New homogenization results for convex integral functionals and their Euler-Lagrange equations

We study stochastic homogenization for convex integral functionals $$u\mapsto \int_D W(\omega,\tfrac{x}\varepsilon,\nabla u)\,\mathrm{d}x,\quad\mbox{where}\quad u:D\subset \mathbb{R}^d\to\mathbb{R}^m,$$ defined on Sobolev spaces. Assuming only stochastic integrability of the map $\omega\mapsto W(\omega,0,\xi)$, we prove homogenization results under two different sets of assumptions, namely $\bullet_1\quad$ $W$ satisfies superlinear growth quantified by the stochastic integrability of the Fenchel conjugate $W^*(\cdot,0,\xi)$ and a mild monotonicity condition that ensures that the functional does not increase too much by componentwise truncation of $u$, $\bullet_2\quad$ $W$ is $p$-coercive in the sense $|\xi|^p\leq W(\omega,x,\xi)$ for some $p>d-1$. Condition $\bullet_2$ directly improves upon earlier results, where $p$-coercivity with $p>d$ is assumed and $\bullet_1$ provides an alternative condition under very weak coercivity assumptions and additional structure conditions on the integrand. We also study the corresponding Euler-Lagrange equations in the setting of Sobolev-Orlicz spaces. In particular, if $W(\omega,x,\xi)$ is comparable to $W(\omega,x,-\xi)$ in a suitable sense, we show that the homogenized integrand is differentiable.


Introduction
We revisit the problem of stochastic homogenization of vectorial convex integral functionals. For a bounded Lipschitz domain D ⊂ R d , d ≥ 2, we consider integral functionals of the form where W : Ω × R d × R m×d → [0, +∞) is a random integrand which is stationary in the spatial variable and convex in the last variable (see Section 2 for the precise setting). Homogenization of (1.1) (for convex or nonconvex integrands) in terms of Γ-convergence is a classical problem in the calculus of variations, see for instance [4,19] for textbook references. Assuming qualitative mixing in the form of ergodicity, Dal Maso and Modica proved in [10] that in the scalar case m = 1 the sequence of functionals (1.1) Γ-converges almost surely towards a deterministic and autonomous integral functional provided that W satisfies standard p-growth, that is, there exist 1 < p < ∞ and 0 < c 1 , c 2 < ∞ such that W is p-coerciv in the sense that c 1 |z| p − c 2 ≤ W (ω, x, z) for all z ∈ R d and a.e. (ω, x) ∈ Ω × R d , (1.3) and satisfies p-growth in the form W (ω, x, z) ≤ c 2 (|z| p + 1) for all z ∈ R d and a.e. (ω, x) ∈ Ω × R d . (1.4) The result was extended to the vectorial (quasiconvex) case in [24]. By now there are many (classical and more recent) contributions on homogenization where the p-growth conditions (1. 3) and (1.4) are relaxed in various ways: for instance nonstandard (e.g. p(x), (p, q) or unbounded) growth conditions [4,14,19,28,32] or degenerate p-growth (that is c 1 , c 2 depend on x and inf c 1 = 0 and sup c 2 = ∞) [13,29,30] (see also [31] for the case p = 1). In this manuscript, we relax (1.3) and (1.4) in the way that, instead of (1.4), we only assume that W is locally bounded in the second variable, that is -roughly speaking -we assume E[W (·, x, ξ)] < +∞. (1.5) In the periodic setting, Müller proved in [28], among other things, Γ-convergence of (1.1) assuming the stronger boundedness condition esssup x∈R d W (x, ξ) < +∞ for all ξ ∈ R m×d (1. 6) and p-coercivity with p > d, that is, This result was significantly extended [2,14] to cover unbounded integrands and certain non-convex integrands with convex growth -still assuming (1.7). In the scalar case m = 1, condition (1.7) can be significantly relaxed to p > 1, see [14,19]. Note that condition (1.7) has to two effects: on the one hand it proves compactness of energy-bounded sequences (for this any p > 1 suffices), but at the same time the Sobolev embedding turns energy-bounded sequences to compact sequences in L ∞ , which is crucial for adjusting boundary values of energy-bounded sequences in absence of the so-called fundamental estimate.
