Γ-convergence and stochastic homogenization of degenerate integral functionals in weighted Sobolev spaces

We study the $\Gamma$-convergence of nonconvex vectorial integral functionals whose integrands satisfy possibly degenerate growth and coercivity conditions. The latter involve suitable scale-dependent weight functions. We prove that under appropriate uniform integrability conditions on the weight functions, which shall belong to a Muckenhoupt class, the corresponding functionals $\Gamma$-converge, up to subsequences, to a degenerate integral functional defined on a limit weighted Sobolev space. The general analysis is then applied to the case of random stationary integrands and weights to prove a stochastic homogenization result for the corresponding functionals.


Introduction
In this paper, we study the effective behaviour of scale-dependent integral functionals with possibly degenerate integrands. Functionals of this kind typically model the energy of a heterogeneous material whose physical properties (elastic, thermal, electrical, etc.) may both deteriorate and vary significantly from point to point, on a mesoscopic scale.
The energy functionals we consider are of the form where A ⊂ R n is an open, bounded, Lipschitz set, k ∈ N is a parameter related to some material property (e.g. the size of the microstructure) and u : A → R m represents a physical variable (e.g. the elastic deformation of the body). The degeneracy of the integrands f k : R n × R m×n → [0, +∞) is expressed in terms of growth and coercivity conditions which can depend both on the parameter k and on the spatial variable x. These are given by introducing weight functions λ k : R n → [0, +∞) which modulate the typical superlinear growth in the gradient variable. That is, for every x ∈ R n , ξ ∈ R m×n , and k ∈ N the integrands f k satisfy αλ k (x)(|ξ| p − 1) f k (x, ξ) βλ k (x)(|ξ| p + 1), (1.2) where p > 1, and 0 < α β < +∞. If the weight functions λ k are bounded in L ∞ uniformly in k, then (1.2) reduces to the standard growth and coercivity of order p > 1. In this case, the limit behaviour of F k is well understood and can be described using the language of Γconvergence. Namely, if k → ∞, the functionals F k Γ-converge (up to subsequences), on W 1,p (A; R m ), to an integral functional of the form with f 0 satisfying the same (nondegenerate) growth conditions satisfied by f k (see [9]). Moreover, if ε k → 0 + and f k (x, ξ) = f (x/ε k , ξ) for some nondegenerate f , then the limit integrand f 0 is x-independent and subsequence-independent both in the periodic [6,30] and in the stationary random case [15,16,28], and given by a so-called homogenization formula. As a result, in this case, the whole sequence (F k ) Γ-converges to F . In this paper, we consider sequences of weight functions (λ k ) which are not bounded in general. Specifically, for every k ∈ N we assume that λ k , λ −1/(p−1) k ∈ L 1 loc (R n ), (1.4) moreover, we additionally require the existence of a constant K 1 such that for every k ∈ N there holds (1.5) for every cube Q ⊂ R n . The uniform integrability condition (1.5) is known as Muckenhoupt condition and the functions satisfying it are referred to as Muckenhoupt A p (K)-weights [29]. In this case, the growth conditions (1.2) satisfied by f k naturally set the problem in the parameter-dependent weighted Sobolev space W 1,p λ k (A; R m ) where, for a given A p (K)-weight λ we have The limit behaviour of functionals F k with integrands satisfying (1.2) was studied for the first time in [10], in the convex, scalar case and under the sole integrability condition (1.4). Assuming that λ k converges weakly to some λ ∞ in L 1 , in [10] the authors proved a Γ-convergence and integral representation result for the Γlimit of F k , on the space of Lipschitz functions. The latter, though, in general is smaller than the domain of the Γ-limit. Moreover, in the setting considered in [10] the functionals F k are not equi-coercive and therefore a convergence result for the associated minimization problems cannot be derived from the Γ-convergence analysis.
In order to extend the Γ-convergence result in [10] to the domain of the Γ-limit and to gain compactness, in [17] the Muckenhoupt condition (1.5) was also required together with the additional bound where 0 < c 1 c 2 < +∞ and Q 0 ⊂ R n is a given cube. The Muckenhoupt condition (1.5) guarantees the continuous embedding of W 1,p λ k (A) in the Sobolev space W 1,1+δ (A), for some δ > 0. Then, a combination of (1.5) and (1.6) ensures that sequences with equi-bounded energy are bounded in W 1,1+δ (A), and hence precompact in L 1 (A) (whenever A ⊂ Q 0 ). Therefore, in the setting considered in [17] the equi-coerciveness of the functionals F k can be recovered. Moreover, again thanks to (1.5)-(1.6) a lower bound on the Γ-limit can be established, which shows that its domain is the weighted Sobolev space W 1,p λ∞ (A), where λ ∞ belongs to a Muckenhoupt class and is the weak L 1 -limit of (a subsequence of) λ k .
Besides the contributions [10,17], Γ-convergence and relaxation results for functionals of type (1.1)-(1.2) defined on weighted Sobolev spaces were also established in [3,11,18,19,21,22,31] without departing, though, from the convex/monotone operator, scalar setting, with the only exception of [31]. More specifically, in [31] the authors proved a stochastic homogenization result for a sequence of discrete nonconvex, vectorial energy functionals with degenerate integrands. Under suitable assumptions on the random weights, which are weaker than (1.5) in the scalar case but not really comparable to (1.5) in the vectorial case, the authors showed that in the stationary ergodic case the energies homogenize to a nondegenerate deterministic integral functional. We observe that the case of homogenization is somehow special since in this case the limit functional is always nondegenerate and thus defined on the space W 1,p .
In the present paper, we extend the analysis in [17] to the nonconvex, vectorial setting, without assuming any periodicity or stationarity of the integrands f k . Namely, we assume that f k satisfies (1.2), together with some mild continuity condition in ξ (cf. (3.3)), and that the weight functions λ k are as in (1.4)- (1.6). Under these assumptions we show the existence of a subsequence (k h ), a limit Muckenhoupt weight λ ∞ , with λ k h λ ∞ in L 1 (Q 0 ), and a degenerate integrand f ∞ satisfying a.e. in Q 0 and for every ξ ∈ R m×n , such that the functionals F k h Γ-converge, with respect to the strong L 1 (A; R m )-convergence, to the integral functional We also show that the Γ-convergence holds true, with the same subsequence (k h ), for every open, bounded, Lipschitz set A ⊂ R n , with A ⊂⊂ Q 0 . Moreover, we derive an asymptotic formula for the limit integrand f ∞ which can be expressed as a (double) limit of sequences of scaled minimization problems as follows: f∞(x, ξ) := lim sup Qρ (x) f k h (y, ∇u + ξ) dy : u ∈ W 1,p 0,λ k h (Qρ(x); R m ) , (1.8) where Q ρ (x) ⊂ R n denotes the cube centred in x with side-length ρ > 0, and The proof of this result is carried out in a number of intermediate steps. Namely, we first prove the Γ-convergence and integral representation result on the space W 1,∞ (A; R m ) W 1,p λ∞ (A; R m ). To do so, we use the localization method of Γconvergence and adapt the approach in [7,14] to our setting to get an integral representation result for functionals with degenerate integrands. We remark here that the most delicate part in the implementation of the localization method is the proof of the subadditivity of the Γ-limit, which requires to combine a fundamental estimate for the functionals F k together with an ad hoc vectorial truncation argument, in the same spirit as, e.g. [8, lemma 3.5]. We then extend the Γ-convergence and integral representation result to the limit weighted Sobolev space W 1,p λ∞ (A; R m ). The latter coincides with the domain of F ∞ , thanks to (1.7); hence we get a complete description of the Γ-limit of F k h . The passage from W 1,∞ (A; R m ) to W 1,p λ∞ (A; R m ) is performed by resorting to classical approximation argument (see [1,theorem II.4]) which exploits the property of the maximal function in relation to the Muckenhoupt weights. More precisely, we can adapt [17, theorem 3.1] to the vectorial setting to show that in the liminf inequality, we can replace a sequence (u k ), with u k → u in L 1 (A; R m ) and equi-bounded W 1,p λ k (A; R m )-norm, with a sequence of Lipschitz functions converging to a W 1,∞ (A; R m )-function which differs from u on a set with vanishing measure. Eventually, the asymptotic formula for f ∞ is obtained by combining a convergence result for minimization problems with prescribed Dirichlet conditions together with a derivation formula for f ∞ which is obtained by extending to the weighted Sobolev setting the method developed in [4,5].
Finally, the general Γ-convergence analysis is complemented by an application to the case of stationary random weights and integrands, thus generalizing the classical stochastic homogenization result in [15,16,28] to the degenerate setting.
That is, we specialize our general result to the choice where ω belongs to the sample space of a given probability space (Ω, F, P ), λ is a random Muckenhoupt weight (cf. assumption 8.5), and f is a degenerate stationary random integrand (cf. definition 8.7). Then, following the same approach as in [16], we combine the deterministic analysis and the subadditive ergodic theorem [2, theorem 2.9] to show that, almost surely, the random functionals Γ-converge to a nondegenerate (spatially) homogeneous random functional where f hom satisfies standard growth conditions of order p > 1 with random coefficients (cf. (8.10)) and is given by the following asymptotic cell formula (1.9) If, moreover, λ and f are ergodic, we show that f hom is deterministic and given by the expected value of the right hand side of (1.9). Furthermore, in the ergodic case f hom satisfies the following deterministic growth and coercivity conditions of order p > 1: for every ξ ∈ R m×n .
Outline of the paper. The paper is organized as follows. In § 2 we recall the notions of Muckenhoupt classes and weights and of weighted Sobolev spaces. Moreover, we recall here some well-known related results which will be used throughout. In § 3 we introduce the functionals we study and state the main result of this paper, theorem 3.2. The proof of theorem 3.2 is then carried out in § 4-7. Namely, in § 4 we prove a Γ-convergence and integral representation result in the space W 1,∞ , theorem 4.1. In § 5 we establish theorem 5.2 which extends the results in theorem 4.1 to the weighted Sobolev space W 1,p λ∞ , also showing that the latter coincides with the domain of the Γ-limit. On account of theorem 5.2, in § 6 we prove that in this setting Γ-convergence is stable under the addition of Dirichlet boundary conditions and we derive a convergence result for the associated minimization problems. In § 7 we prove a derivation formula for the integrand of the Γ-limit, theorem 7.1 (see also corollary 7.2). Eventually, in § 8 we prove a stochastic homogenization result for stationary random weights and integrands, theorem 8.12.

