Stochastic homogenization of nonconvex integrals in the space of functions of bounded deformation

Abstract

We study stochastic homogenization by Γ-convergence of nonconvex integrals of the calculus of variations in the space of functions of bounded deformation.

Keywords

Stochastic homogenization Γ-convergence nonconvex integrand space of functions of bounded deformation

1. Introduction

The space of functions of bounded deformation has been introduced by [31,36,37,40] to study variational problems of plasticity theory (see [22,26,34,38,39]). This space is made of vectorial $L^{1}$ -functions u whose symmetric part of the distributional derivative, i.e. $E u : = \frac{1}{2} (D u + D u^{T})$ , is a vector Radon measure. For such functions u we have $E u = E u d x + E^{s} u$ with $E u : = \frac{1}{2} (\nabla u + \nabla u^{T})$ the symmetrized gradient of u, where $(E u, E^{s} u)$ is the Lebesgue decomposition of $E u$ with respect to the Lebesgue measure $d x$ . In the context of the hyperelastic-plastic theory, at the macroscopic scale, the energy of deformation of a hyperelastic-plastic material occupying in a reference configuration a bounded open set O is of the form $\begin{matrix} (1.1) & \int_{O} W_{macro} (E u) \end{matrix}$ where $W_{macro}$ is the energy density of the hyperelastic-plastic material at the macroscopic scale. From the point of view of homogenization, an important problem is to look for an effective formula for $W_{macro}$ which takes the heterogeneities of the material at the microscopic scale into account. To do this, a classical procedure consists of considering periodic or stochastic energy integrals on regular deformations representing the material at small scales $ε > 0$ , i.e. $\begin{matrix} (1.2) & \int_{O} W (\frac{x}{ε}, E u, ω) d x \end{matrix}$ where $W (\frac{\cdot}{ε}, ξ, ω)$ is the energy density of the material at the scale ε, and to pass to the limit, in the sense of Γ-convergence of De Giorgi, as ε tends to 0. So, under suitable assumptions on W, the problem is to know whether the Γ-limit of (1.2) is of type (1.1) and to find the formula of the energy density $W_{macro}$ which will depend on W. In the periodic and convex case, i.e. when $W (\frac{\cdot}{ε}, ξ)$ is convex with respect to ξ, this Γ-convergence problem was solved by Bouchitté in [10, Theorem 3.2] (see also Ansini and Ebobisse [4, Theorem 5.1]). The object of the present paper is to deal with the stochastic and nonconvex case.

The plan of the paper is as follows. The main result of the paper is stated in Section 2 (see Theorem 2.3). In Section 3 we give auxiliary results needed to prove Theorem 2.3. The proof of Theorem 2.3 follows the classical strategy of Dal Maso and Modica (see [24,25]) in the convex case, extended to the nonconvex case by Messaoudi and Michaille (see [32,33]), and combines a deterministic analysis via blow-up, in the spirit of Fonseca and Müller (see [20,21]), with the subadditive ergodic theorem of Akcoglu and Krengel (see [2,29]). The key tool of subadditive process is recalled in §3.1 together with the properties of the homogenized density, which is defined as the almost sure limit of a subadditive process. In §3.2 we recall some properties of the functions of bounded deformation that we use in the proof of the lower bound and the upper bound. As we deal with integral functionals defined on the space of functions of bounded deformation, to implement the blow-up analysis, in the upper bound we also need a relaxation theorem in the space of functions of bounded deformation due to Kosiba and Rindler (see [28]) and the use of the Vitali envelope of a set function: these are recalled in §3.3 and §3.4 respectively. Finally, Theorem 2.3 is proved in Section 4. Its proof is divided into two propositions: the lower bound (see Proposition 4.1) and the upper bound (see Proposition 4.3).

2. Main result

Let $(Ω, F, P, {τ_{z}}_{z \in Z^{N}})$ be a dynamical system, let $N \in N^{*}$ , let $O \subset R^{N}$ be a bounded open set, let $O (O)$ be the class of open subsets of O and let $BD (O)$ be the space of functions of bounded deformation on O, i.e. $\begin{matrix} BD (O) : = {u \in L^{1} (O; R^{N}) : E u : = \frac{1}{2} (D u + D u^{T}) \in M (O; R_{sym}^{N \times N})}, \end{matrix}$ where $M (O; R_{sym}^{N \times N})$ is the space of $N \times N$ matrix-valued bounded Radon measures on O and $D u$ is the distributional derivative of u. For each $u \in BD (O)$ , $E u = E u d x + E^{s} u$ with $E u (x) : = \frac{1}{2} (\nabla u (x) + \nabla u {(x)}^{T})$ the symmetrized gradient, where $(E u, E^{s} u)$ is the Lebesgue decomposition of $E u$ with respect to the Lebesgue measure on O that we denote by $d x$ . Let $LD (O) \subset BD (O)$ be given by $\begin{matrix} LD (O) : = {u \in BD (O) : E^{s} u = 0} . \end{matrix}$ In this paper we are concerned with stochastic integrals $I_{ε} : BD (O) \times Ω \to [0, \infty]$ , depending on a parameter $ε > 0$ , defined by $\begin{matrix} I_{ε} (u, ω) : = \{\begin{matrix} \int_{O} W (\frac{x}{ε}, E u (x), ω) d x & if u \in LD (O) \\ \infty & if u \in BD (O) ∖ LD (O) \end{matrix} \end{matrix}$ where $W : R^{N} \times R_{sym}^{N \times N} \times Ω \to [0, \infty [$ is a Borel measurable stochastic integrand1

¹
By a Borel measurable stochastic integrand $W : R^{N} \times R_{sym}^{N \times N} \times Ω \to [0, \infty [$ we mean that W is $(B (R^{N}) \otimes B (R_{sym}^{N \times N}) \otimes F, B (R))$ -measurable, where $B (R^{N})$ , $B (R_{sym}^{N \times N})$ and $B (R)$ denote the Borel σ-algebra on $R^{N}$ , $R_{sym}^{N \times N}$ and $R$ respectively.

satisfying the following conditions:

W is ${τ_{z}}_{z \in Z^{N}}$ -covariant or ${τ_{z}}_{z \in Z^{N}}$ -stationary, i.e. $\begin{matrix} W (x + z, ξ, ω) = W (x, ξ, τ_{z} (ω)) \end{matrix}$ for $x \in R^{N}$ , all $ξ \in R_{sym}^{N \times N}$ , all $z \in Z^{N}$ and all $ω \in Ω$ ;

W has 1-growth, i.e. there exist $α, β > 0$ such that for every $ω \in Ω$ , one has $\begin{array}{l} (2.1) & α | ξ | ⩽ W (x, ξ, ω) ⩽ β (1 + | ξ |) \end{array}$ for all $x \in R^{N}$ and all $ξ \in R_{sym}^{N \times N}$ with $R_{sym}^{N \times N}$ denoting the space of $N \times N$ symmetric real matrices;

W is Lipschitz continuous, i.e. there exists $C > 0$ such that for every $ω \in Ω$ , one has $\begin{array}{l} | W (x, ξ, ω) - W (x, ζ, ω) | ⩽ C | ξ - ζ | \end{array}$ for all $x \in R^{N}$ and all $ξ, ζ \in R_{sym}^{N \times N}$ .

The object of the paper is to compute the almost sure Γ-limit of

{I_{ε}}_{ε > 0}

ε \to 0

with respect to the strong convergence of

L^{1} (O; R^{N})

. By the almost sure

Γ (L^{1})

-limit of

{I_{ε}}_{ε > 0}

ε \to 0

we mean a functional

I_{hom} : BD (O) \times Ω \to [0, \infty]

such that for

P

-a.e.

ω \in Ω

, one has:

Γ-

lim_{_}

for every $u \in BD (O)$ , $Γ (L^{1}) - {lim_{_}}_{ε \to 0} I_{ε} (u, ω) ⩾ I_{hom} (u, ω)$ with $\begin{matrix} Γ (L^{1}) - \underset{ε \to 0}{lim_{_}} I_{ε} (u, ω) : = inf {\underset{ε \to 0}{lim_{_}} I_{ε} (u_{ε}, ω) : u_{ε} \to u in L^{1} (O; R^{N})}, \end{matrix}$ or equivalently, for every $u \in BD (O)$ and every ${u_{ε}}_{ε > 0} \subset LD (O)$ such that $u_{ε} \to u$ in $L^{1} (O; R^{N})$ , $\begin{matrix} \underset{ε \to 0}{lim_{_}} I_{ε} (u_{ε}, ω) ⩾ I_{hom} (u, ω); \end{matrix}$

Γ-

lim^{‾}

for every $u \in BD (O)$ , $Γ (L^{1}) - {lim^{‾}}_{ε \to 0} I_{ε} (u, ω) ⩽ I_{hom} (u, ω)$ with $\begin{matrix} Γ (L^{1}) - \underset{ε \to 0}{lim^{‾}} I_{ε} (u, ω) : = inf {\underset{ε \to 0}{lim^{‾}} I_{ε} (u_{ε}, ω) : u_{ε} \to u in L^{1} (O; R^{N})}, \end{matrix}$ or equivalently, for every $u \in BD (O)$ there exists ${u_{ε}}_{ε > 0} \subset LD (O)$ such that $u_{ε} \to u$ in $L^{1} (O; R^{N})$ and $\begin{matrix} \underset{ε \to 0}{lim^{‾}} I_{ε} (u_{ε}, ω) ⩽ I_{hom} (u, ω) . \end{matrix}$

We then write $Γ (L^{1})$ - ${lim}_{ε \to 0} I_{ε} = I_{hom}$ almost surely. (For more details on the theory of Γ-convergence we refer to [23].)

Let $(Ω, F, P)$ be a probability space and let ${τ_{z}}_{z \in Z^{N}}$ be satisfying the following three properties:

$τ_{z} : Ω \to Ω$ is $F$ -measurable for all $z \in Z^{N}$ ;

$τ_{z} \circ τ_{z^{'}} = τ_{z + z^{'}}$ and $τ_{- z} = τ_{z}^{- 1}$ for all $z, z^{'} \in Z^{N}$ ;

$P (τ_{z} (A)) = P (A)$ for all $A \in F$ and all $z \in Z^{N}$ .

Definition 2.1.

Such a ${τ_{z}}_{z \in Z^{N}}$ is said to be a group of $P$ -preserving transformation on $(Ω, F, P)$ and the quadruplet $(Ω, F, P, {τ_{z}}_{z \in Z^{N}})$ is called a measurable dynamical system.

Let $I : = {A \in F : P (τ_{z} (A) Δ A) = 0 for all z \in Z^{N}}$ be the σ-algebra of invariant sets with respect to $(Ω, F, P, {τ_{z}}_{z \in Z^{N}})$ .

Definition 2.2.

When $P (A) \in {0, 1}$ for all $A \in I$ , the measurable dynamical system $(Ω, F, P, {τ_{z}}_{z \in Z^{N}})$ is said to be ergodic.

In what follows, for each bounded open set $A \subset R^{N}$ , ${LD}_{0} (A)$ denotes the subspace of functions in $LD (A)$ that vanish at the boundary of A. The main result of the paper is the following.

Theorem 2.3.