Here, we propose two ways for relaxing condition (1.7) in the vectorial setting. In particular, in Theorems 3.1 and 3.3 below, we provide Γ-convergence results for (1.1) with our without Dirichlet boundary conditions essentially (the precise statements can be found in Section 3) assuming (1.5) and one of the following two conditions: (a) a 'mild monotonicity condition' which requires that for any matrix ξ and any matrix ξ that is obtained from ξ by setting some (or equivalently one of the ) rows to zero we have W (ω, x, ξ) W (ω, x, ξ) (1.8) (see Assumption (A4) below for the details), together with superlinear growth from below lim |ξ|→∞ W (ω, x, ξ) |ξ| = +∞ quantified by the stochastic integrability of the Fenchel conjugate of ξ → W (ω, x, ξ) (see Assumption (A3)). In particular, we do not assume p-growth from below for some p > 1 and thus improve even previous results in the scalar case (where the 'mild monotonicity condition' is always satisfied; see Remark 2.4 (iii)). Considering for instance the integrand |ξ| p(ω,x) , it becomes clear that the integrability of the conjugate is a very weak condition as it allows for exponents p arbitrarily close to 1 (see Example 2.5 for details) (b) p-coercivity (1.3) for some p with p > d−1. This improves the findings of [28] and (in parts) of [14], where corresponding statements are proved under the more restrictive assumption p > d. It also enlarges the range of admissible exponents considered in [19,Chapter 15], where homogenization results were proven under (p, q)-growth conditions with q < p * (the critical Sobolev exponent associated to p > 1) Some comments are in order: (i) To the best of our knowledge, the degenerate lower bound via the integrability of the Fenchel conjugate is the most general considered so far in the literature on homogenization of multiple integrals (at least when compactness in Sobolev spaces was established). For instance, letting W (ω, x, ξ) = λ(ω, x) p |ξ| p for some weight λ, it reads that λ(·, x) −p/(p−1)) ∈ L 1 (Ω), which coincides with the assumption in [30], where it was shown that in general this integrability is necessary to have a non-degenerate value of the multi-cell formula with this class of integrands. While in the proof of the compactness statement in Lemma 4.1 we use the convexity of W , this is not needed as one can replace W by its biconjugate W * * in the same proof when applying the Fenchel-Young inequality. Hence this assumption also yields compactness in the non-convex setting.
(ii) While the condition (1.8) yields that componentwise truncation does not increase the energy too much, it does not imply that recovery sequences can assumed to be bounded (at least not using standard arguments). Moreover, truncation does only yield boundedness instead of compactness in L ∞ . It is for this reason that our approach is restricted to the convex case, where we can directly work with a corrector that has equi-integrable energy due to the strengthened version of the ergodic theorem proven in [30]. (iii) Condition (1.8) is tailor-made for componentwise truncation. Taking truncations of the full field u of the form ϕ(|u| 0 ) u |u|0 (with a norm | · | 0 and a suitable cut-off ϕ) leads to a more complicated condition on W since the value of gradient of this map additionally depends on the value of u. In case of the Euclidean norm, a particular choice of ϕ ensures that the norm of each partial derivative of the truncated map does not increase (cf. [7,Section 4]), so (1.8) for all matrices ξ with | ξe i | ≤ |ξe i | for all i ∈ {1, . . . , d} would suffice to perform non-componentwise truncations. They have the potential to reduce the Γ-convergence analysis to bounded sequences. We expect such a condition to pave the way to non-convex energies without condition (1.7). Note that in the scalar case m = 1, we also obtain homogenization results in the non-convex setting by applying known relaxation results to the non-convex functional, see Remark 3.4 (iii). (iv) The p-coercivity with exponent p > d − 1 needs a fine choice of cut-off functions in combination with the compact embedding of W 1,p (S 1 ) ⊂ L ∞ (S 1 ), where S 1 denotes the d− 1-dimensional unit sphere. This idea has already been used for example in [5,6] in context of div-curl lemmas and deterministic homogenization, and in [3] in the context of regularity and stochastic homogenization of non-uniformly elliptic linear equations.