Preliminaries
In this section, we collect some useful definitions and preliminary results which will be used throughout.

Muckenhoupt classes
We start by recalling the definition of the so-called Muckenhoupt classes. An introduction to the theory of Muckenhoupt classes can be found in [24].
for every cube Q ⊂ R n with faces parallel to the coordinate hyperplanes. Moreover, we set A p : The elements of the class A p (resp. A p (K)) are usually referred to as A p -weights (resp. A p (K)-weights). Simple examples of A p -weights are radially symmetric functions of the type Further examples can be found, e.g. in [25].
We recall the following 'reverse Hölder inequality' which holds for functions in A p and whose proof can be found in [13,theorem IV].

2)
for every cube Q and for every λ ∈ A p (K).
In this paper, we will deal with sequences of A p (K)-weights. The following result is a consequence of theorem 2.2 and its proof can be found in [17, proposition 4.1].
Proposition 2.4. Let K 1, p > 1, and let (λ k ) be a sequence of functions in A p (K). Let Q 0 ⊂ R n be a cube and assume that there exist two constants c 1 , c 2 with 0 < c 1 c 2 such that for some σ > 0. Moreover, there holds then (2.4) and (2.5) holds true for every cube Q ⊂ R n , (2.6) holds a.e. in R n , and λ ∞ ∈ A p (K).
The equi-integrability estimate below is another immediate consequence of theorem 2.2.
Proposition 2.5. Let p > 1, K 1, and let (λ k ) be a sequence of functions in A p (K) satisfying (2.3). Then there exist σ = σ(K, p, n) > 0 and c = c(K, p, n) > 0 such that , for every measurable set E ⊂ Q 0 and every k ∈ N.
Proof. Let σ > 0 and c > 0 be the constants given by theorem 2.2. By (2.1) and for every k ∈ N. Therefore, the Hölder inequality easily gives

Weighted Sobolev spaces
In this short subsection, we recall the definition and the basic properties of weighted Sobolev spaces. For a comprehensive treatment of this subject we refer the reader to the monographs [25,34]. For further relevant results concerning weighted Sobolev spaces, we will provide a precise reference to the literature whenever these results are used in the paper.
Let p > 1, let λ ∈ A p . In all that follows A ⊂ R n denotes an open and bounded set with Lipschitz boundary. Let m ∈ N, m 1; we define the weighted Lebesgue space we recall that L p λ (A; R m ) equipped with the norm is a reflexive Banach space. Moreover, we define the weighted Sobolev space the latter is also a reflexive Banach space when endowed with the norm We recall that the embedding of W 1,p λ (A; R m ) in L p λ (A; R m ) is compact (see, e.g. [23, lemma 1]). Furthermore, we have the following continuous embeddings: for some δ > 0. Throughout the paper, we will also use the fact that To prove (2.7) we will establish the following Poincaré-type inequality: there exists C > 0 such that We will obtain (2.8) arguing by contradiction. Were (2.8) false, then for every j ∈ N there would exist u j ∈ C ∞ (A; R m ) such that , for every j ∈ N.
for every j ∈ N. Hence, in particular, the sequence (v j ) is bounded in W 1,p λ (A; R m ). Therefore, by the compact embedding of W 1,p λ (A; R m ) in L p λ (A; R m ), up to subsequences (not relabelled), v j → v in L p λ (A; R m ), for some v ∈ L p λ (A; R m ). Moreover, since the embedding of L p λ (A; R m ) in L 1 (A; R m ) is continuous, we also have v j → v in L 1 (A; R m ). Therefore, (2.9) entails both v L p λ (A;R m ) = 1 and v = 0 a.e. in A and hence a contradiction. Now let u ∈ W 1,p λ (A; R m ); by [11, proposition 3.5] (see also [12, theorem 6 Moreover, in view of (2.8) the sequence (u j ) is bounded in W 1,p λ (A; R m ), therefore again appealing to the compact embedding of Remark 2.7. We note that by the density of C ∞ (A; R m ) in W 1,p λ (A; R m ) inequality (2.8) actually holds in the whole space W 1,p λ (A; R m ). That is, there exists a constant C > 0 such that for every u ∈ W 1,p λ (A; R m ).
Finally, in this paper, we will also consider the space We recall that W 1,p 0,λ (A; R m ) agrees with the closure of C ∞ 0 (A; R m ) in W 1,p λ (A; R m ) (see, e.g. [32, theorem 1.4] or [17, proposition 2.1]).