Assume that (C₁), (C₂) and (C₃) hold. Then, $Γ (L^{1}) - {lim}_{ε \to 0} I_{ε} = I_{hom}$ almost surely with $I_{hom} : BD (O) \times Ω \to [0, \infty]$ given by $\begin{matrix} I_{hom} (u, ω) : = \int_{O} W_{hom} (E u (x), ω) d x + \int_{O} W_{hom}^{\infty} (\frac{d E^{s} u}{d | E^{s} u |} (x), ω) d | E^{s} u | (x), \end{matrix}$ where $W_{hom}, W_{hom}^{\infty} : R_{sym}^{N \times N} \times Ω \to [0, \infty [$ are defined by: $\begin{array}{l} W_{hom} (ξ, ω) : = inf_{k \in N^{*}} \frac{1}{k^{N}} E^{I} [inf {\int_{] 0, k [^{N}} W (x, ξ + E v (x), \cdot) d x : v \in {LD}_{0} (] 0, k [^{N})}] (ω); \\ W_{hom}^{\infty} (ξ, ω) : = \underset{t \to \infty}{lim^{‾}} \frac{W_{hom} (t ξ, ω)}{t}, \end{array}$ where $E^{I}$ denotes the conditional expectation over $I$ with respect to $P$ , with $I$ being the σ-algebra of invariant sets with respect to $(Ω, F, P, {τ_{z}}_{z \in Z^{N}})$ . If in addition $(Ω, F, P, {τ_{z}}_{z \in Z^{N}})$ is ergodic, then $W_{hom}$ is deterministic and is given by $\begin{matrix} W_{hom} (ξ) : = inf_{k \in N^{*}} \frac{1}{k^{N}} E [inf {\int_{] 0, k [^{N}} W (y, ξ + E v (y), \cdot) d y : v \in {LD}_{0} (] 0, k [^{N})}], \end{matrix}$ where $E$ denotes the expectation with respect to $P$ .

Periodic homogenization by Γ-convergence for nonconvex Hencky plasticity functionals has been recently studied by Jesenko and Schmidt (see [27]). Analogue results of Theorem 2.3 in the space of functions of bounded variation were obtained by De Arcangelis and Gargiulo in the periodic case (see [16]) and by Abddaimi, Licht and Michaille in the stochastic case (see [1]).

3. Auxiliary results

3.1. Definition and properties of the homogenized density

In what follows, we assume that $(Ω, F, P, {τ_{z}}_{z \in Z^{N}})$ is a measurable dynamical system and we denote the class of bounded Borel subsets of $R^{N}$ by $B_{b} (R^{N})$ . We begin with the definition of a subadditive process.

Definition 3.1.
We say that $S : B_{b} (R^{N}) \to L^{1} (Ω, F, P)$ is a subadditive process if the following four conditions holds:
$S$ is subadditive, i.e. $\begin{matrix} S (B \cup B^{'}) ⩽ S (B) + S (B^{'}) \end{matrix}$ for all $B, B^{'} \in B_{b} (R^{N})$ such that $B \cap B^{'} = \emptyset$ ;

$S$ is ${τ_{z}}_{z \in Z^{N}}$ -covariant, i.e. $\begin{matrix} S (B + z) = S (B) \circ τ_{z} \end{matrix}$ for all $B \in B_{b} (R^{N})$ and all $z \in Z^{N}$ ;

$γ (S) : = inf \{\int_{Ω} \frac{S (I) (ω)}{| I |} d P (ω) : I is an interval in Z^{N}\} > - \infty$ ;

there exists $h \in L^{1} (Ω, F, P)$ such that $| S (Q) | ⩽ h$ for all $Q \in B_{b} (R^{N})$ such that $Q \subset [0, 1 [^{N}$ .

Let $ξ \in R_{sym}^{N \times N}$ and let $S^{ξ} : B_{b} (R^{N}) \to L^{1} (Ω, F, P)$ be defined by $\begin{matrix} (3.1) & S^{ξ} (B) (ω) : = inf {\int_{\overset{˚}{B}} W (x, ξ + E v (x), ω) d x : v \in {LD}_{0} (\overset{˚}{B})} \end{matrix}$ with $W : R^{N} \times R_{sym}^{N \times N} \times Ω \to [0, \infty [$ satisfying (C₁) and (C₂). Then, it is easily seen that $S^{ξ}$ is a subadditive process in the sense of Definition 3.1, and the following proposition is a straightforward consequence of the subadditive ergodic theorem of Akcoglu and Krengel (see [2,29] and Licht and Michaille [30, Theorem 4.1]), the Lipschitz continuity of W, i.e. (C₃), and the fact that $Q_{sym}^{N \times N}$ is dense in $R_{sym}^{N \times N}$ , where $Q_{sym}^{N \times N}$ denotes the space of $N \times N$ symmetric rational matrices.
Proposition 3.2.
Assume that (C₁), (C₂) and (C₃) hold. Then, there exists $\hat{Ω} \in F$ with $P (\hat{Ω}) = 1$ such that for every $ω \in \hat{Ω}$ , one has $\begin{matrix} lim_{ε \to 0} \frac{S^{ξ} (\frac{1}{ε} Q) (ω)}{| \frac{1}{ε} Q |} = inf_{k \in N^{}} \frac{1}{k^{N}} E^{I} [S^{ξ} ([0, k [^{N})] (ω) \end{matrix}$ for all* $ξ \in R_{sym}^{N \times N}$ and all cube Q in $R^{N}$ . If in addition $(Ω, F, P, {τ_{z}}_{z \in Z^{N}})$ is ergodic, then for every $ω \in \hat{Ω}$ , one has $\begin{matrix} lim_{ε \to 0} \frac{S^{ξ} (\frac{1}{ε} Q) (ω)}{| \frac{1}{ε} Q |} = inf_{k \in N^{}} \frac{1}{k^{N}} E [S^{ξ} ([0, k [^{N})] . \end{matrix}$
Definition 3.3.
According to Proposition 3.2 we define $W_{hom} : R_{sym}^{N \times N} \times Ω \to [0, \infty [$ by $\begin{array}{rcl} W_{hom} (ξ, ω) & : = & lim_{ε \to 0} \frac{S^{ξ} (\frac{1}{ε} Q) (ω)}{| \frac{1}{ε} Q |} \\ = & inf_{k \in N^{}} \frac{1}{k^{N}} E^{I} [inf {\int_{] 0, k [^{N}} W (x, ξ + E v (x), \cdot) d x : v \in {LD}_{0} (] 0, k [^{N})}] (ω) . \end{array}$ When $(Ω, F, P, {τ_{z}}_{z \in Z^{N}})$ is ergodic, $W_{hom}$ is deterministic, i.e. $W_{hom} : R_{sym}^{N \times N} \to [0, \infty [$ is given by $\begin{array}{rcl} W_{hom} (ξ) & = & inf_{k \in N^{}} \frac{1}{k^{N}} E [inf {\int_{] 0, k [^{N}} W (x, ξ + E v (x), \cdot) d x : v \in {LD}_{0} (] 0, k [^{N})}] . \end{array}$

Finally, here are some elementary properties of $W_{hom}$ that will be useful in the proof of Proposition 4.3. Proposition 3.4.
Assume that* (C₁), (C₂) and (C₃) hold. Then, $W_{hom}$ has 1-growth and is Lipschitz continuous.

3.2. Some properties of functions of bounded deformation

Here we recall some properties of functions of bounded deformation that we use in the proof of Theorem 2.3. (For more details on the space of bounded deformation, we refer to [3,9,19,41] and the references therein.)

Let $O \subset R^{N}$ , let $M (O; R_{sym}^{N \times N})$ be the space of $N \times N$ matrix-valued bounded Radon measures on O and let $BD (O)$ the space of functions of bounded deformation on O, i.e. $\begin{matrix} BD (O) : = {u \in L^{1} (O; R^{N}) : E u : = \frac{1}{2} (D u + D u^{T}) \in M (O; R_{sym}^{N \times N})}, \end{matrix}$ $D u$ denotes the distributional derivative of u. For each $u \in BD (O)$ we have $\begin{matrix} E u = E u d x + E^{s} u, \end{matrix}$ where $(E u = \frac{d E u}{d x}, E^{s} u)$ is the Lebesgue decomposition of $E u$ with respect to $d x$ the Lebesgue measure on O. Moreover, $E u$ is the approximate symmetrized gradient of u (see Theorem 3.5) and the analogue of Alberti’s rank-one theorem holds (see Theorem 3.6).

Theorem 3.5.
For $L^{N}$ -a.e. $x_{0} \in O$ , $E u (x_{0}) = \frac{1}{2} (\nabla u (x_{0}) + \nabla u {(x_{0})}^{T})$ and $\begin{matrix} lim_{ρ \to 0} \frac{1}{| Q_{ρ} (x_{0}) |} \int_{Q_{ρ} (x_{0})} \frac{| u - u_{x_{0}} |}{ρ} d x = 0, \end{matrix}$ where $u_{x_{0}}$ is the affine function defined by $u_{x_{0}} (x) : = u (x_{0}) + \nabla u (x_{0}) (x - x_{0})$ and $Q_{ρ} (x_{0}) : = x_{0} + ρ Q$ with Q the unit cell centered at the origin.
Theorem 3.6.
Let $u \in BD (O)$ . Then, for $| E^{s} u |$ -a.e. $x_{0} \in O$ there exist $a (x_{0}) \in R^{N}$ and $b (x_{0}) \in R^{N}$ with $| a (x_{0}) | = | b (x_{0}) | = 1$ such that $\begin{matrix} (3.2) & \frac{d E u}{d | E u |} (x_{0}) = a (x_{0}) ⊙ b (x_{0}), \end{matrix}$ where $a (x_{0}) ⊙ b (x_{0}) : = \frac{1}{2} (a (x_{0}) \otimes b (x_{0}) + b (x_{0}) \otimes a (x_{0}))$ is the symmetric tensor product of the vectors $a (x_{0})$ and $b (x_{0})$ .

Theorem 3.6, which will be used in the proof of Proposition 4.1, has recently been established by De Philippis and Rindler in [17]. The following two lemmas will be also useful in the proof of Proposition 4.1.
Lemma 3.7.
Let $u \in BD (O)$ , let $x_{0} \in supp (| E^{s} u |)$ be such that ( 3.2 ) holds and let Q be the unit cube centered at the origin whose the sides are either orthogonal or parallel to $a (x_{0})$ .Then: $\begin{array}{rcl} (3.3) & \begin{array}{l} lim_{ρ \to 0} \frac{| E u | (Q_{ρ} (x_{0}))}{ρ^{N}} = \infty; \\ \underset{ρ \to 0}{lim^{‾}} \frac{| E u | (Q_{δ ρ} (x_{0}))}{| E u | (Q_{ρ} (x_{0}))} ⩾ δ^{N} for all δ \in] 0, 1 [. \end{array} \end{array}$
Lemma 3.8.
Let $u \in BD (O)$ , let $x_{0} \in supp (| E^{s} u |)$ be such that ( 3.2 ) holds and, for each $ρ > 0$ , let $v_{ρ} \in BD (Q)$ be defined by $\begin{matrix} (3.4) & v_{ρ} (x) : = \frac{ρ^{N - 1}}{| E u | (Q_{ρ} (x_{0}))} (u (x_{0} + ρ x) - \frac{1}{ρ^{N}} \int_{Q_{ρ} (x_{0})} u (y) d y) + R_{ρ} (x), \end{matrix}$ where $R_{ρ} : R^{N} \to R^{N}$ is a rigid deformation.2
²
By a rigid deformation we mean a map $R : R^{N} \to R^{N}$ defined by $R (x) = S x + σ$ for all $x \in R^{N}$ , where S is a $N \times N$ skew-symmetric matrix and $σ \in R^{N}$ .