For the proof of our main Γ-convergence result we follow the strategy laid out in [28]: the lower bound, which does not require (1.8) or p-coercivity with p > d − 1, is achieved via truncation of the energy. However, due to the degenerate lower bound, the truncation of the integrand is not straightforward, but needs to be done carefully (see Lemma 4.5). Once the energies are suitably truncated, their Γ-convergence follows via standard arguments using blow-up and the multi-cell formula. In order to pass to the limit in the truncation parameter, we show that the multi-cell formula agrees with the single-cell formula given by a corrector on the probability space (see Lemma 4.7) as this formula passes easily to the limit (see Lemma 4.8). The new assumption (1.8) or the relaxed p-coercivity enter in the proof of the upper bound when we try to provide a recovery sequence for affine functions that agree with the affine function on the boundary. While under (1.8) we truncate peaks of the corrector and carefully analyze the error due to this truncation using the equi-integrability of the energy density of the corrector, the p-growth coercivity with p > d − 1 makes use of the Sobolev embedding on spheres into L ∞ that we quantify in a suitable way in Lemma 4.10 via the existence of ad hoc cut-off functions that allow us to control the energetic error when we impose boundary conditions on a non-constrained recovery sequence.
In addition to the Γ-convergence results of Theorems 3.1 and 3.3, we also consider homogenization of the Euler-Lagrange system corresponding to the functional (1.1). In order to formulate the latter problem in a convenient sense, i.e., without using merely the abstract notion of subdifferential of convex functionals, we need to investigate two issues: • given that the integrand W is differentiable with respect to the last variable, does a minimizer of the heterogeneous functional u → F ε (ω, u, D) + {boundary conditions} satisfy any PDE? • is the homogenized integrand W hom differentiable?
For both points, the general growth conditions rule out deriving a PDE or the differentiability of W hom by differentiating under the integral sign. Therefore, we rely on convex analysis. The subgradient for convex integral functionals is well-known on L p -spaces. In order to capture the dependence on the gradient the standard way is to rely on the chain rule for subdifferentials. However, in general this only holds when the functional under examination has at least one continuity point. Hence, working on L p -spaces is not feasible except for the choice p = ∞, which however does not necessarily coincide with the domain of the functional. As it turns out, the correct framework are generalized Orlicz spaces (cf. Section 2.3). It should be noted that the integrand defining a generalized Orlicz space has to be even (otherwise the corresponding Luxemburg-norm fails to be a norm). Since on the one hand we need that the domain of our functional F ε (ω, ·, D) is contained in some generalized Sobolev-Orlicz space, while on the other hand the domain of the functional F ε (ω, ·, D) should have interior points on that space, the integrand W (ω, x, ·) has to be comparable to an even function. For this reason, we are only able to prove the above two points under the additional assumption that see Assumption 2 for the detailed formulation. A possible approach to remove this assumption would be a theory of subdifferentials on so-called Orlicz-cones (see [12,Section 2], where such a theory is initiated in a very special setting). This is however beyond the scope of this work. Nevertheless, to the best of our knowledge this is the first time that the issue of global differentiability of W hom and of the convergence of Euler-Lagrange equations (in a random setting) is settled without any polynomial growth from above, see the recent textbook [8] for results on periodic homogenization in Orlicz spaces. The corresponding results are stated in Theorem 3.5. The paper is structured as follows: in Section 2 we first recall the basic notions from ergodic theory and state the properties of generalized Orlicz spaces that will be used in the paper. Then we formulate precisely our assumptions. In Section 3 we present the main results of the paper, while we postpone the proofs to Section 4. In the appendix we prove a representation result for the convex envelope of radial functions, a very general measurability result for the optimal value of Dirichlet problems of integral functionals with jointly measurable integrand and extend an approximation-in-energy result of [16] that we need to treat the convergence of Dirichlet problems in the vectorial setting.

