds for a.e. xα ∈ ω. Therefore, we have Wεj(u;A) = Whom(u;A) for all

A ∈ A(ω) and all u ∈ W 1,p(A; R3). Since the result does not depend upon the specific

choice of the subsequence, we obtain by Proposition 8.3 in Dal Maso [35] that the whole

sequence Wε(· ;A) Γ(Lp(A× I))-converges to Whom(· ;A). The proof of Theorem 5.2.1 fol-

lows by taking A = ω.

To conclude we state an interesting consequence of Theorem 5.2.1 that could not be obtained

from the analysis made in Subsection 5.1.

Corollary 5.2.12. Let W : Ω × R3 × R3×3 → R be a function satisfying (A4), and such

that W (x, · ; ξ) is Q-periodic for all ξ ∈ R3×3 and a.e. x ∈ Ω. Define the functional Wε :

Lp(Ω; R3) → R by

Wε(u) :=

∫

ΩW

(

x,x

ε;∇αu(x)

∣

∣

∣

1

ε∇3u(x)

)


∞ otherwise.

Then the Γ(Lp(Ω))-limit of the family Wεε is given by Whom : Lp(Ω; R3) → R with

Whom(u) :=

2

∫

ωW hom(xα,∇αu(xα)) dxα if u ∈W 1,p(ω; R3),

∞ otherwise,

where, for all ξ ∈ R3×2 and a.e. xα ∈ ω

W hom(xα, ξ) := limT→∞

infφ

1

2T 2

∫

(0,T )2×IWhom(xα, y3, yα; ξ + ∇αφ(y)|∇3φ(y)) dy :

φ ∈W 1,p((0, T )2 × I; R3), φ = 0 on ∂(0, T )2 × I

and, for all yα ∈ R2 and a.e. x ∈ Ω

Whom(x, yα; ξ) := limT→∞

infφ

1

T 3

∫

(0,T )3W (x, yα, z3; ξ + ∇φ(z)) dz : φ ∈W 1,p

0 ((0, T )3; R3)

.

6. GENERALIZATIONS AND FURTHER WORK

There are several ways to generalize or improve the results we have presented. First we

could think of energies whose integrands depend also on the function u and not only on its

gradient ∇u. In this direction, we have extended Theorem 4.1.1 by proving the following

result.

Theorem 6.0.13. Let f : Ω × RN × Rd × Rd×N → R be a function such that

(H1) f(x, ·, ·, ·) is continuous a.e. x ∈ Ω;

(H2) f(·, y, s, ξ) is measurable for all y ∈ RN , s ∈ Rd and ξ ∈ Rd×N ;

(H3) f(x, ·, s, ξ) is Q-periodic for a.e. x ∈ Ω, all s ∈ Rd and ξ ∈ Rd×N ;

(H4) there exist number p > 1 and a constant β > 0 such that

|ξ|pβ

− β 6 f(x, y, s, ξ) 6 β(1 + |s|p + |ξ|p),

for a.e. x ∈ Ω, all y ∈ RN , s ∈ Rd and ξ ∈ Rd×N .


Iε(u) :=

∫

Ω f(

x, xε , u(x),∇u(x))


+∞ otherwise.



Iε(u) =

∫

Ω fhom(x, u(x),∇u(x)) dx if u ∈W 1,p(Ω; Rd),

+∞ otherwise,

6. Generalizations and further work 170


fhom(x, s, ξ) := limT→+∞

infφ

1

TN

∫

(0,T )N

f(x, y, s, ξ + ∇φ(y)) dy, φ ∈W 1,p0 ((0, T )N ; Rd)

for a.e. x ∈ Ω, all s ∈ RN and ξ ∈ Rd×N . It turns out that fhom is (equivalent to) a

Caratheodory function and satisfies p-coercivity and p-growth conditions similar to those of

f . Moreover fhom(x, s, · ) is quasiconvex for a.e. x ∈ Ω and all s ∈ Rd.

Another possible generalization is to prove a similar result for integrands that are continuous

with respect to the first variable x and measurable with respect to the second variable y.

In this direction, we were able to prove the following version of Theorem 4.1.1.