Maximal function and measure theory
In this subsection, we recall the definition of maximal function and some of its properties which are useful for our purposes. Moreover, for the readers' convenience we also recall some classical result in measure theory which we are going to employ in the paper.
For the theory of maximal functions we refer to [33]. Let u ∈ L 1 loc (R n ), then the Hardy maximal function of u at x is defined as where Q r (x) is the cube centred at x, with side length r and faces parallel to the coordinate planes. The following property will be useful for our purposes: there exists a constantc =c(n) > 0 depending only on n such that for every u ∈ L 1 (R n ) and every l > 0.
The following result is proven in [29, theorem 9].
Theorem 2.8. Let p > 1, K 1, and let λ ∈ A p (K). Then there exists a constant c 4 = c 4 (K, p, n) > 0 such that for every u ∈ L 1 loc (R n ).
We observe that theorem 2.8 in particular implies that if u ∈ L p λ (R n ) then Mu ∈ L p λ (R n ). For the following lemma we refer to [1, lemma I.11].
Lemma 2.9. Let u ∈ C ∞ 0 (R n ) and let l > 0. Set Then u is Lipschitz continuous in H l ; i.e. there exists a constant c 5 = c 5 (n) > 0 such that for every x, y ∈ H l .
We recall the following result which can be found in [20].
Eventually, we state the following technical lemma whose proof can be found in [1, lemma I.7].
Lemma 2.11. Let (φ h ) be a bounded sequence in L 1 (R n ). Then for every τ > 0 there exist a measurable set E τ with |E τ | < τ, δ τ > 0, and a sequence (h τ j ) such that for for every measurable set B such that B ∩ E τ = ∅ and |B| < δ τ .

Setting of the problem and statement of the main result
In this section, we introduce the functionals we are going to study and state the main result of the paper. • λ k ∈ A p (K), for every k ∈ N; • there exists a cube Q 0 ⊂ R n such that for every k ∈ N there holds for some constants 0 < c 1 c 2 < +∞.
Let (λ k ) be a sequence of weights satisfying assumption 3.1; in this paper, we consider Borel integrands f k : R n × R m×n → [0, +∞) satisfying the two following conditions: (1) (degenerate growth conditions) there exist two constants 0 < α β < +∞ such that for almost every for every ξ ∈ R m×n and every k ∈ N; (2) (continuity in ξ) there exists L > 0 such that for almost every x ∈ R n for every ξ 1 , ξ 2 ∈ R m×n , and every k ∈ N.
Let A(Q 0 ) denote the collection of all open subsets of Q 0 with Lipschitz boundary. We consider the sequence of localized integral functionals F k : We endow W 1,1 (Q 0 ; R m ) with the strong L 1 (Q 0 ; R m )-topology. If not otherwise specified, throughout the paper the Γ-limits will all be computed with respect to this topology. The following theorem is the main result of this paper.
Theorem 3.2. Let F k be the functionals defined in (3.4). Then there exists a subse- .
is a Borel function and for a.e. x ∈ Q 0 and every ξ ∈ R m×n is given by the following asymptotic formula where, for every A ∈ A(Q 0 ), Moreover, f ∞ satisfies the following properties for almost every x ∈ Q 0 :
Remark 3.3. We observe that if we replace (3.1) with the following stronger condition: where A 0 is the collection of open, bounded, and Lipschitz subsets of R n , thanks to a diagonal argument, it can be shown that the functionals We note that (3.8) holds true in the case of admissible periodic or stationary weights (cf. § 8 and remark 8.11).
The proof of theorem 3.2 will be broken up in several intermediate results and it will be carried out in § 4-7. Namely, in § 4 we prove that (up to subsequences) the functionals F k Γ-converge to the integral functional F ∞ on the space W 1,∞ (Q 0 ; R m ). Moreover, in this section we also prove that the limit integrand f ∞ satisfies the desired growth conditions as well as the continuity property. By means of an approximation argument, in § 5 we extend the Γ-convergence result established in § 4 to the whole W 1,1 (Q 0 ; R m ), also showing that the domain of F ∞ coincides with the 'limit' weighted Sobolev space W 1,p λ∞ (Q 0 ; R m ). Eventually, by combining the analysis in § 6 and § 7, we derive the asymptotic formula (3.5) for f ∞ .