Preliminaries and notation
2.1. General notation. We fix d ≥ 2. Given a measurable set S ⊂ R d , we denote by |S| its d-dimensional Lebesgue measure. For x ∈ R d we denote by |x| the Euclidean norm and B ρ (x) denotes the open ball with radius ρ > 0 centered at x. The real-valued m × d-matrices are equipped with the operator-norm | · | induced from the Euclidean norm on R d , while we write ·, : for the Euclidean scalar product between two m × d-matrices. Given a function f : T × R m×d → R, where T is an arbitrary set, we denote by f * the Legendre-Fenchel conjugate of f with respect to the last variable, that is, For a measurable set with positive measure, we define − S = 1 |S| S . We use standard notation for L p -spaces and Sobolev spaces W 1,p . The Borel σ-algebra on R d will be denoted by B d , while we use L d for the σ-algebra of Lebesgue-measurable sets. Throughout the paper, we use the continuum parameter ε, but statements like ε → 0 stand for an arbitrary sequence ε n → 0. Finally, the letter C stands for a generic positive constant that may change every time it appears.

Stationarity and ergodicity.
Let Ω = (Ω, F , P) be a complete probability space. Here below we recall some definitions from ergodic theory. Definition 2.1 (Measure-preserving group action). A measure-preserving additive group action on (Ω, F , P) is a family {τ z } z∈R d of measurable mappings τ z : Ω → Ω satisfying the following properties: (2) (invariance) P(τ z F ) = P(F ), for every F ∈ F and every z ∈ R d ; (3) (group property) τ 0 = id Ω and τ z1+z2 = τ z2 • τ z1 for every z 1 , z 2 ∈ R d . If, in addition, {τ z } z∈R d satisfies the implication then it is called ergodic. Remark 2.2. As noted in [19,Lemma 7.1], the joint measurability of the group action implies that for every set Ω 0 of full probability there exists a subset Ω 1 ⊂ Ω 0 of full probability that is invariant under τ z for a.e. z ∈ R d . In particular, iff : Ω → R is a function that satisfies a given property almost surely, then the stationary extension f : Ω × R d → R defined by f (ω, x) =f (τ x ω) satisfies the same property almost surely for a.e. x ∈ R d .
We recall the following version of the ergodic theorem that is crucial for our setting (see [30,Lemma 4.1]). Lemma 2.3. Let g ∈ L 1 (Ω) and {τ z } z∈R d be a measure-preserving, ergodic group action. Then for a.e. ω ∈ Ω and every bounded, measurable set B ⊂ R d the sequence of functions x → g(τ x/ε ω) converges weakly in L 1 (B) as ε → 0 to the constant function E[g].