Then:
$E v_{ρ} (Q) = \frac{E u (Q_{ρ} (x_{0}))}{| E u | (Q_{ρ} (x_{0}))}$ and $\begin{matrix} (3.5) & lim_{ρ \to 0} E v_{ρ} (Q) = lim_{ρ \to 0} \frac{E u (Q_{ρ} (x_{0}))}{| E u | (Q_{ρ} (x_{0}))} = a (x_{0}) ⊙ b (x_{0}); \end{matrix}$

up to a subsequence, $v_{ρ} \to v$ in $L^{1} (Q; R^{N})$ and $E v_{ρ} ⇀ E v$ weakly in $M (O; R_{sym}^{N \times N})$ with $v \in BD (Q)$ defined by $\begin{matrix} (3.6) & v (x) : = \bar{v} (⟨ b (x_{0}), x ⟩) a (x_{0}) + c (a (x_{0}) \otimes b (x_{0})) x + R (x), \end{matrix}$ where $\bar{v} :] - \frac{1}{2}, \frac{1}{2} [\to R$ is bounded and increasing, $c > 0$ and $R : R^{N} \to R^{N}$ is a rigid deformation. Moreover, for a.e. $δ \in] 0, 1 [$ , one has $\begin{matrix} (3.7) & lim_{ρ \to 0} E v_{ρ} (δ Q) = E v (δ Q) . \end{matrix}$

For a proof of Lemmas 3.7 and 3.8 we refer to [18,28].
3.3. A relaxation theorem in the space of functions of bounded deformation

The following result has been recently established by Kosiba and Rindler (see [28, Theorem 1.3] and also [8,14,15,35]).

Theorem 3.9.
Let $O \subset R^{N}$ be a bounded open set, let $V : R_{sym}^{N \times N} \to [0, \infty [$ be a continuous and symmetric quasiconvex integrand having 1-growth, let $J : BD (O) \to [0, \infty]$ defined by $\begin{matrix} J (u) : = \{\begin{matrix} \int_{O} V (E u (x)) d x & if u \in LD (O) \\ \infty & if u \in BD (O) ∖ LD (O) \end{matrix} \end{matrix}$ and let $\overline{J} : BD (O) \to [0, \infty]$ be the $L^{1}$ -lower semicontinuous envelope of J, i.e. $\begin{matrix} \overline{J} (u) : = inf \{\underset{n \to \infty}{lim_{_}} J (u_{n}) : u_{n} \to u in L^{1} (O; R^{N})\} . \end{matrix}$ Then, for every $u \in BD (O)$ , one has $\begin{matrix} \overline{J} (u) = \int_{O} V (E u (x)) d x + \int_{O} V^{\infty} (\frac{d E^{s} u}{d | E^{s} u |} (x)) d | E^{s} u | (x) \end{matrix}$ with $V^{\infty} : R_{sym}^{N \times N} \to [0, \infty [$ given by $V^{\infty} (ξ) : = {lim^{‾}}_{t \to \infty} \frac{V (t ξ)}{t}$ .

3.4. Integral representation of the Vitali envelope of a set function

What follows was first developed in [11,12] (see also [5–7]). Let $O \subset R^{N}$ be a bounded open set and let $O (O)$ be the class of open subsets of O. We begin with the concept of the Vitali envelope of a set function.

Definition 3.10.
Given $S : O (O) \to [0, \infty]$ , for each $δ > 0$ we define $S^{δ} : O (O) \to [0, \infty]$ by $\begin{matrix} (3.8) & S^{δ} (A) : = inf {\sum_{i \in I} S (Q_{i}) : {Q_{i}}_{i \in I} \in V_{δ} (A)} . \end{matrix}$ By the Vitali envelope of $S$ we call the set function $S^{} : O (O) \to [- \infty, \infty]$ defined by $\begin{matrix} (3.9) & S^{} (A) : = sup_{δ > 0} S^{δ} (A) = lim_{δ \to 0} S^{δ} (A) . \end{matrix}$

The interest of Definition 3.10 comes from the following integral representation result. (For a proof we refer to [5, §A.4].) Theorem 3.11.
Let $S : O (O) \to [0, \infty]$ be a set function satisfying the following two conditions:
there exists a finite Radon measure ν on O which is absolutely continuous with respect to $d x$ such that $S (A) ⩽ ν (A)$ for all $A \in O (O)$ ;

$S$ is subadditive, i.e. $S (A) ⩽ S (B) + S (C)$ for all $A, B, C \in O (O)$ with $B, C \subset A$ , $B \cap C = \emptyset$ and $| A ∖ B \cup C | = 0$ .
Then ${lim}_{ρ \to 0} \frac{S (Q_{ρ} (\cdot))}{| Q_{ρ} (\cdot) |} \in L^{1} (O)$ and for every $A \in O (O)$ , one has $\begin{matrix} S^{*} (A) = \int_{A} lim_{ρ \to 0} \frac{S (Q_{ρ} (x))}{| Q_{ρ} (x) |} d x . \end{matrix}$

4. Proof of the homogenization theorem

Theorem 2.3 is a direct consequence of the following two propositions (see Proposition 4.1 in §4.1 and Proposition 4.3 in §4.2).

4.1. The lower bound

Here we establish that $Γ (L^{1}) - {lim_{_}}_{ε \to 0} I_{ε} ⩾ I_{hom}$ .

Proposition 4.1 (lower bound).

Under the assumption of Theorem 2.3 , for $P$ -a.e. $ω \in Ω$ , one has $\begin{matrix} (4.1) & \underset{ε \to 0}{lim_{_}} I_{ε} (u_{ε}, ω) ⩾ I_{hom} (u, ω) \end{matrix}$ for all $u \in BD (O)$ and all ${u_{ε}}_{ε > 0} \subset LD (O)$ such that $u_{ε} \to u$ in $L^{1} (O; R^{N})$ .

Proof of Proposition 4.1.
The proof of this proposition follows the same method as in [1, Theorem 3.1] and uses blow-up techniques in the spirit of Fonseca and Müller (see [20,21]). Let $ω \in \hat{Ω}$ where $\hat{Ω} \in F$ is given by Proposition 3.2. Let $u \in BD (O)$ and let ${u_{ε}}_{ε > 0} \subset LD (O)$ be such that $u_{ε} \to u$ in $L^{1} (O; R^{N})$ . Without loss of generality we can assume that $\begin{matrix} (4.2) & \underset{ε \to 0}{lim_{_}} I_{ε} (u_{ε}, ω) = lim_{ε \to 0} I_{ε} (u_{ε}, ω) < \infty, and so sup_{ε > 0} I_{ε} (u_{ε}, ω) < \infty . \end{matrix}$ For each $ε > 0$ , we define the (positive) Radon measure $μ_{ε}$ on O by $\begin{matrix} μ_{ε} : = W (\frac{\cdot}{ε}, E u_{ε} (\cdot), ω) d x . \end{matrix}$ From (4.2) we see that ${sup}_{ε > 0} μ_{ε} (O) < \infty$ , and so there exists a (positive) Radon measure μ on O such that (up to a subsequence) $μ_{ε} ⇀ μ$ weakly. By Lebesgue’s decomposition theorem, we have $μ = μ^{a} + μ^{s}$ where $μ^{a}$ and $μ^{s}$ are (positive) Radon measures on O such that $μ^{a} ≪ d x$ and $μ^{s} ⊥ d x$ . Thus, to prove (4.1) it suffices to show that: $\begin{array}{l} (4.3) & μ^{a} ⩾ W_{hom} (E u (\cdot), ω) d x; \\ (4.4) & μ^{s} ⩾ W_{hom}^{\infty} (\frac{d E^{s} u}{d | E^{s} u |} (\cdot), ω) | E^{s} u | . \end{array}$ Proof of ( 4.3 ). It suffices to prove that $\begin{matrix} (4.5) & lim_{ρ \to 0} \frac{μ (Q_{ρ} (x_{0}))}{| Q_{ρ} (x_{0}) |} ⩾ W_{hom} (E u (x_{0}), ω) \end{matrix}$ for $d x$ -a.a. $x_{0} \in O$ with $Q_{ρ} (x_{0}) : = x_{0} + ρ Q$ where Q is the unit cell centered at the origin. As $μ (O) < \infty$ without loss of generality we can assume that $μ (\partial Q_{ρ} (x_{0})) = 0$ for all $ρ > 0$ , and so to prove (4.5) it is sufficient to establish that $\begin{matrix} (4.6) & lim_{ρ \to 0} lim_{ε \to 0} \frac{μ_{ε} (Q_{ρ} (x_{0}))}{| Q_{ρ} (x_{0}) |} ⩾ W_{hom} (E u (x_{0}), ω) . \end{matrix}$ Fix any $ε > 0$ , any $ρ > 0$ and any $s \in] 0, 1 [$ . Fix any $q \in N^{}$ and consider ${Q_{i}}_{i \in {0, \dots, q}} \subset Q_{ρ} (x_{0})$ given by $\begin{matrix} Q_{i} : = \{\begin{matrix} Q_{s ρ} (x_{0}) & if i = 0 \\ Q_{s ρ + \frac{i}{q} ρ (1 - s)} (x_{0}) & if i \in {1, \dots, q} . \end{matrix} \end{matrix}$ For every $i \in {1, \dots, q}$ , consider a cut-off function3
³
By a cut-off function from O to $R$ for the pair $(O ∖ V, K)$ , where $K \subset V \subset O$ with K compact and V open, we mean $φ \in C^{\infty} (O)$ such that $φ (x) \in [0, 1]$ for all $x \in O$ , $φ (x) = 0$ for all $x \in O ∖ V$ and $φ (x) = 1$ for all $x \in K$ .