Γ-convergence and integral representation in W 1,∞
In this section, we show that on W 1,∞ (Q 0 ; R m ) the sequence (F k ) Γ-converges (up to subsequences) to a limit functional which can be represented in an integral form.
The following theorem is the main result of the present section.
Moreover, the function f ∞ satisfies the following properties for almost every x ∈ Q 0 : for some L > 0.
The proof of theorem 4.1 will be achieved in a number of intermediate steps by means of the so-called localization method of Γ-convergence (see, e.g. [7, chapters 9-11] or [14, chapters 16-19]).
To this end, we consider the localized Γ-liminf and the Γ-limsup of F k ; i.e. we consider the functionals F , F : . Then, the aim of this section is to show that, up to subsequences, for every where F is as in (4.1).
Remark 4.2. We observe that F and F are lower semicontinuous with respect to the strong topology of L 1 (Q 0 ; R m ) [14, proposition 6.8]. They also inherit some of the properties of the functionals F k . Namely, as set functions they are both increasing [14, proposition 6.7], moreover, F is superadditive on pairwise-disjoint sets [14, proposition 16.12]; while as functionals they are both local [14, proposition 16.15].
Thanks to assumption 3.1 we can invoke proposition 2.4 to deduce the existence . Then in the following lemma we show that the domain of F (and hence also the domain of F ) contains the space Proof. Let (λ k h ) ⊂ (λ k ) be the subsequence whose existence is established by proposition 2.4. Hence, in particular, Hence, thanks to (4.7), to the fact that u j → u in L 1 (Q 0 ; R m ), and to the lower semicontinuity of F with respect to the strong and thus the claim.
The following lemma shows that F k (almost) decreases by smooth truncations.
Then for every η > 0, M > 0 and for every k ∈ N there exists a Lipschitz function ϕ k : R m → R m with Lipschitz constant less than or equal to 1 satisfying for every k ∈ N. Moreover, the function ϕ k can be chosen in a finite family independent of k.
Proof. The proof of this lemma is classical and follows the line of, e.g. [8, lemma 3.5] with minor modifications. However, since we work in a different functional setting, we repeat the proof here for the readers' convenience. Let η > 0 and M > 0 be fixed. Let (a j ) be a strictly increasing sequence of positive real numbers such that for every j ∈ N there exists a Lipschitz function ϕ j : R m → R m with Lipschitz constant less than or equal to 1 satisfying For every k ∈ N and every j ∈ N set w j k := ϕ j (u k ). We have where to establish the last inequality we have used the nonnegativity of f k , (3.2), and the fact that ϕ j has Lipschitz constant less than or equal to 1.
Let N ∈ N be arbitrary; we now want to estimate 1/N In view of (3.1) and (4.8) we can find a constant C > 0 such that for every k ∈ N. Moreover, thanks to proposition 2.5 there exist c, σ > 0 such that for every k ∈ N and every j ∈ 1, . . . , N. Therefore, we define the sequence (a j ) recursively by imposing the following condition on a 1 : (4.12) which is clearly possible thanks to the boundedness of (u k ) in L 1 (A; R m ). Eventually, by choosing N ∈ N in a way such that C/N η/2, gathering (4.9)-(4.12) we Therefore, for every k ∈ N we can find j(k) ∈ {1, . . . , N} such that hence the proof is accomplished by setting ϕ k := ϕ j(k) . Finally, we note that N is independent of k.
where the last inequality follows by lemma 4.3.
Let η > 0 be fixed: by applying lemma 4.4 to the sequence (u k ) with M : for every A ∈ A(Q 0 ). Then, taking the lim sup as k → ∞ in (4.14) and appealing to (4.13) we obtain Eventually, the claim follows by the definition of F and the arbitrariness of η.
The following proposition shows that the functionals F k satisfy the fundamental estimate, uniformly in k. Proposition 4.6 (Fundamental estimate). Let F k be the functionals defined in (3.4) and let A ∈ A(Q 0 ). For every η > 0 and for every A , A , B ∈ A(Q 0 ) with A ⊂⊂ A ⊂⊂ A there exists a constant M η > 0 with the following property: for every k ∈ N and for every u,ũ ∈ W 1,p Proof. Let η > 0, A, A , A , B and S be as in the statement. We start observing that by (3.1) there exists a constant C > 0 such that For every k ∈ N and for i = 1, . . . , N we have where F * k denotes the extension of F k to the Borel subsets of Q 0 .
Denote by I k,i the last term in (4.17). For every k ∈ N and for i = 1, . . . , N, using (3.2) we obtain for every k ∈ N and for i = 1, . . . , N. Hence, there exists i 0 ∈ {1, . . . , N} such that for every k ∈ N; thus by (4.15) we get Eventually, in view of (4.16) and (4.17) the proof is accomplished choosing With the help of propositions 4.5 and 4.6 we can deduce the following result which will eventually lead to the inner regularity and subadditivity of the set function F (u, ·), for every u ∈ W 1,∞ (Q 0 ; R m ).
Let η > 0 be fixed; then, in view of proposition 4.6 we can find a constant M η > 0 and a sequence (ϕ k ) of cut-off functions between A and A such that Now let σ > 0 be the exponent as in theorem 2.2, using the Hölder inequality and recalling (3.1) we get . Therefore, since u k −ũ k L q (Q0;R m ) → 0 for every q 1, we immediately obtain Hence, (4.18) follows by the arbitrariness of η > 0.
The proof of the following proposition is classical, for this reason we only sketch it here, while we refer the reader to the monographs [7,14] for further details.
Proposition 4.8 (Γ-convergence and measure property of the Γ-limit). Let F k be the functionals defined in (3.4). Then there exist a subsequence (k h ) and a functional F : where F and F are as in (4.4) and (4.5), respectively, with k replaced by k h .
Moreover, for every u ∈ W 1,∞ (Q 0 ; R m ) the set function F (u, ·) is the restriction to A(Q 0 ) of a Radon measure on Q 0 .
Proof. Let (k h ) be the subsequence whose existence is established by proposition 2.4. Thanks to the compactness of Γ-convergence [14, theorem 8.5], a standard diagonal argument gives the existence of a further subsequence (not relabelled), such that the corresponding functionals F and F satisfy for every u ∈ W 1,∞ (Q 0 ; R m ) and for every A ∈ A(Q 0 ). We note that the set function F (u, ·) is inner regular by definition.
Moreover, by virtue of lemma 4.7 we can reason as in [14, proposition 18.4] to deduce that F (u, ·) is subadditive.
We now prove that (4.20), which will ensure that F is the Γ-limit of F k on W 1,∞ (Q 0 ; R m ).
Since by definition of F we have F F F , to get (4.20) it suffices to show that for every u ∈ W 1,∞ (Q 0 ; R m ) and A ∈ A(Q 0 ). To prove (4.21) we consider the localized functional H : Therefore, by lemma 4.3 we immediately obtain that F (u, A) H(u, A), for every u ∈ W 1,∞ (Q 0 ; R m ) and A ∈ A(Q 0 ). For every fixed u ∈ W 1,∞ (Q 0 ; R m ) the set function H(u, ·) defines a Radon measure on Q 0 , hence for every η > 0 fixed there exists a compact set Then by definition of F we readily obtain thus (4.21) follows by the arbitrariness of η > 0.
Finally, the inner regularity and subadditivity of F (u, ·) together with remark 4.2 allow us to apply the De Giorgi-Letta measure criterion (see, e.g. [14, theorem 14.23]) to deduce that F (u, ·) is the restriction to A(Q 0 ) of a Radon measure on Q 0 , and thus to conclude.
Remark 4.9. We observe that for every A ∈ A(Q 0 ) the functional F (·, A) is invariant under translations in u. Indeed, for given u ∈ W 1,∞ (Q 0 ; R m ) and A ∈ A(Q 0 ) let (u k ) ⊂ W 1,1 (Q 0 ; R m ) be such that u k → u in L 1 (Q 0 ; R m ) and lim k→∞ F k (u k , A) = F (u, A). Let now s ∈ R m , then clearly (u k + s) converges to u + s in L 1 (Q 0 ; R m ) and by (4.20) On the other hand, the argument above also gives + s, A) and thus the claim.
Proof. Proposition 4.8 ensures the existence of a subsequence (F k h ) of (F k ) such that F k h (u, A) Γ-converges to a functional F (u, A) for every u ∈ W 1,∞ (Q 0 ; R m ) and every A ∈ A(Q 0 ). Then, it remains to prove that the functional F admits an integral representation as in (4.22). We will break up the proof of the integral representation into a number of steps.
Step 1. Definition of f ∞ . Let ξ ∈ R m×n be fixed and set u ξ (x) := ξx. By the measure property of F established in proposition 4.8, the set function F (u ξ , ·) can be extended to a Radon measure on Q 0 . Moreover, thanks to lemma 4.3, F (u ξ , ·) is absolutely continuous with respect to the Lebesgue measure. For every x ∈ Q 0 define where Q ρ (x) is the cube centred at x, with side length ρ > 0, and faces parallel to the coordinate planes. Then, f ∞ is a Borel function and the Lebesgue differentiation theorem guarantees that We now show that f ∞ satisfies the growth and coercivity conditions as in (4.2). To this end, we start observing that the growth condition from above readily follows from lemma 4.3. In fact, choosing in (4.6) u = u ξ , A = Q ρ (x), with x Lebesgue point for λ ∞ , the estimate from above in (4.2) follows by dividing both sides of (4.6) by |Q ρ (x)|, and eventually passing to the limit as ρ → 0 + .
To derive the growth condition from below on f ∞ let u ∈ W 1,1 (Q 0 ; R m ) and A ∈ A(Q 0 ) be fixed. By the Hölder inequality and by the growth condition from below in (3.2) we get therefore the following lower bound for every k ∈ N. Now let u ∈ W 1,∞ (Q 0 ; R m ) and let (u h ) ⊂ W 1,1 (Q 0 ; R m ) be such that Hence, by the lower semicontinuity of u → A |∇u| dx with respect to the L 1 (Q 0 ; R m )-topology and by proposition 2.4, evaluating (4.23) in (u h ) and passing to the limit as h → ∞ we find for every u ∈ W 1,∞ (A; R m ) and every A ∈ A(Q 0 ). Now let x ∈ Q 0 be a Lebesgue point for λ ∞ andλ ∞ and choose in (4.24) u = u ξ and A = Q ρ (x); then, dividing both sides of (4.24) by |Q ρ (x)| and passing to the limit as ρ → 0 + give for a.e. x ∈ Q 0 and every ξ ∈ R m×n . Eventually, (2.6) entails the desired bound from below.
Step 2. Integral representation on piecewise affine functions. Let A ∈ A(Q 0 ) and u ∈ W 1,∞ (Q 0 ; R m ) be piecewise affine on A; i.e. there exists a finite family of pairwise disjoint open sets A j such that |A \ N j=1 A j | = 0 and for every x ∈ A with ξ j ∈ R m×n , z j ∈ R m for j = 1, . . . , N. By remark 4.9 and step 1, taking into account the locality of F , we have that is, the integral representation (4.22) on piecewise affine functions.
We observe that the local Lipschitz condition (4.3) satisfied by f ∞ ensures that, for every A ∈ A(Q 0 ), the functional (4.26) is continuous with respect to the strong W 1,p λ∞ (A; R m )-convergence. Indeed, using Hölder's inequality we easily get for every u 1 , u 2 ∈ W 1,p λ∞ (Q 0 ; R m ). Moreover, arguing as in the proof of lemma 4.7 we can deduce that (4.26) is also continuous with respect to the strong convergence of W 1,q (Q 0 ; R m ), for q p(1 + σ)/σ.
Let u ∈ W 1,∞ (Q 0 ; R m ) and A ∈ A(Q 0 ) be given; then there exists a sequence (u j ) ⊂ W 1,q (Q 0 ; R m ) strongly converging to u in W 1,q (Q 0 ; R m ) for any q ∈ [1, ∞) such that its restrictions to A are piecewise affine. Since F is lower semicontinuous with respect to the strong topology of L 1 (Q 0 ; R m ), appealing to step 2 and to the continuity of (4.26) we then obtain Hence, to represent F in an integral form it only remains to prove the opposite inequality. To this end fix u ∈ W 1,∞ (Q 0 ; R m ) and consider the functional We observe that F satisfies the same properties as F , hence there exists a Carathéodory function h ∞ : for every v ∈ W 1,∞ (Q 0 ; R m ) and every A ∈ A(Q 0 ). Note that the equality holds whenever v is piecewise affine on A. Let (u j ) be the sequence of piecewise affine functions considered above. Then hence the equality in (4.22) holds for every u ∈ W 1,∞ (Q 0 ; R m ) and every A ∈ A(Q 0 ).
Remark 4.11. From (4.25) it can be seen that actually f ∞ satisfies the growth conditions for a.e. x ∈ Q 0 and every ξ ∈ R m×n , which then reduce to those established in [6,9,30] when λ k ≡ 1.