Theorem 6.0.14. Let f : Ω × RN × Rd×N → R be a function such that

(H1) f(x, ·, ξ) is measurable for a.e. x ∈ Ω and all ξ ∈ Rd×N ;

(H2) f(·, y, ξ) ∈ C(Ω) for all y ∈ RN and all ξ ∈ Rd×N ;

(H3) f(x, ·, ξ) is Q-periodic for a.e. x ∈ Ω and all ξ ∈ Rd×N ;

(H4) there exist a real number p > 1 and a constant β > 0 such that

|ξ|pβ

− β 6 f(x, y, s, ξ) 6 β(1 + |ξ|p),

for a.e. x ∈ Ω, all y ∈ RN and ξ ∈ Rd×N .


Iε(u) :=

∫

Ωf(

x,x

ε,∇u(x)

)


∞ otherwise.



Iε(u) =

∫


∞ otherwise,

6. Generalizations and further work 171



infφ

1

TN

∫

(0,T )N


for a.e. x ∈ Ω and all ξ ∈ Rd×N . It turns out that fhom is (equivalent to) a Caratheodory

function and satisfies p-coercivity and p-growth conditions similar to those of f . Moreover

fhom(x, · ) is quasiconvex for a.e. x ∈ Ω.

Concerning future work, it would be interesting to extend this study to multiscale gradient

Young measures, and to apply this characterization in different settings, including relaxation

problems. This is the objective of a current collaboration with J. F. Babadjian.

APPENDIX

Appendix 175

A Auxiliary lemmas for periodic homogenization

We start by recalling an auxiliary lemma by Licht and Michaille [59] that allowed us to

justify that the function fhom given in (4.3) is well defined (see Lemma 4.1.4 above).

Lemma A.1. Let N ∈ N with N > 1 and let S : A(RN ) → R+ be such that

i) S(A) 6 βLN (A), for all A ∈ A(RN ), where β is a positive constant,

ii) S(C) 6 S(A) + S(B) for all A,B,C ∈ A(RN ), with A ∩B 6= ∅, C = A ∪B,

iii) there exists T ⊂ RN and M > 0 such that T + [0,M)N = RN and S(A + τ) = S(A)

for all A ∈ A(RN ) and τ ∈ T .

Then, for any cube A of the form [a, b)N there exists the limit of the sequence

S(sA)LN (sA)

as

s→ +∞ and

lims→+∞

S(sA)

LN (sA)= lim

s→+∞

S([0, s)N )

sN.

Furthermore, if SLL is a family of set functions satisfying i), ii), iii) for C, T and M

independent of L, then the above limits are attained uniformly with respect to L.

Next, we recall an auxiliary lemma that was stated in [20] and is useful when diagonalization

arguments are required (see Lemma 4.1.11).

Lemma A.2. Let ak,j be a doubly indexed sequence of real numbers (k, j ր +∞). If

limk→∞

limj→∞

ak,j = L,

then there exists a subsequence k(j)j ր +∞ such that

limj→∞

ak(j),j = L.

B Continuous extension results for the applications to thin

films

We prove here a technical result of extension of Caratheodory functions useful in the proof of

Lemma 5.1.13. It was obtained in collaboration with J. F. Babadjian [9] and uses arguments

analogous to those of Theorem 1, Section 1.2 in Evans and Gariepy [46].

Appendix 176

Lemma B.1. Let Ω ⊂ RN be a bounded open set and f : Ω × Rm × Rd×N → R a function

such that

f(x, · ; · ) is continuous for a.e. x ∈ Ω;

f( · , y ; ξ) is LN -measurable for all y ∈ RN and ξ ∈ RN ;

f(x, · ; ξ) is (0, 1)m-periodic for a.e. x ∈ Ω and all ξ ∈ Rd×N .

Assume also that there exists β > 0 and 1 6 p <∞ such that

1

β|ξ|p − β 6 f(x, y; ξ) 6 β(1 + |ξ|p), for a.e. x ∈ Ω and all (y, ξ) ∈ Rm × Rd×N .

Then for any η > 0 and λ > 0 there exist a compact set Kη ⊂ Ω and a function fη,λ :

RN × Rm × Rd×N → R such that

LN (Ω \Kη) < η,

fη,λ(x, y; ξ) = f(x, y; ξ) for all (x, y; ξ) ∈ Kη × Rm ×B(0, λ),

fη,λ( · , y ; · ) is continuous for all y ∈ Rm,

fη,λ(x, · ; ξ) is continuous and (0, 1)m-periodic for all (x, ξ) ∈ RN × Rd×N ,

and

−β 6 fη,λ(x, y; ξ) 6 β(1 + λp) for all (x, y, ξ) ∈ RN × Rm × Rd×N . (0.1)

Proof. Since f is a Caratheodory function, by Scorza-Dragoni’s Theorem for all η > 0

there exists a compact set Kη ⊂ Ω satisfying LN (Ω\Kη) < η and such that f is continuous

in Kη × Rm × Rd×N . Let Cη,λ := Kη × B(0, λ) ≡ C (to simplify notation) and Uη,λ :=

(RN × Rd×N ) \ Cη,λ ≡ U . Fix (s, γ) ∈ C, and for all (x, ξ) ∈ U set

uη,λ(s,γ)(x, ξ) := max

2 − |(s, γ) − (x, ξ)|dist((x, ξ), C)

, 0

≡ u(s,γ)(x, ξ).