$φ_{i} \in C^{\infty} (O)$ for the pair $(O ∖ Q_{i}, {\overline{Q}}_{i - 1})$ such that $\begin{matrix} ‖ \nabla φ_{i} ‖_{L^{\infty} (O; R^{N})} ⩽ \frac{q}{ρ (1 - s)} \end{matrix}$ and define $u_{ε}^{i} \in u_{x_{0}} + {LD}_{0} (Q_{ρ} (x_{0}))$ by $\begin{matrix} u_{ε}^{i} : = u_{x_{0}} + φ_{i} (u_{ε} - u_{x_{0}}) \end{matrix}$ with $u_{x_{0}} (x) : = u (x_{0}) + \nabla u (x_{0}) (x - x_{0})$ . Fix any $i \in {1, \dots, q}$ . We then have $\begin{matrix} E u_{ε}^{i} = \{\begin{matrix} E u_{ε} & in Q_{i - 1} \\ E u (x_{0}) + φ_{i} E (u_{ε} - u_{x_{0}}) + \nabla φ_{i} ⊙ (u_{ε} - u_{x_{0}}) & in Q_{i} ∖ Q_{i - 1} \\ E u (x_{0}) & in Q_{ρ} (x_{0}) ∖ Q_{i}, \end{matrix} \end{matrix}$ and so $\begin{array}{rcl} \int_{Q_{ρ} (x_{0})} W (\frac{x}{ε}, E u_{ε}^{i}, ω) d x & = & \int_{Q_{i - 1}} W (\frac{x}{ε}, E u_{ε}, ω) d x + \int_{Q_{i} ∖ Q_{i - 1}} W (\frac{x}{ε}, E u_{ε}^{i}, ω) d x \\ + \int_{Q_{ρ} (x_{0}) ∖ Q_{i}} W (\frac{x}{ε}, E u (x_{0}), ω) d x . \end{array}$ Taking the right inequality in (2.1) into account, we see that: $\begin{array}{l} \begin{matrix} \frac{1}{| Q_{ρ} (x_{0}) |} \int_{Q_{i - 1}} W (\frac{x}{ε}, E u_{ε}, ω) d x & ⩽ \frac{1}{| Q_{ρ} (x_{0}) |} \int_{Q_{ρ} (x_{0})} W (\frac{x}{ε}, E u_{ε}, ω) d x \\ ⩽ \frac{μ_{ε} (Q_{ρ} (x_{0}))}{| Q_{ρ} (x_{0}) |}; \end{matrix} \\ \begin{matrix} \frac{1}{| Q_{ρ} (x_{0}) |} \int_{Q_{i} ∖ Q_{i - 1}} W (\frac{x}{ε}, E u_{ε}^{i}, ω) d x ⩽ & c (1 - s^{N}) + \frac{β}{| Q_{ρ} (x_{0}) |} \int_{Q_{i} ∖ Q_{i - 1}} | E (u_{ε} - u_{x_{0}}) | d x \\ + \frac{β q}{(1 - s)} \frac{1}{| Q_{ρ} (x_{0}) |} \int_{Q_{i} ∖ Q_{i - 1}} \frac{| u_{ε} - u_{x_{0}} |}{ρ} d x; \end{matrix} \\ \frac{1}{| Q_{ρ} (x_{0}) |} \int_{Q_{ρ} (x_{0}) ∖ Q_{i}} W (\frac{x}{ε}, E u (x_{0}), ω) d x ⩽ c (1 - s^{N}), \end{array}$ where $c : = β (1 + | E u (x_{0}) |)$ . Consider $S^{E u (x_{0})} (\cdot) (ω)$ defined by (3.1) with $ξ = E u (x_{0})$ . From the above we deduce that $\begin{array}{rcl} \frac{S^{E u (x_{0})} (\frac{1}{ε} Q_{ρ} (x_{0})) (ω)}{| \frac{1}{ε} Q_{ρ} (x_{0}) |} & ⩽ & \frac{1}{| Q_{ρ} (x_{0}) |} \int_{Q_{ρ} (x_{0})} W (\frac{x}{ε}, E u_{ε}^{i}, ω) d x \\ ⩽ & \frac{μ_{ε} (Q_{ρ} (x_{0}))}{| Q_{ρ} (x_{0}) |} + 2 c (1 - s^{N}) + \frac{β}{| Q_{ρ} (x_{0}) |} \int_{Q_{i} ∖ Q_{i - 1}} | E (u_{ε} - u_{x_{0}}) | d x \\ + \frac{β q}{(1 - s)} \frac{1}{| Q_{ρ} (x_{0}) |} \int_{Q_{i} ∖ Q_{i - 1}} \frac{| u_{ε} - u_{x_{0}} |}{ρ} d x, \end{array}$ and averaging these inequalities, we obtain $\begin{array}{rcl} \frac{S^{E u (x_{0})} (\frac{1}{ε} Q_{ρ} (x_{0})) (ω)}{| \frac{1}{ε} Q_{ρ} (x_{0}) |} & ⩽ & \frac{μ_{ε} (Q_{ρ} (x_{0}))}{| Q_{ρ} (x_{0}) |} + 2 c (1 - s^{N}) \\ + \frac{1}{q} \frac{β}{| Q_{ρ} (x_{0}) |} \int_{Q_{ρ} (x_{0}) ∖ Q_{s ρ} (x_{0})} | E (u_{ε} - u_{x_{0}}) | d x \\ + \frac{β}{(1 - s)} \frac{1}{| Q_{ρ} (x_{0}) |} \int_{Q_{ρ} (x_{0})} \frac{| u_{ε} - u_{x_{0}} |}{ρ} d x . \end{array}$ Taking Proposition 3.2 (and Definition 3.3) and Theorem 3.5, letting $q \to \infty$ and then $ε \to 0$ and $ρ \to 0$ , we conclude that $\begin{array}{rcl} W_{hom} (E u (x_{0}), ω) & ⩽ & lim_{ρ \to 0} lim_{ε \to 0} \frac{μ_{ε} (Q_{ρ} (x_{0}))}{| Q_{ρ} (x_{0}) |} + 2 c (1 - s^{N}), \end{array}$ (4.6) follows by letting $s \to 1$ .