Γ-convergence and integral representation in W 1,p λ∞
Consider now the integral functional F ∞ : with f ∞ as in theorem 4.10.
The purpose of this section is to show that (up to subsequences) there holds for every u ∈ W 1,1 (Q 0 ; R m ) and every A ∈ A(Q 0 ), where F and F are as in (4.4) and (4.5), respectively. In other words we will show that, up to subsequences, the functionals F k defined in (3.4) Γ-converge on the whole space W 1,1 (Q 0 ; R m ) to the functional F ∞ , whose domain is the (limit) weighted Sobolev space W 1,p λ∞ (Q 0 ; R m ). To do so, we will make use of the following approximation result whose proof follows the line of that of [1,theorem II.4] (see also [17, Then, for every τ > 0 there exist: with Lipschitz constant c(n)L τ , for some c(n) > 0 depending only on n, such that: (2) |{x ∈ A : v τ (x) = u(x)}| (m + 1)τ ; (3) the following estimate holds for every τ > 0: Proof. Without loss of generality, we can assume that liminf in the left-hand side of (5.4) is actually a limit. Moreover, we can also assume that (u k ) ⊂ C ∞ 0 (R n ; R m ), supp(u k ) ⊂⊂ Q 0 , and  ) with a sequence of functions in W 1,p λ k (R n ; R m ), whose support is compactly contained in Q 0 , and such that (5.5) holds. Moreover, since for fixed k the space C ∞ 0 (R n ; R m ) is dense in W 1,p λ k (R n ; R m ) (see, e.g. [34, corollary 2.1.6]), a diagonal argument provides us with a sequence (w k ) ⊂ C ∞ 0 (R n ; R m ), with supp(w k ) ⊂⊂ Q 0 , such that Then, we readily get and by the compact embedding of W 1,p Furthermore, we observe that u k and w k are close in energy so that once we establish the estimate (5.4) along (w k ), the same estimate will hold true along (u k ). In fact, (3.3) gives hence gathering (5.6)-(5.8) yields for some constant C > 0. Therefore, in all that follows, with a little abuse of notation, (u k ) denotes a sequence in C ∞ 0 (R n ; R m ), with supp(u k ) ⊂⊂ Q 0 , and such that (5.5) holds. For k ∈ N and i ∈ {1, . . . , m}, let u (i) k denote the i-th component of the vectorvalued function u k . By applying theorem 2.8 to |∇u for every k ∈ N, every i = 1, . . . , m, and for some c 4 > 0. Hence, by combining (5.5) and (5.9) it follows that the sequence (λ k (M |∇u , for every i = 1, . . . , m. Let now τ > 0, then lemma 2.11 ensures the existence of a measurable set E τ , with of a constant δ τ > 0, and a subsequence (k τ j ) such that for every j ∈ N, every i = 1, . . . , m, and for every measurable set B with B ∩ E τ = ∅ and |B| < δ τ .
To simplify the notation we drop the dependence of the sequence on j and τ , thus we write (5.11) for every k ∈ N, every i = 1, . . . , m, and every measurable B with B ∩ E τ = ∅ and |B| < δ τ . By the Hölder inequality we deduce hence by (3.1), (5.5) and (5.12), since λ k belongs to A p (K) we get for every k ∈ N, i = 1, . . . , m, and some C > 0. In its turn (5.13) together with (2.11) provide us with a constant L τ (c/τ )(CK/c 1 ) 1/p such that for every k ∈ N, For k ∈ N, and i = 1, . . . , m define the sets Then lemma 2.9 yields Appealing to McShane's theorem [27] we can extend u (i) k from H τ k ∩ Q 0 to R n keeping the same Lipschitz constant c 5 (n)L τ . We denote this extension with v τ,(i) k and note that we can assume that v Now, let x ∈ R n be such that dist(x , Q 0 ) > 1, then To prove (5.17) we start observing that there exists a subsequence (k j ) such that if then |A \ E| = 0; hence, as a consequence, for every x ∈ A and hence, in particular, for every x ∈ B τ . Assume by contradiction that |B τ | > (m + 1)τ , then by (5.14) we obtain (5.19) for every j ∈ N. Therefore, by (5.19) and lemma 2.10 there exists ( Thus, if x belongs to the set above by (5.18) we get which is a contradiction in view of the definition of B τ . Therefore, (5.17) holds.