2.3. Generalized Orlicz spaces. We recall here the framework for generalized Orlicz spaces tailored to our setting. Let (T, T , µ) be a finite measure space. Given a jointly measurable function ϕ : T × R m×d → [0, +∞) satisfying for a.e. t ∈ T the properties i) ϕ(t, 0) = 0, (ii) ϕ(t, ·) is convex and even, (iii) lim |ξ|→+∞ ϕ(t, ξ) = +∞, we define the generalized Orlicz space L ϕ (T ) m×d by where we identify as usual functions that agree a.e. We equip this space with the Luxemburg norm which then becomes a Banach space [21,Theorem 2.4]. We further assume the integrability conditions sup |ζ|≤r ϕ(·, ζ), sup |ζ|≤r ϕ * (·, ζ) ∈ L 1 (T ) for all r > 0, (2.1) where we recall that ϕ * (t, ζ) denotes the Legendre-Fenchel conjugate with respect to the last variable. Then L ϕ (T ) m×d embeds continuously into L 1 (T ) m×d . Indeed, in this case the Fenchel-Young inequality and the definition of the Luxemburg-norm yield that Denoting further by (L ϕ (T ) m×d ) * the dual space, (2.1) allows us to apply [22, Proposition 2.1 and Theorem 2.2] to characterize the dual space as follows: every ℓ ∈ (L ϕ (T ) m×d ) * can be uniquely written as a sum ℓ = ℓ a + ℓ s , with ℓ a ∈ L ϕ * (T ) m×d (here the * denotes the Legendre-Fenchel conjugate) and ℓ s ∈ S ϕ (T ), where we set Fur our analysis it will be crucial that (2.1) further implies that for any element ℓ ∈ S ϕ (T ) it holds ℓ| L ∞ (T ) m×d = 0. To see this, note that (2.1) implies that T ϕ(t, h(t)) dµ < +∞ for all h ∈ L ∞ (T ) m×d . Now let (A n ) n∈N ∈ A be a sequence as in the above definition for the element ℓ. Since T has finite measure, it follows that χ An converges in measure to 0. By linearity, for all n ∈ N we have that and so it suffices to show that hχ An → 0 in L ϕ (T ) m×d . Given σ > 0, the sequence ϕ(·, σhχ An ) also converges in measure to 0 and is uniformly bounded by the integrable function ϕ(·, σh). Hence Vitali's convergence theorem yields lim n→+∞ T ϕ(t, σh(t)χ An (t)) dµ = 0, so that lim sup n→+∞ hχ An ϕ ≤ σ −1 . Since σ can be made arbitrarily large, we obtain the claimed convergence to 0.
Finally, we shall make use of the following representation formula for the subdifferential of convex integral functionals: let f : T × R m×d → R be a jointly measurable function that is convex in its second variable. Assume that g ∈ L ϕ (T ) m×d such that that I f (g) ∈ R, where I f (g) = T f (t, g(t)) dµ. Then the subdifferential of I f at g is given by cf. [22,Theorem 3.1] which can be applied to due (2.1).

2.4.
Framework and assumptions. Let D ⊂ R d be an open, bounded set with Lipschitz boundary and let (Ω, F , P) be a complete probability space equipped with a measure-preserving, ergodic group action {τ z } z∈R d . For ε > 0, we consider integral functionals defined on L 1 (D) m with domain contained in W 1,1 (D) m , taking the form with the integrand W satisfying the following assumptions: where W * denotes the Legendre-Fenchel transform of W with respect to its last variable.
Moreover, W satisfies at least one of the following two conditions (A4) or (A5) below.

Remark 2.4.
(i) Due to Remark 2.2, the above Assumptions are indeed just assumptions on W . Note that the local suprema in (A3) can be replaced by pointwise integrability of W (·, ξ) and W * (·, ξ) since convex functions on cubes attain their maximum at the finitely many corners and moreover W ≥ 0, while W * can be bounded from below by W * (ω, ξ) ≥ −W (ω, 0). Another advantage of the definition of W via stationary extension is that the function W * (ω, x, ξ) = W * (τ x ω, ξ) remains F ⊗ L d ⊗ B m×d -measurable due to the completeness of the probability space. Indeed, following verbatim the proof of [18,Proposition 6.43], one can show that for any jointly measurable function h : Ω × R m×d → R the Fenchel-conjugate with respect to the second variable is still jointly measurable. However, note that completeness of Ω is essential for the proof when one only assumes joint measurability.
1 It is a Carathéodory-function. Indeed, for a.e. ω ∈ Ω the convexity and finiteness of W imply continuity with respect to ξ, which can be used to deduce continuity with respect to r, while measurability with respect to ω can be shown as follows: for every t > 0 the set {(ω, ξ) ∈ Ω × {|ξ| = r} : W (ω, ξ) < t} is F ⊗ B m×d -measurable, so that by the measurable projection theorem the projection onto Ω is measurable. This projection is exactly {w ∈ Ω : inf |ξ|=r W (ω, ξ) < t}.