Proof of ( 4.4 ). It suffices to prove that $\begin{matrix} (4.7) & lim_{ρ \to 0} \frac{μ (Q_{ρ} (x_{0}))}{| E u | (Q_{ρ} (x_{0}))} ⩾ W_{hom}^{\infty} (a (x_{0}) ⊙ b (x_{0}), ω) \end{matrix}$ for $| E^{s} u |$ -a.a. $x_{0} \in O$ such that (3.2) holds with $Q_{ρ} (x_{0}) : = x_{0} + ρ Q$ where Q is the unit cube centered at the origin whose sides are either orthogonal or parallel to $b (x_{0})$ . Fix such a $x_{0}$ . As $μ (O) < \infty$ , without loss of generality we can assume that $μ (\partial Q_{ρ} (x_{0})) = 0$ for all $ρ > 0$ , and so to prove (4.7) it is sufficient to establish that $\begin{matrix} (4.8) & lim_{ρ \to 0} lim_{ε \to 0} \frac{μ_{ε} (Q_{ρ} (x_{0}))}{| E u | (Q_{ρ} (x_{0}))} ⩾ W_{hom}^{\infty} (a (x_{0}) ⊙ b (x_{0}), ω) . \end{matrix}$ For each $ρ > 0$ and each $ε > 0$ , let $v_{ρ} \in BD (Q)$ be given by (3.4), let $v_{ρ, ε} \in LD (Q)$ be defined by $\begin{array}{rcl} v_{ρ, ε} (x) : = \frac{ρ^{N - 1}}{| E u | (Q_{ρ} (x_{0}))} (u_{ε} (x_{0} + ρ x) - \frac{1}{ρ^{N}} \int_{Q_{ρ} (x_{0})} u_{ε} (y) d y) + R_{ρ} (x) \end{array}$ and set $\begin{array}{rcl} t_{ρ} : = \frac{| E u | (Q_{ρ} (x_{0}))}{ρ^{N}} . \end{array}$ Then, as $u_{ε} \to u$ in $L^{1} (O; R^{N})$ and by using Lemma 3.7, we have: $\begin{array}{l} (4.9) & lim_{ε \to 0} ‖ v_{ρ, ε} - v_{ρ} ‖_{L^{1} (O; R^{N})} = 0 for all ρ > 0; \\ (4.10) & lim_{ρ \to 0} t_{ρ} = \infty . \end{array}$ From Lemma 3.8(ii), up to a subsequence, we have $\begin{matrix} (4.11) & κ_{ρ} : = ‖ v_{ρ} - v ‖_{L^{1} (Q; R^{N})}^{\frac{1}{2}} \to 0 as ρ \to 0 \end{matrix}$ with $v \in BD (Q)$ given by (3.6). Fix any $δ \in] 0, 1 [$ such that (3.7) holds. Fix any $ρ > 0$ and any $ε > 0$ . Let $u_{ρ, ε} \in LD (Q)$ be defined by $\begin{matrix} (4.12) & u_{ρ, ε} : = t_{ρ} v_{ρ, ε} . \end{matrix}$ First of all, it is easy to see that $\begin{array}{rcl} (4.13) & \frac{1}{| E u | (Q_{ρ} (x_{0}))} \int_{Q_{δ ρ} (x_{0})} W (\frac{x}{ε}, E u_{ε}, ω) d x = \frac{1}{t_{ρ}} \int_{δ Q} W (\frac{x_{0} + ρ x}{ε}, E u_{ρ, ε}, ω) d x . \end{array}$ On the other hand, fix any $q \in N^{}$ and consider ${Q_{i}}_{i \in {0, \dots, q}} \subset δ Q$ given by $\begin{matrix} Q_{i} : = \{\begin{matrix} (1 - κ_{ρ}) δ Q & if i = 0 \\ (1 - κ_{ρ} + i \frac{κ_{ρ}}{q}) δ Q & if i \in {1, \dots, q} . \end{matrix} \end{matrix}$ For every $i \in {1, \dots, q}$ , consider a cut-off function $φ_{i} \in C^{\infty} (O)$ for the pair $(O ∖ Q_{i}, {\overline{Q}}_{i - 1})$ such that $\begin{matrix} (4.14) & ‖ \nabla φ_{i} ‖_{L^{\infty} (O; R^{N})} ⩽ \frac{q}{κ_{ρ}} \end{matrix}$ and define $u_{ρ, ε}^{i} \in t_{ρ} Θ_{δ} + {LD}_{0} (δ Q)$ by $\begin{matrix} u_{ρ, ε}^{i} : = t_{ρ} Θ_{δ} + φ_{i} (u_{ρ, ε} - t_{ρ} Θ_{δ}) \end{matrix}$ where $Θ_{δ}$ is the affine function defined by $\begin{matrix} Θ_{δ} (x) : = \frac{E v (δ Q)}{δ^{N}} x + \frac{Ψ ({(\frac{δ}{2})}^{-}) + Ψ ({(- \frac{δ}{2})}^{+})}{2} a (x_{0}) + R (x) \end{matrix}$ with $Ψ :] - \frac{1}{2}, \frac{1}{2} [\to R$ given by $Ψ (r) : = \overline{v} (r) + c r$ , $Ψ ({(\frac{δ}{2})}^{-}) : = {lim}_{r ↑ \frac{δ}{2}} Ψ (r)$ and $Ψ ({(- \frac{δ}{2})}^{+}) : = {lim}_{r ↓ - \frac{δ}{2}} Ψ (r)$ , where $\overline{v} :] - \frac{1}{2}, \frac{1}{2} [\to R$ , $c > 0$ and R (a rigid deformation) are given by Lemma 3.8(ii). (Note that the trace of v and $Θ_{δ}$ are equal on the faces of $δ Q$ orthogonal to $b (x_{0})$ .) Fix any $i \in {1, \dots, q}$ . We then have $\begin{matrix} E u_{ρ, ε}^{i} = \{\begin{matrix} E u_{ρ, ε} & in Q_{i - 1} \\ \frac{t_{ρ}}{δ^{N}} E v (δ Q) + φ_{i} E (u_{ρ, ε} - t_{ρ} Θ_{δ}) + \nabla φ_{i} ⊙ (u_{ρ, ε} - t_{ρ} Θ_{δ}) & in Q_{i} ∖ Q_{i - 1} \\ \frac{t_{ρ}}{δ^{N}} E v (δ Q) & in δ Q ∖ Q_{i} . \end{matrix} \end{matrix}$ Hence $\begin{array}{rcl} \frac{1}{t_{ρ}} \int_{δ Q} W (\frac{x_{0} + ρ x}{ε}, E u_{ρ, ε}^{i}, ω) d x & ⩽ & \frac{1}{t_{ρ}} \int_{δ Q} W (\frac{x_{0} + ρ x}{ε}, E u_{ρ, ε}, ω) d x \\ + \frac{1}{t_{ρ}} \int_{Q_{i} ∖ Q_{i - 1}} W (\frac{x_{0} + ρ x}{ε}, E u_{ρ, ε}^{i}, ω) d x \\ (4.15) & + \frac{1}{t_{ρ}} \int_{δ Q ∖ Q_{i}} W (\frac{x_{0} + ρ x}{ε}, \frac{t_{ρ}}{δ^{N}} E v (δ Q), ω) d x . \end{array}$ Moreover, taking (4.12) and (4.14) into account, from the right inequality in (2.1) we see that: $\begin{array}{l} (4.16) & \frac{1}{t_{ρ}} \int_{δ Q ∖ Q_{i}} W (\frac{x_{0} + ρ x}{ε}, \frac{t_{ρ}}{δ^{N}} E v (δ Q), ω) d x ⩽ Δ_{1} (ρ, δ); \\ (4.17) & \begin{matrix} \frac{1}{t_{ρ}} \int_{Q_{i} ∖ Q_{i - 1}} W (\frac{x_{0} + ρ x}{ε}, E u_{ρ, ε}^{i}, ω) d x ⩽ & Δ_{1} (ρ, δ) + \frac{β}{t_{ρ}} \int_{Q_{i} ∖ Q_{i - 1}} | E (u_{ρ, ε} - t_{ρ} Θ_{δ}) | d x \\ + \frac{β q}{κ_{ρ}} \int_{Q_{i} ∖ Q_{i - 1}} | v_{ρ, ε} - Θ_{δ} | d x \end{matrix} \end{array}$ with $Δ_{1} (ρ, δ) : = \frac{β δ^{N}}{t_{ρ}} + \frac{β (1 - {(1 - κ_{ρ})}^{N})}{δ^{N}} | E v (δ Q) |$ . Note that by (4.10) and (4.11) we have $\begin{matrix} (4.18) & Δ_{1} (ρ, δ) \to 0 as ρ \to 0 . \end{matrix}$ Set $ξ_{ρ, δ} : = \frac{t_{ρ}}{δ^{N}} E v (δ Q) \in R_{sym}^{N \times N}$ and consider $S^{ξ_{ρ, δ}} (\cdot) (ω)$ defined by (3.1) with $ξ = ξ_{ρ, δ}$ . From (4.13), (4.15), (4.16) and (4.17) we deduce that $\begin{array}{rcl} \frac{1}{| E u | (Q_{ρ} (x_{0}))} \int_{Q_{δ ρ} (x_{0})} W (\frac{x}{ε}, E u_{ε}, ω) d x & ⩾ & \frac{δ^{N}}{t_{ρ}} \frac{S^{ξ_{ρ, δ}} (\frac{1}{ε} Q_{ρ} (x_{0})) (ω)}{| \frac{1}{ε} Q_{ρ} (x_{0}) |} - 2 Δ_{1} (ρ, δ) \\ - \frac{β}{t_{ρ}} \int_{Q_{i} ∖ Q_{i - 1}} | E (u_{ρ, ε} - t_{ρ} Θ_{δ}) | d x \\ - \frac{β q}{κ_{ρ}} \int_{Q_{i} ∖ Q_{i - 1}} | v_{ρ, ε} - Θ_{δ} | d x \end{array}$ for all $i \in {1, \dots, q}$ , and averaging these inequalities, it follows that $\begin{array}{rcl} \frac{1}{| E u | (Q_{ρ} (x_{0}))} \int_{Q_{δ ρ} (x_{0})} W (\frac{x}{ε}, E u_{ε}, ω) d x & ⩾ & \frac{δ^{N}}{t_{ρ}} \frac{S^{ξ_{ρ, δ}} (\frac{1}{ε} Q_{ρ} (x_{0})) (ω)}{| \frac{1}{ε} Q_{ρ} (x_{0}) |} - 2 Δ_{1} (ρ, δ) \\ - \frac{1}{q} \frac{β}{t_{ρ}} \int_{δ Q ∖ (1 - κ_{ρ}) δ Q} | E (u_{ρ, ε} - t_{ρ} Θ_{δ}) | d x \\ - \frac{β}{κ_{ρ}} \int_{δ Q ∖ (1 - κ_{ρ}) δ Q} | v_{ρ, ε} - Θ_{δ} | d x . \end{array}$ But $\begin{array}{rcl} \frac{β}{κ_{ρ}} \int_{δ Q ∖ (1 - κ_{ρ}) δ Q} | v_{ρ, ε} - Θ_{δ} | d x & ⩽ & \frac{β}{κ_{ρ}} ‖ v_{ρ, ε} - v ‖_{L^{1} (Q; R^{N})} + \frac{β}{κ_{ρ}} \int_{δ Q ∖ (1 - κ_{ρ}) δ Q} | v - Θ_{δ} | d x, \end{array}$ and so, for every $ε > 0$ and every $q \in N^{*}$ , one has $\begin{array}{rcl} \frac{1}{| E u | (Q_{ρ} (x_{0}))} \int_{Q_{δ ρ} (x_{0})} W (\frac{x}{ε}, E u_{ε}, ω) d x & ⩾ & \frac{δ^{N}}{t_{ρ}} \frac{S^{ξ_{ρ, δ}} (\frac{1}{ε} Q_{ρ} (x_{0})) (ω)}{| \frac{1}{ε} Q_{ρ} (x_{0}) |} - 2 Δ_{1} (ρ, δ) \\ - \frac{1}{q} \frac{β}{t_{ρ}} \int_{δ Q ∖ (1 - κ_{ρ}) δ Q} | E (u_{ρ, ε} - t_{ρ} Θ_{δ}) | d x \\ - \frac{β}{κ_{ρ}} ‖ v_{ρ, ε} - v ‖_{L^{1} (Q; R^{N})} \\ - \frac{β}{κ_{ρ}} \int_{δ Q ∖ (1 - κ_{ρ}) δ Q} | v - Θ_{δ} | d x . \end{array}$ Taking Proposition 3.2 (and Definition 3.3) and (4.9) into account and recalling that $κ_{ρ} = ‖ v_{ρ} - v ‖_{L^{1} (Q; R^{N})}^{\frac{1}{2}}$ and $ξ_{ρ, δ} = \frac{t_{ρ}}{δ^{N}} E v (δ Q)$ , letting $q \to \infty$ and then $ε \to 0$ , we obtain $\begin{array}{rcl} lim_{ε \to 0} \frac{μ_{ε} (Q_{ρ} (x_{0}))}{| E u | (Q_{ρ} (x_{0}))} & = & lim_{ε \to 0} \frac{1}{| E u | (Q_{ρ} (x_{0}))} \int_{Q_{ρ} (x_{0})} W (\frac{x}{ε}, E u_{ε}, ω) d x \\ ⩾ & \underset{ε \to 0}{lim^{‾}} \frac{1}{| E u | (Q_{ρ} (x_{0}))} \int_{Q_{δ ρ} (x_{0})} W (\frac{x}{ε}, E u_{ε}, ω) d x \\ ⩾ & \frac{δ^{N}}{t_{ρ}} W_{hom} (\frac{t_{ρ}}{δ^{N}} E v (δ Q), ω) - 2 Δ_{1} (ρ, δ) - β κ_{ρ} \\ (4.19) & - \frac{β}{κ_{ρ}} \int_{δ Q ∖ (1 - κ_{ρ}) δ Q} | v - Θ_{δ} | d x . \end{array}$ As $W_{hom}$ is Lipschitz continuous (see Proposition 3.4) we have $\begin{array}{rcl} \frac{δ^{N}}{t_{ρ}} W_{hom} (\frac{t_{ρ}}{δ^{N}} E v_{ρ} (δ Q), ω) & ⩽ & C^{'} | E v_{ρ} (δ Q) - E v (δ Q) | + \frac{δ^{N}}{t_{ρ}} W_{hom} (\frac{t_{ρ}}{δ^{N}} E v (δ Q), ω), \end{array}$ with $C^{'} > 0$ (which depends on α and β), and noticing that $E v_{ρ} (δ Q) = \frac{E u (Q_{δ ρ} (x_{0}))}{| E u | (Q_{ρ} (x_{0}))}$ we see that $\begin{array}{rcl} \frac{δ^{N}}{t_{ρ}} W_{hom} (\frac{t_{ρ}}{δ^{N}} a (x_{0}) ⊙ b (x_{0}), ω) & ⩽ & C^{'} | a (x_{0}) ⊙ b (x_{0}) - \frac{E u (Q_{ρ} (x_{0}))}{| E u | (Q_{ρ} (x_{0}))} | \\ + C^{'} | \frac{E u (Q_{ρ} (x_{0}))}{| E u | (Q_{ρ} (x_{0}))} - \frac{E u (Q_{δ ρ} (x_{0}))}{| E u | (Q_{ρ} (x_{0}))} | \\ + \frac{δ^{N}}{t_{ρ}} W_{hom} (\frac{t_{ρ}}{δ^{N}} E v_{ρ} (δ Q), ω) \\ ⩽ & C^{'} | a (x_{0}) ⊙ b (x_{0}) - \frac{E u (Q_{ρ} (x_{0}))}{| E u | (Q_{ρ} (x_{0}))} | + C^{'} | 1 - \frac{| E u | (Q_{δ ρ} (x_{0}))}{| E u | (Q_{ρ} (x_{0}))} | \\ + \frac{δ^{N}}{t_{ρ}} W_{hom} (\frac{t_{ρ}}{δ^{N}} E v_{ρ} (δ Q), ω), \end{array}$ where $C > 0$ is the Lipschitz constant. Hence $\begin{array}{rcl} \frac{δ^{N}}{t_{ρ}} W_{hom} (\frac{t_{ρ}}{δ^{N}} E v (δ Q), ω) & ⩾ & \frac{δ^{N}}{t_{ρ}} W_{hom} (\frac{t_{ρ}}{δ^{N}} a (x_{0}) ⊙ b (x_{0}), ω) - C^{'} | E v_{ρ} (δ Q) - E v (δ Q) | \\ - C^{'} | a (x_{0}) ⊙ b (x_{0}) - \frac{E u (Q_{ρ} (x_{0}))}{| E u | (Q_{ρ} (x_{0}))} | \\ (4.20) & - C^{'} | 1 - \frac{| E u | (Q_{δ ρ} (x_{0}))}{| E u | (Q_{ρ} (x_{0}))} | . \end{array}$ As the trace of v and $Θ_{δ}$ are equal on the faces of $δ Q$ orthogonal to $b (x_{0})$ , from Poincaré’s inequality we can assert that $\begin{array}{rcl} (4.21) & \int_{δ Q ∖ (1 - κ_{ρ}) δ Q} | v - Θ_{δ} | d x & ⩽ & C^{″} κ_{ρ} Δ_{2} (ρ, δ), \end{array}$ where $C^{″} > 0$ does not depend on ρ and $Δ_{2} (ρ, δ) : = \int_{δ Q ∖ (1 - κ_{ρ}) δ Q} | E v - \frac{E v (δ Q)}{δ^{N}} | d x$ . Note that by (4.11) we have $\begin{matrix} (4.22) & Δ_{2} (ρ, δ) \to 0 as ρ \to 0 . \end{matrix}$ From (4.19), (4.20) and (4.21) we deduce that $\begin{array}{rcl} lim_{ε \to 0} \frac{μ_{ε} (Q_{ρ} (x_{0}))}{| E u | (Q_{ρ} (x_{0}))} & ⩾ & \frac{δ^{N}}{t_{ρ}} W_{hom} (\frac{t_{ρ}}{δ^{N}} a (x_{0}) ⊙ b (x_{0}), ω) - C^{'} | E v_{ρ} (δ Q) - E v (δ Q) | \\ - C^{'} | a (x_{0}) ⊙ b (x_{0}) - \frac{E u (Q_{ρ} (x_{0}))}{| E u | (Q_{ρ} (x_{0}))} | - C^{'} | 1 - \frac{| E u | (Q_{δ ρ} (x_{0}))}{| E u | (Q_{ρ} (x_{0}))} | \\ - 2 Δ_{1} (ρ, δ) - β κ_{ρ} - β C^{″} Δ_{2} (ρ, δ) . \end{array}$ Letting $ρ \to 0$ and taking (3.3), (3.5), (3.7), (4.10), (4.11), (4.18) and (4.22) into account, we conclude that $\begin{array}{rcl} lim_{ρ \to 0} lim_{ε \to 0} \frac{μ_{ε} (Q_{ρ} (x_{0}))}{| E u | (Q_{ρ} (x_{0}))} & ⩾ & W_{hom}^{\infty} (a (x_{0}) ⊙ b (x_{0}), ω) - C^{'} (1 - δ^{N}), \end{array}$ and (4.8) follows by letting $δ \to 1$ . □
Remark 4.2.
The Lipschitz continuity condition on W seems to play a crucial role in the proof. We do not know whether this condition could be replaced by a weaker one which is also closed by homogenization as, for example, in [13].