Now let A τ ⊂ A be an open set containing A \ E τ and such that
Aτ We note that this choice is always possible thanks to the growth conditions satisfied by f k (3.2), to (5.15), and in view of proposition 2.5. Indeed, we have Eventually, by combining (5.20), (5.22) and (5.23) we deduce and hence the claim follows with β τ := α τ + τ (βm p−1 c 5 (n) p + 1).
We are now in a position to show that, up to subsequences, the functionals F k Γ-converge to F ∞ .
Theorem 5.2. Let F k and F ∞ be the functionals defined in (3.4) and (5.1), respectively. Then there exists a subsequence (k h ) such that for every u ∈ W 1,1 (Q 0 ; R m ) and for every A ∈ A(Q 0 ) with A ⊂⊂ Q 0 there holds (5.24) where F and F are, respectively, as in (4.4) and (4.5) with k replaced by k h .
Proof. In all that follows (k h ) denotes the subsequence provided by theorem 4.10.
We divide the proof into two main steps.
Step 1: Lower bound. In this step, we prove that  A). (5.26) We observe that lemma 4.3 guarantees that F (u, A) < +∞; therefore, (u h ) ⊂ W 1,p λ h (A; R m ) and (up to possibly passing to a subsequence) by (3.2) we get Now let τ > 0 be fixed and arbitrary; theorem 5.1 provides us with (β τ ), infinitesimal as  27) where the last inequality follows by theorem 4.10, since v τ ∈ W 1,∞ (Q 0 ; R m ). Now let B τ := {x ∈ A : v τ (x) = u(x)} be as in the proof of theorem 5.1 and recall that |B τ | (m + 1)τ. (5.28) By (5.27) and the nonnegativity of f ∞ we have for every τ > 0. Now, since |A \ A τ | < τ, using (5.28) we get thus, thanks to (4.2) and (5.30), we can pass to the limit as τ → 0 + in (5.29) and obtain If this is the case, we may argue exactly as in substep 1.1 and get By the Fatou lemma, (4.2) and proposition 2.6 this yields u ∈ W 1,p λ∞ (A; R m ) and hence a contradiction.
Step 2. Upper bound. In this step, we prove that for every u ∈ W 1,1 (Q 0 ; R m ) and every A ∈ A(Q 0 ) with A ⊂⊂ Q 0 . To this end, let u ∈ W 1,1 (Q 0 ; R m ) and A ∈ A(Q 0 ), A ⊂⊂ Q 0 be fixed. We start observing that by the definition of F ∞ , if u / ∈ W 1,p λ∞ (A; R m ) then there is nothing to prove. Therefore, we only consider the case u ∈ W 1,p λ∞ (A; R m ). Since A is Lipschitz, by [34, theorem 2.1.13] we can find a functionũ ∈ W 1,p λ∞ (Q 0 ; R m ) with u =ũ a.e. in A. Then, by density (see, e.g. [34, corollary 2.1.6]) there exists (u j ) ⊂ W 1,∞ (Q 0 ; R m ) such that u j →ũ in W 1,p λ∞ (Q 0 ; R m ). Then, by the locality of F and F ∞ , the continuity of F ∞ in W 1,p λ∞ (Q 0 ; R m ), and the L 1 (Q 0 ; R m )-lower semicontinuity of F , invoking theorem 4.10 we deduce thus the upper bound.