Appendix 177

Clearly,

u(s,γ) is continuous in U,

0 6 u(s,γ) 6 1,

u(s,γ)(x, ξ) = 0 if and only if |(s, γ) − (x, ξ)| > 2dist((x, ξ), C).

Let sηjj>1 ≡ sjj>1 and γλj j>1 ≡ γjj>1 be a countable dense family in Kη and

B(0, λ), respectively. Define

ση,λ(x, ξ) :=∑

j>1

2−ju(sj ,γj)(x, ξ) ≡ σ(x, ξ) for all (x, ξ) ∈ U.

Since σ is the uniform limit of a sequence of continuous functions in U , then σ is continuous

in U . Moreover, for all (x, ξ) ∈ U ,

0 < σ(x, ξ) 6 1.

Indeed, assume that σ(x, ξ) = 0 for some (x, ξ) ∈ U . Then, for all j > 1, u(sj ,γj)(x, ξ) = 0

and thus |(sj , γj) − (x, ξ)| > 2 dist((x, ξ), C). The density of sj , γj in C yields that

|(s, γ) − (x, ξ)| > 2 dist((x, ξ), C)

for all (s, γ) ∈ C. We obtain a contradiction if we choose (s, γ) to be a point of C such that

dist((x, ξ), C) = dist((x, ξ), (s, γ))

so σ(x, ξ) > 0 for all (x, ξ) ∈ U . Consequently, the function

(x, ξ) 7→ vk(x, ξ) ≡ vη,λk (x, ξ) :=2−ku(sk,γk)(x, ξ)

σ(x, ξ)

is well defined and continuous in U . Moreover it satisfies

0 6 vk(x, ξ) 6 1,∑

k>1

vk(x, ξ) = 1 for all (x, ξ) ∈ U. (0.2)

Fix y ∈ Rm and define the continuous extension of f(·, y; · ) outside C as

fη,λ(x, y; ξ) =

f(x, y, ξ) if (x, ξ) ∈ C,∑

k>1

vk(x; ξ) f(sk, y; γk) if (x, ξ) ∈ U.

Appendix 178

Obviously, we have fη,λ(x, y; ξ) = f(x, y; ξ) for all (x, y; ξ) ∈ Kη × Rm × B(0, λ). On the

other hand, if (x, y, ξ) is such that (x, ξ) ∈ U , in view of the p-growth and the p-coercivity

condition on f we get

−β 6 fη,λ(x, y; ξ) 6∑

k>1

vk(x; ξ)β(1 + |γk|p) 6 β(1 + λp).

Since we have

supy∈Rm, (x,ξ)∈U

[

∑

k>n

∣

∣

∣2−ku(sk,γk)(x, ξ)f(sk, y; γk)

∣

∣

∣

]

6 β(1 + λp)∑

k>n

2−k −−−−−→n→+∞

0, (0.3)

then the function

(x, y; ξ) 7→∑

k>1

2−ku(sk,γk)(x, ξ)f(sk, y; γk)

is continuous on (x, y, ξ) : (x, ξ) ∈ U, y ∈ Rm. In particular, for all (x, ξ) ∈ RN × Rd×N

the function fη,λ(x, · ; ξ) is continuous. Further, fη,λ(x, · ; ξ) it is (0, 1)m-periodic because

if i ∈ Zm then for (x, ξ) ∈ U

fη,λ(x, y + i; ξ) =∑

k>1

vk(x; ξ) f(sk, y + i; γk) =∑

k>1

vk(x; ξ) f(sk, y; γk) = fη,λ(x, y; ξ).

Finally, we prove the continuity of fη,λ(·, y;·). By (0.3) it suffices to show that for all

(a,A) ∈ C

limU∋(x,ξ)→(a,A)

fη,λ(x, y; ξ) = f(a, y;A).