4.2. The upper bound

Here we establish that $Γ (L^{1}) - {lim^{‾}}_{ε \to 0} I_{ε} ⩽ I_{hom}$ .

Proposition 4.3 (Upper bound).

Under the assumption of Theorem 2.3 , for $P$ -a.e. $ω \in Ω$ , one has $\begin{matrix} (4.23) & Γ (L^{1}) - \underset{ε \to 0}{lim^{‾}} I_{ε} (u, ω) ⩽ I_{hom} (u, ω) \end{matrix}$ for all $u \in BD (O)$ .

Proof of Proposition 4.3.

In what follows, for each $ε > 0$ , we consider $J_{ε} : BD (O) \times O (O) \times Ω \to [0, \infty]$ defined by $\begin{matrix} (4.24) & J_{ε} (u, A, ω) : = \{\begin{matrix} \int_{A} W (\frac{x}{ε}, E u (x), ω) d x & if u \in LD (O) \\ \infty & if u \in BD (O) ∖ LD (O) . \end{matrix} \end{matrix}$ (Then $I_{ε} = J_{ε} (\cdot, O, \cdot)$ for all $ε > 0$ .)

Let $ω \in \hat{Ω}$ where $\hat{Ω} \in F$ is given by Proposition 3.2. The proof is divided into two steps.

Step 1: establishing the upper bound on $LD (O)$ . We prove that (4.23) holds on $LD (O)$ , i.e. for every $u \in LD (O)$ , one has $\begin{matrix} Γ (L^{1}) - \underset{ε \to 0}{lim^{‾}} I_{ε} (u, ω) ⩽ \int_{O} W_{hom} (E u, ω) d x . \end{matrix}$ For this we proceed into four substeps. Fix $u \in LD (O)$ .

Substep 1-1: using the Vitali envelope. Consider the set function $S_{u} : O (O) \to [0, \infty]$ defined by $\begin{array}{rcl} (4.25) & S_{u} (A) : = \underset{ε \to 0}{lim^{‾}} inf {J_{ε} (v, A, ω) : v \in u + {LD}_{0} (A)} . \end{array}$

For each $δ > 0$ and each $A \in O (O)$ , denote the class of countable families ${Q_{i} = Q_{ρ_{i}} (x_{i})}_{i \in I}$ (where $Q_{ρ_{i}} (x_{i}) : = x_{i} + ρ_{i} Q$ where Q is the unit cell centered at the origin) of disjoint open cubes of A with $x_{i} \in A$ , $ρ_{i} > 0$ and $diam (Q_{i}) \in] 0, δ [$ such that $| A ∖ ⋃_{i \in I} Q_{i} | = 0$ by $V_{δ} (A)$ , consider $S_{u}^{δ} : O (O) \to [0, \infty]$ given by $\begin{matrix} S_{u}^{δ} (A) : = inf {\sum_{i \in I} S_{u} (Q_{i}) : {Q_{i}}_{i \in I} \in V_{δ} (A)}, \end{matrix}$ and define $S_{u}^{*} : O (O) \to [0, \infty]$ by $\begin{matrix} S_{u}^{*} (A) : = sup_{δ > 0} S_{u}^{δ} (A) = lim_{δ \to 0} S_{u}^{δ} (A) . \end{matrix}$ The set function $S_{u}^{*}$ is called the Vitali envelope of $S_{u}$ (see §3.4). We prove that for every $A \in O (O)$ , one has $\begin{matrix} (4.26) & Γ (L^{1}) - \underset{ε \to 0}{lim^{‾}} J_{ε} (u, A, ω) ⩽ S_{u}^{*} (A) . \end{matrix}$ Fix $A \in O (O)$ such that $S_{u}^{*} (A) < \infty$ . Fix any $δ > 0$ . By definition of $S_{u}^{δ} (A)$ , there exists ${Q_{i}}_{i \in I} \in V_{δ} (A)$ such that $\begin{matrix} (4.27) & \sum_{i \in I} S_{u} (Q_{i}) ⩽ S_{u}^{δ} (A) + \frac{δ}{2} . \end{matrix}$ Fix any $ε > 0$ and define $S_{u, ε} : O (O) \to [0, \infty]$ by $\begin{matrix} (4.28) & S_{u, ε} (A) : = inf {J_{ε} (v, A, ω) : v \in u + {LD}_{0} (A)} . \end{matrix}$ (Thus $S_{u} = {lim^{‾}}_{ε \to 0} S_{u, ε}$ .) Given any $i \in I$ , by definition of $S_{u, ε} (Q_{i})$ , there exists $v_{ε}^{i} \in u + {LD}_{0} (Q_{i})$ such that $\begin{matrix} (4.29) & J_{ε} (v_{ε}^{i}, Q_{i}, ω) ⩽ S_{u, ε} (Q_{i}) + \frac{δ | Q_{i} |}{2 | A |} . \end{matrix}$ Define $u_{ε}^{δ} \in u + {LD}_{0} (A)$ by $\begin{matrix} u_{ε}^{δ} : = \{\begin{matrix} u & in O ∖ A \\ v_{ε}^{i} & in Q_{i} . \end{matrix} \end{matrix}$ From (4.29) we see that $\begin{matrix} J_{ε} (u_{ε}^{δ}, A, ω) ⩽ \sum_{i \in I} S_{u, ε} (Q_{i}) + \frac{δ}{2}, \end{matrix}$ hence ${lim^{‾}}_{ε \to 0} J_{ε} (u_{ε}^{δ}, A, ω) ⩽ S_{u}^{δ} (A) + δ$ by using (4.27), and consequently $\begin{matrix} (4.30) & \underset{ε \to 0}{lim^{‾}} \underset{ε \to 0}{lim^{‾}} J_{ε} (u_{ε}^{δ}, A, ω) ⩽ S_{u}^{*} (A) . \end{matrix}$ On the other hand, we have $\begin{array}{rcl} {‖ u_{ε}^{δ} - u ‖}_{L^{1} (O; R^{N})} = \int_{A} | u_{ε}^{δ} - u | d x = \sum_{i \in I} \int_{Q_{i}} | v_{ε}^{i} - u | d x, \end{array}$ and so, as $diam (Q_{i}) \in] 0, δ [$ for all $i \in I$ , from Poincaré’s inequality (see [39, Chapter II, Remark 2.5]) we can assert that $\begin{array}{rcl} {‖ u_{ε}^{δ} - u ‖}_{L^{1} (O; R^{N})} & ⩽ & C δ \sum_{i \in I} \int_{Q_{i}} | E v_{ε}^{i} - E u | d x \\ (4.31) & ⩽ & C δ (\sum_{i \in I} \int_{Q_{i}} | E v_{ε}^{i} | d x + \int_{A} | E u | d x), \end{array}$ where $C > 0$ is independent of δ, ε and i. Taking the left inequality in (2.1), (4.29) and (4.27) into account, from (4.31) we deduce that $\begin{matrix} \underset{ε \to 0}{lim^{‾}} {‖ u_{ε}^{δ} - u ‖}_{L^{1} (O; R^{N})} ⩽ C δ (\frac{1}{α} (S_{u}^{δ} (A) + ε) + \int_{A} | E u | d x) \end{matrix}$ which gives $\begin{matrix} (4.32) & \underset{δ \to 0}{lim^{‾}} \underset{ε \to 0}{lim^{‾}} {‖ u_{ε}^{δ} - u ‖}_{L^{1} (O; R^{N})} = 0 \end{matrix}$ because ${lim}_{δ \to 0} S_{u}^{δ} (A) = S_{u}^{*} (A) < \infty$ . According to (4.30) and (4.32), by diagonalization there exists a mapping $ε \mapsto δ_{ε}$ , with $δ_{ε} \to 0$ as $ε \to 0$ , such that: $\begin{array}{l} lim_{ε \to 0} ‖ w_{ε} - u ‖_{L^{1} (O; R^{N})} = 0; \\ \underset{ε \to 0}{lim^{‾}} J_{ε} (w_{ε}, A, ω) ⩽ S_{u}^{*} (A) \end{array}$ with $w_{ε} : = u_{ε}^{δ_{ε}}$ , and (4.26) follows because $Γ (L^{1}) - {lim^{‾}}_{ε \to 0} J_{ε} (u, A, ω) ⩽ {lim^{‾}}_{ε \to 0} J_{ε} (w_{ε}, A, ω)$ .

Substep 1-2: differentiation with respect to $L^{N}$ . We prove that $\begin{matrix} (4.33) & Γ (L^{1}) - \underset{ε \to 0}{lim^{‾}} I_{ε} (u, ω) ⩽ \int_{O} lim_{ρ \to 0} \frac{S_{u} (Q_{ρ} (x))}{| Q_{ρ} (x) |} d x \end{matrix}$ with $S_{u}$ given by (4.25) and $Q_{ρ} (x) : = x + ρ Q$ where Q is the unit cell centered at the origin. Recalling that $I_{ε} = J_{ε} (\cdot, O, \cdot)$ for all $ε > 0$ , from (4.26) we have $\begin{matrix} Γ (L^{1}) - \underset{ε \to 0}{lim^{‾}} I_{ε} (u, ω) ⩽ S_{u}^{*} (O) . \end{matrix}$ But, it is clear that $S_{u} ⩽ (β + | E u |) d x$ , i.e. the assumption (a) of Theorem 3.11 is satisfied, and it is easily seen that $S_{u}$ is subadditive in the sense of the assumption (b) of Theorem 3.11. Consequently (4.33) follows from Theorem 3.11.