Convergence of minimization problems
In this section, we modify the domain of the functionals F k by prescribing boundary conditions of Dirichlet type. We then study the Γ-convergence of the corresponding functionals and prove a convergence result for the associated minimization problems.
We start by proving a preliminary energy bound.
Proposition 6.1. Let F k be the functionals defined in (3.4). Then there exist an exponent δ > 0 and a constant C > 0 such that for every A ∈ A(Q 0 ), every u ∈ W 1,p λ k (A; R m ), and every k ∈ N.
Proof. By theorem 2.2 we can deduce the existence of an exponent σ > 0 and a constant c > 0 such that for every cube Q and for every k ∈ N. Now, let A ∈ A(Q 0 ) and u ∈ W 1,p λ k (A; R m ) be arbitrary, and let δ > 0 to be chosen later. By the Hölder inequality we have For δ := (p − 1)σ/(p + σ) it is immediate to check that hence by (6.2) we readily get Moreover, since λ k belongs to A p (K), by (3.1) we also deduce that Eventually, gathering (3.1), (3.2) and (6.3) gives for every k ∈ N. Hence, (6.1) immediately follows by choosing C := max{C 1 , C 2 } with Let F k be functionals defined in (3.4). We consider F ψ k : We are now in a position to prove a Γ-convergence result for the functionals F ψ k .
Theorem 6.2 (Γ-convergence with boundary data). Let F ψ k be the functionals defined in (6.4). Then there exists a subsequence (k h ) such that for every A ∈ A(Q 0 ),
Proof. Let u ∈ W 1,1 (Q 0 ; R m ) and A ∈ A(Q 0 ), with A ⊂⊂ Q 0 be fixed and let (k h ) be the subsequence whose existence is guaranteed by theorem 5.2. We divide the proof into two main steps.
Step 1: Lower bound. Let (u h ) ⊂ W 1,1 (Q 0 ; R m ) be such that u h → u in L 1 (Q 0 ; R m ). In this step, we want to show that We note that we can always assume that otherwise there is nothing to prove. Moreover, without loss of generality, we may also assume that the liminf in (6.7) is actually a limit. Then, by the definition , to conclude it is enough to show that u belongs to W 1,1 0 (A; R m ) + ψ. To this end, we start observing that thanks to (6.7), proposition 6.1 yields the existence of an exponent δ > 0 and of a constant C > 0 such that for every h ∈ N. Then, by Poincaré's inequality the sequence (u h ) is bounded in W 1,1+δ (A; R m ). This readily implies that, up to subsequences, u h u in W 1,1+δ (A; R m ). Since (u h ) ⊂ W 1,1+δ 0 (A; R m ) + ψ and this space is weakly closed, we immediately get u ∈ W 1,1+δ 0 (A; R m ) + ψ, and therefore the claim.
Step 2: Upper bound. We start by considering the case u ∈ C ∞ 0 (A; R m ) + ψ. By proposition 4.5 and theorem 5.2 there exists a sequence (u h ) ⊂ W 1,p Starting from u h we now want to construct a recovery sequence which also satisfies the boundary condition. To this purpose, let η > 0 be fixed. By the equi-integrability of the sequence (λ k h ) (cf. proposition 2.5) there exists a compact set K η ⊂ A such that (6.9) for every h ∈ N.
Then, proposition 4.6 ensures the existence of a positive constant M η and a sequence (ϕ h ) of cut-off functions between A and A such that for every 1 q < +∞. Moreover, by (6.10) and (3.2) we get Hence, by (6.8), (6.9) and (6.11) we have Therefore, by the arbitrariness of η > 0 we conclude that for every u ∈ C ∞ 0 (A; R m ) + ψ. Now, let u ∈ W 1,p 0,λ∞ (A; R m ) + ψ. We extend u to ψ outside A; we clearly have that the extended function (still denoted by u) belongs to (6.12), and by the lower semicontinuity of the Γ-limsup with respect to the strong topology of L 1 (Q 0 ; R m ) we get for every u ∈ W 1,p 0,λ∞ (A; R m ) + ψ, and therefore the upper bound.
The following result shows that the functionals F ψ k are equi-coercive with respect to the strong L 1 (Q 0 ; R m )-topology. Proposition 6.3 (Equi-coerciveness). Let F ψ k be functionals defined in (6.4), let A ∈ A(Q 0 ), A ⊂⊂ Q 0 , and let (u k ) ⊂ W 1,1 (A; R m ) be such that Then there exist a subsequence (u k h ) ⊂ (u k ) and an exponent δ > 0 such that with u ∈ W 1,p 0,λ∞ (A; R m ) + ψ. Moreover, if we extend u k h and u to Q 0 by setting u k h := ψ and u := ψ in Q 0 \ A, respectively, then u k h → u in L 1 (Q 0 ; R m ).
Proof. By (6.13) and by (6.4) we have u k ∈ W 1,p 0,λ k (A; R m ) + ψ, for every k ∈ N. Then, arguing exactly as in the proof of theorem 6.2 we may deduce the existence of a subsequence (u k h ) ⊂ (u k ) which weakly converges in W 1,1+δ (A; R m ) to a function u ∈ W 1,1 0 (A; R m ) + ψ. Furthermore, by the compact embedding of Hence, by theorem 6.2 and by (6.13) there holds thus by (6.5) we get u ∈ W 1,p 0,λ∞ (A; R m ) + ψ.
Thanks to the fundamental property of Γ-convergence, by combining theorem 6.2 and proposition 6.3 we obtain the following convergence result for the associated minimization problems.

Asymptotic formula for f ∞
In this section, we derive an asymptotic formula for the integrand of the Γ-limit, f ∞ . This formula will be particularly useful when proving the homogenization result in § 8. In all that follows F ∞ : where for a.e. x ∈ Q 0 and for every ξ ∈ R m×n moreover, f ∞ satisfies (4.2) and (4.3) (cf. theorem 4.1). We also recall that, being F ∞ a Γ-limit, it is lower semicontinuous with respect to the strong L 1 (Q 0 ; R m )convergence.
The following theorem is the main result of this section.
Theorem 7.1. For almost every x ∈ Q 0 and every ξ ∈ R m×n there holds where, for every A ∈ A(Q 0 ), The proof of theorem 7.1 will be achieved by combining lemmas 7.3-7.5 below, by following the same strategy as in [4, § 3] (see also [5, § 2.2]).
As an immediate corollary of theorems 6.4 and 7.1 we also obtain the following asymptotic formula for f ∞ .
Corollary 7.2 (Asymptotic formula for f ∞ ). For almost every x ∈ Q 0 and every ξ ∈ R m×n there holds where, for every A ∈ A(Q 0 ), We now turn to the proof of theorem 7.1; to this end, we need to introduce the following notation. Set A * := {Q ρ (x): x ∈ Q 0 , ρ > 0} and let δ > 0. For We note that m δ F∞ is decreasing in δ; hence for every A ∈ A(Q 0 ) we can consider We start by proving the following technical lemma which is an adaptation from [4, lemma 3.3] to the setting of weighted Sobolev spaces.
Proof. We observe that the inequality is an immediate consequence of the definition of m * F∞ . Indeed, let δ > 0 be fixed and let (Q i ) be an admissible sequence in the sense of the definition of m δ F∞ (u ξ , A), then , thus (7.5) follows by taking the limit as δ → 0 + . We now prove the converse inequality; i.e.
To this end, let δ > 0 be fixed and let (Q δ i ) be an admissible sequence in the definition of m δ By definition of m F∞ , for every i ∈ N we can choose v δ i ∈ W 1,p 0,λ∞ (Q δ i ; R m ) + u ξ such that for some C(δ, p) > 0. By (4.2), (7.7) and (7. Hence, for δ > 0 fixed, the sequence (v δ,N ) is bounded in W 1,p λ∞ (Q 0 ; R m ), uniformly in N . Then, by [25, theorem 1.32] v δ belongs to W 1,p λ∞ (Q 0 ; R m ) and the claim is proven. Moreover, we have indeed, by (4.2) By combining (7.7)-(7.9) we deduce that We now claim that v δ → u ξ in L 1 (Q 0 ; R m ). If so, by virtue of the lower semicontinuity of F ∞ with respect to the strong L 1 (Q 0 ; R m )-convergence, passing to the limit as δ → 0 + in (7.10) would give and therefore (7.6). Hence, to conclude the proof it only remains to show that Moreover, arguing similarly as above, by (4.2), (7.7) and (7.8) we deduce Therefore, gathering (7.4), (7.11) and (7.12) gives the desired convergence and completes the proof.
We also need the following lemma.
Proof. The proof follows arguing exactly as in [4, lemma 3.5], now using lemmas 7.3 and 7.4.