As (sj , γj)j>1 is dense in C and f(·, y; ·) is continuous on C, for every ε > 0 there exists

δ > 0 such that

|f(a, y;A) − f(sj , y; γj)| < ε

for all j > 1 with |(a,A) − (sj , γj)| < δ. Assume that |(x, ξ) − (a,A)| < δ/4 and suppose

that j > 1 is such that |(a,A) − (sj , γj)| > δ. Then

δ 6 |(a,A) − (sj , γj)| 6 |(a,A) − (x, ξ)| + |(x, ξ) − (sj , γj)| 6δ

4+ |(x, ξ) − (sj , γj)|,

and thus

|(x, ξ) − (sj , γj)| >3δ

4> 2|(a,A) − (x, ξ)| > 2 dist((x, ξ), C).

Appendix 179

Consequently, vj(x, ξ) = 0 if j is such that |(a,A) − (sj , γj)| > δ, and so by (0.2)

|fη,λ(x, y; ξ) − f(a, y;A)| 6∑

j>1, |(a,A)−(sj ,γj)|<δ

vj(x, ξ)|f(sj , y; γj) − f(a, y;A)| < ε,

since the non zero terms of the sum are those which satisfy |f(a, y;A) − f(sj , y; γj)| < ε.

The continuity of fη,λ(·, y; ·) now follows.

We recall here the Tietze Extension theorem.

Theorem B.2. (Tietze Extension Theorem)(see DiBenedetto [42]) Let X be a normal topo-

logical space. A continuous function f from a closed subset C of X into R has a continuous

extension on X, i.e., there exists a continuous real-valued function f defined on the whole

X, such that f = f on C. Moreover, if f is bounded, i.e., if

|f(x)| 6 C

for all x ∈ C for some C > 0, then f satisfies the same bound.

The following proposition is used in Subsection 5.2.3. It allows us to extend Caratheodory

integrands continuously. It relies on Scorza-Dragoni’s Theorem and on Tietze’s Extension

Theorem.

Lemma B.3. Let W : Ω × R3 × R2 × R3×3 → R satisfy (A1)-(A4) in Section 5.2. Then

for any m ∈ N, there exists a compact set Cm ⊂ Ω and a continuous function Wm :

Ω × R3 × R2 × R3×3 → R such that Wm(x, ·, · ; ·) = W (x, ·, · ; ·) for all x ∈ Cm and

L3(Ω \ Cm) <1

m. (0.4)

Moreover,

- yα 7→Wm(x, yα, y3, zα; ξ) is Q′-periodic for all (zα, y3, ξ) ∈ R3 ×R3×3 and a.e. x ∈ Ω,

- (zα, y3) 7→Wm(x, yα, y3, zα; ξ) is Q-periodic for all (yα, ξ) ∈ R2×R3×3 and a.e. x ∈ Ω;

and for some β > 0, we have

−β 6 Wm(x, y, zα; ξ) 6 β(1+|ξ|p) for all (y, zα, ξ) ∈ R3×R2×R3×3 and a.e x ∈ Ω. (0.5)

Appendix 180

Proof. By Scorza-Dragoni’s Theorem for any m ∈ N there exists a compact set Cm ⊂ Ω

with L3(Ω \ Cm) < 1/m such that W is continuous on Cm × R3 × R2 × R3×3. Since

Cm × R3 × R2 × R3×3 is a closed set, according to Tietze’s Extension Theorem one can

extend W to a continuous function Wm outside Cm×R3 ×R2 ×R3×3. By the construction

of Wm it can be seen that it satisfies the same periodicity and growth condition as W and

that it is bounded from below by −β.

We remark that the above result improves Lemma B.1.

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INDEX

Chebyshev inequality, 14

Caratheodory integrand, 37

Continuous extension theorems, 176, 179

Decomposition lemmas, 28

De Giorgi-Letta’s type criterion, 17

Integral representation theorems, 40, 41

Γ-convergence, 43, 44

Boundary value problems, 49

The fundamental theorem, 46

Jensen inequality, 33

Lebesgue point, 20

Lebesgue-Radon-Nikodym theorem, 19

Measurable Selection criterion, 13

Periodic function, 54

Quasiconvex function, 34

Quasiconvex envelope, 35

Urysohn property of Γ-convergence, 45

Relaxation theorem, 39

Riemann-Lebesgue lemma, 55

Scorza-Dragoni, 37

Two-scale convergence, 56

Vitali Covering theorem, 18

Weierstrass Theorem, 30

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