Substep 1-3: using approximate differentiability. We prove that for $L^{N}$ -a.e. $x_{0} \in O$ , one has $\begin{matrix} (4.34) & lim_{ρ \to 0} \frac{S_{u} (Q_{ρ} (x_{0}))}{| Q_{ρ} (x_{0}) |} ⩽ lim_{ρ \to 0} \frac{S_{u_{x_{0}}} (Q_{ρ} (x_{0}))}{| Q_{ρ} (x_{0}) |} \end{matrix}$ with $u_{x_{0}} (x) : = u (x_{0}) + \nabla u (x_{0}) (x - x_{0})$ . Fix any $δ > 0$ . Fix any $ε > 0$ , any $s \in] 0, 1 [$ and any $ρ \in] 0, δ [$ . By definition of $S_{u_{x_{0}}, ε} (Q_{s ρ} (x_{0}))$ in (4.28) there exists $v \in u_{x_{0}} + {LD}_{0} (Q_{s ρ} (x_{0}))$ such that $\begin{matrix} (4.35) & \int_{Q_{s ρ} (x_{0})} W (\frac{x}{ε}, E v, ω) d x ⩽ S_{u_{x_{0}}, ε} (Q_{s ρ} (x_{0})) + δ | Q_{s ρ} (x_{0}) | . \end{matrix}$ Consider a cut-off function $φ \in C^{\infty} (O)$ for the pair $(O ∖ Q_{ρ} (x_{0}), {\overline{Q}}_{s ρ} (x))$ such that $\begin{matrix} (4.36) & ‖ \nabla φ ‖_{L^{\infty} (O; R^{N})} ⩽ \frac{1}{ρ (1 - s)} . \end{matrix}$ Define $w \in LD (Q_{ρ} (x_{0}))$ by $\begin{matrix} w : = u + φ (v - u) . \end{matrix}$ Then $w \in u + {LD}_{0} (Q_{ρ} (x_{0}))$ and $\begin{matrix} E w = \{\begin{matrix} E u (x_{0}) & in Q_{s ρ} (x_{0}) \\ \nabla φ ⊙ (v - u) + φ E u (x_{0}) + (1 - φ) E u & in Q_{ρ} (x_{0}) ∖ Q_{s ρ} (x_{0}) . \end{matrix} \end{matrix}$ Taking (4.35), the right inequality in (2.1) and (4.36) into account, we see that $\begin{array}{rcl} \frac{S_{u, ε} (Q_{ρ} (x_{0}))}{| Q_{s ρ} (x_{0}) |} & ⩽ & \frac{1}{| Q_{s ρ} (x_{0}) |} \int_{Q_{ρ} (x_{0})} W (\frac{x}{ε}, E w, ω) d x \\ = & \frac{1}{| Q_{s ρ} (x_{0}) |} \int_{Q_{s ρ} (x_{0})} W (\frac{x}{ε}, E w, ω) d x \\ + \frac{1}{| Q_{s ρ} (x_{0}) |} \int_{Q_{ρ} (x_{0}) ∖ Q_{s ρ} (x_{0})} W (\frac{x}{ε}, E w, ω) d x \\ ⩽ & \frac{S_{u_{x_{0}}, ε} (Q_{s ρ} (x_{0}))}{| Q_{s ρ} (x_{0}) |} + δ \\ + β (\frac{1}{(1 - s) s^{N}} \frac{1}{| Q_{ρ} (x_{0}) |} \int_{Q_{ρ} (x_{0})} \frac{| u - u_{x_{0}} |}{ρ} d x + \frac{Δ (ρ, s)}{| Q_{s ρ} (x_{0}) |}) \end{array}$ with $Δ (ρ, s) : = | Q_{ρ} (x_{0}) ∖ Q_{s ρ} (x_{0}) | | E u (x_{0}) | + \int_{Q_{ρ} (x_{0}) ∖ Q_{s ρ} (x_{0})} | E u | d x$ . Noticing that $| Q_{ρ} (x_{0}) | ⩾ | Q_{s ρ} (x_{0}) |$ and letting $ε \to 0$ , we obtain $\begin{array}{rcl} \frac{S_{u} (Q_{ρ} (x_{0}))}{| Q_{ρ} (x_{0}) |} & ⩽ & \frac{S_{u_{x_{0}}} (Q_{s ρ} (x_{0}))}{| Q_{s ρ} (x_{0}) |} + δ \\ (4.37) & + β (\frac{1}{(1 - s)} \frac{| Q_{ρ} (x_{0}) |}{| Q_{s ρ} (x_{0}) |} \frac{1}{| Q_{ρ} (x_{0}) |} \int_{Q_{ρ} (x_{0})} \frac{| u - u_{x_{0}} |}{ρ} d x + \frac{Δ (ρ, s)}{| Q_{s ρ} (x_{0}) |}) . \end{array}$ On the other hand, without loss of generality we can assert that $\begin{matrix} lim_{ρ \to 0} \frac{1}{| Q_{ρ} (x_{0}) |} \int_{Q_{ρ} (x_{0})} | | E u (x) | - | E u (x_{0}) | | d x = 0 . \end{matrix}$ But $\begin{array}{rcl} \frac{Δ (ρ, s)}{| Q_{s ρ} (x_{0}) |} & ⩽ & 2 (\frac{1}{s^{N}} - 1) | E u (x_{0}) | + \frac{1}{s^{N}} \frac{1}{| Q_{ρ} (x_{0}) |} \int_{Q_{ρ} (x_{0})} | | E u (x) | - | E u (x_{0}) | | d x, \end{array}$ and so $\begin{matrix} (4.38) & \underset{ρ \to 0}{lim^{‾}} \frac{Δ (ρ, s)}{| Q_{s ρ} (x_{0}) |} ⩽ 2 (\frac{1}{s^{N}} - 1) | E u (x_{0}) | . \end{matrix}$ Letting $ρ \to 0$ in (4.37) and using Theorem 3.5 and (4.38) we deduce that $\begin{array}{rcl} lim_{ρ \to 0} \frac{S_{u} (Q_{ρ} (x_{0}))}{| Q_{ρ} (x_{0}) |} & ⩽ & lim_{ρ \to 0} \frac{S_{u_{x_{0}}} (Q_{s ρ} (x_{0}))}{| Q_{s ρ} (x_{0}) |} + δ + 2 (\frac{1}{s^{N}} - 1) | E u (x_{0}) | \\ = & lim_{ρ \to 0} \frac{S_{u_{x_{0}}} (Q_{ρ} (x_{0}))}{| Q_{ρ} (x_{0}) |} + δ + 2 (\frac{1}{s^{N}} - 1) | E u (x_{0}) | . \end{array}$ Letting $s \to 1$ we conclude that $\begin{matrix} lim_{ρ \to 0} \frac{S_{u} (Q_{ρ} (x_{0}))}{| Q_{ρ} (x_{0}) |} ⩽ lim_{ρ \to 0} \frac{S_{u_{x_{0}}} (Q_{ρ} (x_{0}))}{| Q_{ρ} (x_{0}) |} + δ \end{matrix}$ and (4.34) follows by letting $δ \to 0$ .

Substep 1-4: end of step 1. Combining (4.33) with (4.34) we deduce that $\begin{matrix} Γ (L^{1}) - \underset{ε \to 0}{lim^{‾}} I_{ε} (u, ω) ⩽ \int_{O} lim_{ρ \to 0} \frac{S_{u_{x}} (Q_{ρ} (x))}{| Q_{ρ} (x) |} d x . \end{matrix}$ On the other hand, taking (4.24) and (4.25) into account, we see that $\begin{matrix} lim_{ρ \to 0} \frac{S_{u_{x}} (Q_{ρ} (x))}{| Q_{ρ} (x) |} = lim_{ρ \to 0} \underset{ε \to 0}{lim^{‾}} \frac{S^{E u (x)} (\frac{1}{ε} Q_{ρ} (x)) (ω)}{| \frac{1}{ε} Q_{ρ} (x) |} \end{matrix}$ with $S^{E u (x)} (\frac{1}{ε} Q_{ρ} (x)) (ω)$ given by (3.1) with $ξ = E u (x)$ . But, by Proposition 3.2 (and Definition 3.3) we have $\begin{matrix} \underset{ε \to 0}{lim^{‾}} \frac{S^{E u (x)} (\frac{1}{ε} Q_{ρ} (x)) (ω)}{| \frac{1}{ε} Q_{ρ} (x) |} = W_{hom} (E u (x), ω) \end{matrix}$ for all $ρ > 0$ , hence $\begin{matrix} Γ (L^{1}) - \underset{ε \to 0}{lim^{‾}} I_{ε} (u, ω) ⩽ \int_{O} W_{hom} (E u (x), ω) d x, \end{matrix}$ which completes the proof of Step 1.