We also study the convergence of the associated Euler-Lagrange equations for F ε (ω, ·, D) under Dirichlet boundary conditions and external forces. To show that the homogenized operator is differentiable under the general growth conditions, we need to impose an additional structural assumption.
Assumption 2. In addition to Assumption 1, assume that for a.e. ω ∈ Ω the map ξ → W (ω, ξ) is differentiable and almost even in the sense that there exists C ≥ 1 such that for all ξ ∈ R m×d we have Note that the corresponding integrand W (ω, x, ξ) then satisfies the same properties for a.e. x ∈ R d with Λ replaced by Λ.
We can further define the corresponding generalized Sobolev-Orlicz space and which embeds continuously into W 1,1 (D) m . We also define the space W 1,ϕε 0,ω (D) m as the subspace with vanishing W 1,1 (D) m -trace. Due to the continuous embedding this subspace is closed. As we shall prove, the homogenized integrand W hom appearing in Theorem 3.1 satisfies the deterministic analogue of (2.6), which allows us to define the associated Sobolev-Orlicz spaces W 1,ϕ hom (D) m and W 1,ϕ hom 0 (D) m for the homogenized model. We need those spaces to properly formulate our results concerning the Euler-Lagrange equations.

Main results
Note that due to the probabilistic nature our main results are only true for a.e. ω ∈ Ω. At the beginning of Section 4 we describe precisely which null sets we have to exclude.
We start our presentation of the main results by stating the Γ-convergence result of the unconstrained functionals.

Remark 3.2.
For an intrinsic formula defining W hom see Lemma 4.3. It follows a posteriori from Theorem 3.3 that one can obtain W hom (ξ) by the standard multi-cell formula Indeed, by the change of variables x → x/ε and Theorem 3.3 the above limit equals where the last equality follows from the convexity of W hom .
Next we discuss the convergence of boundary value problems together with a varying forcing term added to the functionals. Given g ∈ W 1,∞ (R d , R m ) and f ε ∈ L d (D) m , we define the constrained functional Due to the Sobolev embedding the integral involving f ε is finite for u ∈ W 1,1 (D) m .
Then almost surely, as ε → 0, the functionals u → F ε,fε,g (ω, u, D) Γ-converge in L 1 (D) m to the deterministic integral functional F hom,f,g : and +∞ otherwise. The integrand W hom is given by Theorem 3.1. Moreover, any sequence u ε such that is weakly relatively compact in W 1,1 (D) m and strongly relatively compact in L d/(d−1) (D) m .
If (A5) is satisfied, the above result is also valid when f ε ⇀ f in L q (D) m for some q ≥ 1 with 1 q < 1 − 1 p + 1 d , and sequences u ε satisfying (3.2) are weakly relatively compact in W 1,p (D) m , where p > d − 1 is the exponent in (A5).

Remark 3.4.
(i) The condition g ∈ W 1,∞ (R d , R m ) can be weakened to Lipschitz-continuity on ∂Ω. Then one can redefine g on R d \ ∂Ω using Kirszbraun's extension theorem and the definition of the functional F ε,fε,g is not affected. (ii) By the fundamental property of Γ-convergence, Theorem 3.3 and the boundedness of F ε,fε,g (ω, g, D) as ε → 0, imply that up to subsequences the minimizers of F ε,fε,g (ω, ·) converge to minimizers of F hom,f,g . In particular, when W is strictly convex in the last variable, then one can argue verbatim as in [30,Propisition 4.14] to conclude that also W hom is strictly convex. In this case, the minimizers u ε at the ε-level and u 0 of the limit functional are unique and u ε → u 0 as ε → 0 weakly in W 1,1 (D) m and strongly in L d/(d−1) (D) m . (iii) In the scalar case m = 1 non-convexity does not play a major role whenever the assumptions ensure a relaxation result via convexification of the integrand. In particular, this holds when the non-convex integrand h is stationary and ergodic (in the sense of Assumption (A1)) and the corresponding integrand h : Ω × R d → [0, +∞) is jointly measurable, upper semicontinuous in the second variable for a.e. ω ∈ Ω and satisfies the following strengthened growth assumptions: Under these assumptions, one can apply [25,Corollary 3.12] to deduce that the weak Due to