Stochastic homogenization
In this last section, we illustrate an application of the Γ-convergence result theorem 3.2 to the case of stochastic homogenization. We start by recalling some basic notions and results from ergodic theory.

Ergodic theory
Let d 1 be an integer; in all that follows B d denotes the Borel σ-algebra of R d ; if d = 1 we set B := B 1 .
Let (Ω, F, P ) be a probability space and let τ = (τ y ) y∈R n denote a group of Ppreserving transformations on (Ω, F, P ); i.e. τ is a family of measurable mappings τ y : Ω → Ω satisfying the following properties: • τ y τ y = τ y+y , τ −1 y = τ −y , for every y, y ∈ R n ; • the map τ y preserves the probability measure P ; i.e. P (τ y E) = P (E), for every y ∈ R n and every E ∈ F; • for any measurable function ϕ on Ω, the function φ(ω, If in addition every τ -invariant set E ∈ F has either probabilty 0 or 1, then τ is called ergodic.
We also need to recall the notion of subadditive process. In what follows A 0 denotes the family of all open, bounded subsets of R n with Lipschitz boundary.
We now state a version of the subadditive ergodic theorem, originally proven by Akcoglu and Krengel [2], which is suitable for our purposes (see [26, theorem 4.3]). for every ω ∈ Ω and for every cube Q in R n with faces parallel to the coordinate planes. If in addition μ is ergodic, then φ is constant.
For later use we also recall the Birkhoff ergodic theorem. To this end, we preliminarily need to fix some notation. Let ϕ be a measurable function on (Ω, F, P ); we denote with E[ϕ] the expected value of ϕ; i.e.
For every ϕ ∈ L 1 (Ω) and for every σ-algebra F ⊂ F, we denote with E[ϕ|F ] the conditional expectation of ϕ with respect to F . We recall that E[ϕ|F ] is the unique We now state the following version of the Birkhoff ergodic theorem which is convenient for our purposes. for every ω ∈ Ω and for every measurable bounded set B ⊂ R n with |B| > 0.
Remark 8.4. We note that if τ is ergodic, then F τ reduces to the trivial σ-algebra, therefore (8.1) becomes

Setting of the problem and main results
In this section, we introduce the random integral functionals we are going to analyse. To this end, we preliminarily need to define the class of admissible random weights.
Below we introduce the notion of stationary random integrand.
(i) We say that f : Ω × R n × R m×n → [0, +∞) is a random integrand if: • for every ω ∈ Ω and for every x ∈ R n , the two following conditions hold: for every ξ ∈ R m×n and for some 0 < α β < +∞, and for every ξ 1 , ξ 2 ∈ R m×n and for some L > 0.
(ii) We say that a random integrand f is stationary if for every ω ∈ Ω, for every x, y ∈ R n and every ξ ∈ R m×n it holds: • f (ω, x + y, ξ) = f (τ y ω, x, ξ).
(iii) We say that a stationary random integrand f is ergodic if τ = (τ y ) y∈R n is ergodic.
Let f be a stationary random integrand in the sense of definition 8.7. Let ω ∈ Ω be fixed and consider the integral functional F (ω) : W 1 = inf A f (ω, x, ∇u + ξ) dx : u ∈ W 1,p 0,λ (A; R m ) .
The following proposition shows that for every fixed ξ ∈ R m×n , the minimization problem in (8.7) defines a subadditive process.
Proof. Let ξ ∈ R m×n and A ∈ A 0 be fixed. We first show that ω → m F (ω) (u ξ , A) is F-measurable. To this end fix u ∈ W 1,p λ (A; R m ), then the function (ω, x) → f (ω, x, ∇u + ξ) is F ⊗ L n -measurable, hence by Fubini's theorem is F-measurable. Observe now that W 1,p 0,λ (A; R m ) endowed with the norm ∇ · L p λ (A;R m×n ) is a separable Banach space and that, by virtue of (8.5), the map u → F (ω)(u + u ξ , A) is continuous with respect to the same norm. Then there exists a countable dense set D ⊂ W 1,p 0,λ (A; R m ) such that m F (ω) (u ξ , A) = inf u∈D F (ω)(u + u ξ , A), hence the map ω → m F (ω) (u ξ , A) is F-measurable.
Let ξ ∈ R m×n and ω ∈ Ω be fixed. We now prove that the function A → m F (ω) (u ξ , A) is subadditive in the sense of definition 8.1. To this end, let A ∈ A 0 and let (A i ) i∈I be a finite family of pairwise disjoint sets in A 0 such that A i ⊂ A, for every i ∈ I, and |A \ ∪ i∈I A i | = 0. Let η > 0 and choose u i ∈ W 1,p 0,λ (A i , R m ) such that F (ω)(u i + u ξ , A i ) m F (ω) (u ξ , A i ) + η. Define u ∈ W 1,p 0,λ (A; R m ) by setting u := i∈I u i χ Ai . Then by the locality of F (ω) we have which proves the subadditivity thanks to the arbitrariness of η > 0.
Finally, by definition of m F (u ξ , A), choosing u = 0, by (8.4) we have 0 m F (ω) (u ξ , A) β(|ξ| p + 1) A λ(ω, x) dx, (8.9) for every ξ ∈ R m×n , every ω ∈ Ω, and every A ∈ A 0 . Therefore, integrating on Ω both sides of (8.9) and using the stationarity of λ we get where to establish the last equality we have used the Tonelli theorem together with a change of variables in ω. Eventually, we deduce both (8.8) and that (ω, A) → m F (ω) (u ξ , A) is a subadditive process.
By combining proposition 8.8 together with the subadditive ergodic theorem 8.2 we are now able to establish the existence of the homogenization formula which will eventually define the integrand of the Γ-limit (cf. theorem 8.12 below). Proposition 8.9. Let f be a stationary random integrand. Then there exist a set Ω ∈ F with P (Ω ) = 1 and a F ⊗ B m×n -measurable function f hom : Ω × R m×n → Then, appealing to (8.5) and to the Hölder inequality we deduce where C(p) > 0 depends only on p. By using (8.4) (see also (8.9)) we get where C > 0 depends on p, α, β. Hence, by the arbitrariness of η > 0 we get and, as above, the other inequality follows by exchanging the role of ξ 1 and ξ 2 . Therefore, taking the lim sup as t → ∞ and invoking theorem 8.3, we deduce the existence of a set Ω ∈ F with P ( Ω) = 1 such that lim sup for every ω ∈ Ω. We also observe that choosing in (8.19) Q = Q 1 (0), it is immediate to check that f hom (ω, ·) satisfies the local Lipschitz condition (8.20), for every ω ∈ Ω.