Step 2: using a relaxation theorem. From Step 1 we have $\begin{matrix} Γ (L^{1}) - \underset{ε \to 0}{lim^{‾}} I_{ε} (u, ω) ⩽ J_{hom} (u) \end{matrix}$ for all $u \in BD (O)$ with $J_{hom} : BD (O) \to [0, \infty]$ given by $\begin{matrix} J_{hom} (u) : = \{\begin{matrix} \int_{O} W_{hom} (E u (x), ω) d x & if u \in LD (O) \\ \infty & if u \in BD (O) ∖ LD (O) . \end{matrix} \end{matrix}$ As $Γ (L^{1}) - {lim^{‾}}_{ε \to 0} I_{ε} (\cdot, ω)$ is $L^{1}$ -lower semicontinuous we deduce that $\begin{matrix} Γ (L^{1}) - \underset{ε \to 0}{lim^{‾}} I_{ε} (u, ω) ⩽ {\overline{J}}_{hom} (u) \end{matrix}$ for all $u \in BD (O)$ , where ${\overline{J}}_{hom}$ is the $L^{1}$ - lower semicontinuous envelope of $J_{hom}$ . On the other hand, from Proposition 4.1 and Step 1 of the proof of Proposition 4.3 we deduce that for $P$ -a.e. $ω \in Ω$ , one has $\begin{matrix} Γ - lim_{ε \to 0} I_{ε} (u, ω) = \int_{O} W_{hom} (E u, ω) d x \end{matrix}$ for all $u \in LD (O)$ , and so $LD (O) ∋ u \mapsto \int_{O} W_{hom} (E u, ω) d x$ is $L^{1}$ -lsc. Hence (by using the same method as in the non-symmetric case) $W_{hom}$ is symmetric quasiconvex. Indeed, let $ξ \in R_{sym}^{N \times N}$ , let $ϕ \in C_{c}^{1} (Y; R^{N})$ with $Y : =] 0, 1 [^{N}$ and let $\hat{ϕ}$ be the Y-periodic extension of ϕ to $R^{N}$ . Define ${u_{n}} \subset LD (O)$ by $u_{n} (x) : = l_{ξ} (x) + \frac{1}{n} \hat{ϕ} (n x)$ with $l_{ξ} (x) : = ξ x$ . Then $u_{n} ⇀ l_{ξ}$ in $W^{1, 1} (O; R^{N})$ , and so (up to a subsequence) $u_{n} \to l_{ξ}$ in $L^{1} (O; R^{N})$ . On the other hand, $W_{hom} (E u_{n} (x), ω) = W_{hom} (ξ + E \hat{ϕ} (n x), ω) = : \hat{f} (n x) : = {\hat{f}}_{n} (x)$ where $\hat{f}$ is the Y-periodic extension of f to $R^{N}$ with $f (y) : = W_{hom} (ξ + E ϕ (y), ω)$ , and consequently ${\hat{f}}_{n} ⇀ \int_{Y} f (y) d y$ in $L^{1} (O; R^{N})$ . By $L^{1}$ -lsc we deduce that $\begin{array}{rcl} | O | W_{hom} (ξ, ω) & = & \int_{O} W_{hom} (E l_{ξ}, ω) d x ⩽ \underset{n \to \infty}{lim_{_}} \int_{O} W_{hom} (E u_{n} (x), ω) d x \\ = & \underset{n \to \infty}{lim_{_}} \int_{Y} {\hat{f}}_{n} (x) d x \\ = & \int_{O} (\int_{Y} f (y) d y) d x \\ = & | O | \int_{Y} W_{hom} (ξ + E ϕ (y), ω) d y, \end{array}$ and the symmetric quasiconvexity of $W_{hom}$ follows. Consequently, taking Proposition 3.4 into account, we can assert that $W_{hom}$ is continuous, symmetric quasiconvex and has 1-growth. Hence, by Theorem 3.9 we have $\begin{matrix} {\overline{J}}_{hom} (u) = \int_{O} W_{hom} (E u (x), ω) d x + \int_{O} W_{hom}^{\infty} (\frac{d E^{s} u}{d | E^{s} u |} (x), ω) d | E^{s} u | (x) \end{matrix}$ for all $u \in BD (O)$ , and (4.23) follows. □

References

Abddaimi,

Michaille and

Licht, Stochastic homogenization for an integral functional of a quasiconvex function with linear growth, Asymptot. Anal. 15(2) (1997), 183–202.

M.A.

Akcoglu and

Krengel, Ergodic theorems for superadditive processes, J. Reine Angew. Math. 323 (1981), 53–67.

Ambrosio,

Coscia and

Dal Maso, Fine properties of functions with bounded deformation, Arch. Rational Mech. Anal. 139(3) (1997), 201–238. doi:10.1007/s002050050051.

Ansini and

Bille Ebobisse, Homogenization of periodic multi-dimensional structures: The linearly elastic/perfectly plastic case, Adv. Math. Sci. Appl. 11(1) (2001), 203–225.

Anza Hafsa,

Clozeau and

J.-P.

Mandallena, Homogenization of nonconvex unbounded singular integrals, Ann. Math. Blaise Pascal 24(2) (2017), 135–193. doi:10.5802/ambp.367.

Anza Hafsa and

J.-P.

Mandallena, Γ-Convergence of nonconvex integrals in Cheeger-Sobolev spaces and homogenization, Adv. Calc. Var. 10(4) (2017), 381–405. doi:10.1515/acv-2015-0053.

Anza Hafsa and

J.P.

Mandallena, Γ-Limits of functionals determined by their infima, J. Convex Anal. 23(1) (2016), 103–137.

Arroyo-Rabasa,

De Philippis and

Rindler, Lower semicontinuity and relaxation of linear-growth integral functionals under pde constraints, 2017, arXiv:1701.02230.

J.-F.

Babadjian, Traces of functions of bounded deformation, Indiana Univ. Math. J. 64(4) (2015), 1271–1290. doi:10.1512/iumj.2015.64.5601.

10.

Bouchitté, Convergence et relaxation de fonctionnelles du calcul des variations à croissance linéaire. Application à l’homogénéisation en plasticité, Ann. Fac. Sci. Toulouse Math. (5) 8(1) (1986/87), 7–36. doi:10.5802/afst.628.

11.

Bouchitté and

Bellieud, Regularization of a set function – application to integral representation, Ricerche Mat. 49(suppl) (2000), 79–93, Contributions in honor of the memory of Ennio De Giorgi (Italian).

12.

Bouchitté,

Fonseca and

Mascarenhas, A global method for relaxation, Arch. Rational Mech. Anal. 145(1) (1998), 51–98. doi:10.1007/s002050050124.

13.

Cagnetti,

Dal Maso,

Scardia and

Ida Zeppieri, Stochastic homogenisation of free-discontinuity problems, Arch. Ration. Mech. Anal. 233(2) (2019), 935–974. doi:10.1007/s00205-019-01372-x.

14.

Caroccia,

Focardi and

Van Goethem, On the integral representation of variational functionals on

B D

, SIAM J. Math. Anal. 52(4) (2020), 4022–4067. doi:10.1137/19M1277564.

15.

Cristina Barroso,

Fonseca and

Toader, A relaxation theorem in the space of functions of bounded deformation, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 29(1) (2000), 19–49.

16.

De Arcangelis and

Gargiulo, Homogenization of integral functionals with linear growth defined on vector-valued functions, NoDEA Nonlinear Differential Equations Appl. 2(3) (1995), 371–416. doi:10.1007/BF01261182.

17.

De Philippis and

Rindler, On the structure of

A

-free measures and applications, Ann. of Math. (2) 184(3) (2016), 1017–1039. doi:10.4007/annals.2016.184.3.10.

18.

De Philippis and

Rindler, Characterization of generalized Young measures generated by symmetric gradients, Arch. Ration. Mech. Anal. 224(3) (2017), 1087–1125. doi:10.1007/s00205-017-1096-1.

19.

De Philippis and

Rindler, Fine properties of functions of bounded deformation – an approach via linear pdes, 2019, arXiv:1911.01356.

20.

Fonseca and

Müller, Quasi-convex integrands and lower semicontinuity in

L^{1}

, SIAM J. Math. Anal. 23(5) (1992), 1081–1098. doi:10.1137/0523060.

21.

Fonseca and

Müller, Relaxation of quasiconvex functionals in

BV (Ω, R^{p})

for integrands

f (x, u, \nabla u)

, Arch. Rational Mech. Anal. 123(1) (1993), 1–49. doi:10.1007/BF00386367.

22.

G.A.

Francfort and

J.-J.

Marigo, Revisiting brittle fracture as an energy minimization problem, J. Mech. Phys. Solids 46(8) (1998), 1319–1342. doi:10.1016/S0022-5096(98)00034-9.

23.

Dal Maso, An Introduction to Γ-Convergence, Progress in Nonlinear Differential Equations and Their Applications, Vol. 8, Birkhäuser Boston Inc., Boston, MA, 1993.

24.

Dal Maso and

Modica, Nonlinear stochastic homogenization, Ann. Mat. Pura Appl. (4) 144 (1986), 347–389. doi:10.1007/BF01760826.

25.

Dal Maso and

Modica, Nonlinear stochastic homogenization and ergodic theory, J. Reine Angew. Math. 368 (1986), 28–42.

26.

Han and

Daya Reddy, Plasticity: Mathematical Theory and Numerical Analysis, 2nd edn, Interdisciplinary Applied Mathematics, Vol. 9, Springer, New York, 2013.

27.

Jesenko and

Schmidt, Homogenization and the limit of vanishing hardening in Hencky plasticity with non-convex potentials, Calc. Var. Partial Differential Equations 57(1) (2018), 2. doi:10.1007/s00526-017-1261-2.

28.

Kosiba and

Rindler, On the relaxation of integral functionals depending on the symmetrized gradient, Proc. Roy. Soc. Edinburgh Sect. A 151(2) (2021), 473–508. doi:10.1017/prm.2020.22.

29.

Krengel, Ergodic Theorems, De Gruyter Studies in Mathematics, Vol. 6, Walter de Gruyter & Co., Berlin, 1985, With a supplement by Antoine Brunel.

30.

Licht and

Michaille, Global-local subadditive ergodic theorems and application to homogenization in elasticity, Ann. Math. Blaise Pascal 9(1) (2002), 21–62. doi:10.5802/ambp.149.

31.

Matthies,

Strang and

Christiansen, The saddle point of a differential program, in: Energy Methods in Finite Element Analysis, Wiley, Chichester, 1979, pp. 309–318.

32.

Messaoudi and

Michaille, Stochastic homogenization of nonconvex integral functionals. Duality in the convex case, Sém. Anal. Convexe 21:Exp. No. 14 (1991), 32.

33.

Messaoudi and

Michaille, Stochastic homogenization of nonconvex integral functionals, RAIRO Modél. Math. Anal. Numér. 28(3) (1994), 329–356. doi:10.1051/m2an/1994280303291.

34.

Nečas and

Hlaváček, Mathematical Theory of Elastic and Elasto-Plastic Bodies: An Introduction, Studies in Applied Mechanics., Vol. 3, Elsevier Scientific Publishing Co., Amsterdam–New York, 1980.

35.

Rindler, Lower semicontinuity for integral functionals in the space of functions of bounded deformation via rigidity and Young measures, Arch. Ration. Mech. Anal. 202(1) (2011), 63–113. doi:10.1007/s00205-011-0408-0.

36.

P.-M.

Suquet, Sur un nouveau cadre fonctionnel pour les équations de la plasticité, C. R. Acad. Sci. Paris Sér. A–B 286(23) (1978), A1129–A1132.

37.

P.-M.

Suquet, Un espace fonctionnel pour les équations de la plasticité, Ann. Fac. Sci. Toulouse Math. (5) 1(1) (1979), 77–87. doi:10.5802/afst.531.

38.

Temam, Mathematical problems in plasticity theory, in: Variational Inequalities and Complementarity Problems, Proc. Internat. School, Erice, 1978, Wiley, Chichester, 1980, pp. 357–373.

39.

Temam, Problèmes mathématiques en plasticité, Méthodes Mathématiques de l’Informatique [Mathematical Methods of Information Science], Vol. 12, Gauthier-Villars, Montrouge, 1983.

40.

Temam and

Strang, Existence de solutions relaxées pour les équations de la plasticité: étude d’un espace fonctionnel, C. R. Acad. Sci. Paris Sér. A–B 287(7) (1978), A515–A518.

41.

Temam and

Strang, Functions of bounded deformation, Arch. Rational Mech. Anal. 75(1) (1980/81), 7–21. doi:10.1007/BF00284617.

Stochastic homogenization of nonconvex integrals in the space of functions of bounded deformation

Abstract

Keywords

1. Introduction

2. Main result

1 By a Borel measurable stochastic integrand W : R N × R sym N × N × Ω → [ 0 , ∞ [ we mean that W is ( B ( R N ) ⊗ B ( R sym N × N ) ⊗ F , B ( R ) ) -measurable, where B ( R N ) , B ( R sym N × N ) and B ( R ) denote the Borel σ-algebra on R N , R sym N × N and R respectively.

3.1. Definition and properties of the homogenized density

4.1. The lower bound

Proposition 4.1 (lower bound).

Proposition 4.3 (Upper bound).

References