the growth condition (2), one can extend this representation by well-known approximation results (cf. [28, Lemma 3.6] and [16, Chapter X, Proposition 2.10]) to all functions u ∈ W 1,1 (D) (the restriction to D being smooth and strongly-starshaped in [28] is not necessary since the left-hand side functional is lower semicontinuous with respect to weak convergence, so recovery sequences on balls are recovery sequences on all open subsets A with |∂A| = 0). Using (1), one can show that the weak W 1,1 (D)-relaxation agrees with the strong L 1 -relaxation (cf. the proof of Lemma 4.1, where one has to use the Fenchel-Young inequality for the two functions h * and h * * instead), so that by general Γ-convergence theory the Γ-limit of the non-convex energy agrees with the Γ-limit of the convexified functionals. Due (1) and (2), the function h * * satisfies Assumption 1, so the results from the convex case transfer to the non-convex one. Concerning our Γ-convergence result with boundary conditions, the only difference when relaxing the heterogeneous functional comes from the extension of the relaxation from Lipschitz-functions to general functions because on ,∞ is the boundary datum. By convexity this integrand can be bounded via Then we can due the approximation using [16, Chapter X, Propositions 2.6 & 2.10] and therefore can again assume that the integrand is convex. It should be noted that we need (2) only for the convexified integrand h * * , while for the relaxation result in [25] a bound of the form h(ω, z) ≤ W (ω, z) with W is a real-valued function that is convex in z for a.e. ω ∈ Ω and satisfies the integrability condition E[W (·, z)] < +∞ for all z ∈ R d (cf. (A3) and Remark 2.4 (i)) suffices. However, we were not able to find a relaxation result only assuming the last integrability condition (translated to the physical space) instead of (2) and which holds on the whole domain of the integral functional.
Our final result concerns the Euler-Lagrange equations of the functionals F ε,fε,g and F hom,f,g . In particular, we address the differentiability of the function W hom . Here we have to rely on the stronger Assumption 2 to be able to work in (Sobolev-)Orlicz spaces. Theorem 3.5. Let W satisfy Assumption 2 and let g, f ε and f be as in Theorem 3.3. Then the following statements hold true.
i) Almost surely there exists a function u ε ∈ g + W 1,1 where the spaces W 1,ϕε 0,ω (D) m and W 1,ϕ hom 0 (D) m are the Sobolev-Orlicz spaces associated to W and W hom (cf. Remark 2.6). v) If W is strictly convex in its last variable, then the solutions u ε and u 0 are unique and almost surely, as ε → 0, the random solutions Since the Sobolev-Orlicz spaces are vector spaces, this yields that the equations in iv) imply equality in i) and iii). Moreover, the equations in i) and iii) can be extended by approximation to ϕ ∈ W 1,1 0 (D) m , whose gradient can be approximated weakly * in L ϕε ω (D) m×d or L ϕ hom (D) m×d , respectively, where both spaces are are regarded as subspaces of the dual space of L ϕ * ε ε (D) m×d or L ϕ * hom (D) m×d , respectively. b) While the points i) and iii) imply that u ε and u 0 are distributional solutions of the PDEs −div(∂ ξ W (ω, · ε , ∇u)) = f ε and −div(∂ ξ W (∇u)) = f respectively, they are no weak solutions in the corresponding Sobolev-Orlicz space. This problem is strongly related to the lack of density of smooth functions in Sobolev-Orlicz spaces. As a byproduct of our proof the solutions that minimize the energy satisfy , so that in general the weak formulation of the PDE would make sense in duality as in iv), but we are not able to prove it. c) Concerning the integrands in Example 2.5, the condition in iv) is satisfied for p(·)-Laplacians with essentially bounded exponent or double phase functionals W (ω, x, ξ) = |ξ| p + a(ω, x ε )|ξ| q with no additional restrictions on the exponents. In a more abstract form, it suffices to have an estimate of the form W (ω, sξ) ≤ CW (ω, ξ) + Λ(ω) for some s > 1. In this case one can use the formula for W hom given in Lemma 4.3 to show that W hom (sξ) ≤ C(W hom (ξ) + 1).

Proofs
Before we start with the different proofs leading to our main results, let us comment on the null sets of Ω that we need to exclude: besides excluding the set of zero measure, where the properties of W (or W ) in Assumption 1 or 2 fail, • we will frequently apply the ergodic theorem in the form of Lemma 2.3 to the random field W (ω, x ε , ξ) = W (τ x ε ω, ξ). A priori, the null set where convergence may fail could depend on ξ, so let us briefly explain why this is not the case. Considering an element ω ∈ Ω, where the ergodic theorem holds for all rational matrices ξ ∈ Q m×d , we know that for any bounded, measurable set To extend this property to irrational matrices ξ 0 , note that the sequence of maps is still convex and by assumption it is bounded on rational matrices. Since we can write R m×d as the countable union of cubes with rational vertices, this implies that the sequence is locally equibounded and by [18,Theorem 4.36] it is locally equi-Lipschitz. Hence it converges pointwise for all ξ ∈ R m×d and the limit is given by E[W (·, ξ)] as this function is the continuous extension of the limit for rational matrices. • we will also apply Lemma 2.3 to the variables sup |η|≤r W (ω, x ε , η) or sup |ζ|≤r W * (ω, x ε , ζ). These terms only appear in bounds, so that we can tacitly restrict r to positive, rational numbers. Moreover, we will use Lemma 2.3 also for the map Λ(ω, x ε ). • we further exclude the null sets where Lemma 4.3 fails for rational matrices ξ ∈ Q m×d or where Lemma 4.4 for W or the countably many approximations W k given by Lemma 4.5. • finally, we apply Lemma 2.3 to the maps W (ω, If not stated explicitly otherwise, we shall always assume that we have an element ω of the set of full measure such that the above properties hold. Then, as ε → 0, the gradients ∇u ε are relatively weakly compact in L 1 (D) m×d . If moreover u ε is bounded in L 1 (D) m , then, up to subsequences, there exists u ∈ W 1,1 (D) m such that u ε ⇀ u weakly in W 1,1 (D) m . If W satisfies in addition (A5), the above statement is also true with L 1 (D) m×d , L 1 (D) m and W 1,1 (D) m replaced by L p (D) m×d , L p (D) m and W 1,p (D) m , respectively.
Proof. Let A ⊂ D be a measurable set and v ∈ L ∞ (D) m×d satisfy v L ∞ (D) ≤ 1. For any r ≥ 1 the Fenchel-Young inequality and the convexity of W in the last variable yield that Taking the supremum over all such v's, the nonnegativity of W and the global energy bound imply that Note that W * inherits the stationary of W and so does the function sup |η|≤r W * (·, ·, η). Combining (A3) and Lemma 2.3, the functions in the last two integrals in (4.1) are equiintegrable. Hence we deduce that Letting r → +∞, it follows that also ∇u ε is equiintegrable as ε → 0. Since D has finite measure, it remains to show that ∇u ε is bounded in L 1 (D) m×d . To see this, choose A = D and r = 1 in (4.1) and use again Lemma 2.3 to conclude that the functions in the last two integrals in (4.1) are also equi-bounded in L 1 (D). The last assertion is a standard result for bounded sequences in W 1,1 once the gradients are weakly compact. The claim for W satisfying (A5) is simpler and follows from sup and standard results for Sobolev spaces with exponent p > 1.
Next, we adapt the construction of correctors in [14,19] to the superlinear setting without any polynomial growth of order p > 1 from below. Define the set Even though d = 3 in general, we refer to the property ∂ i h j − ∂ j h i = 0 as being curl-free. The following result is [30,Lemma 4.11].
almost surely as maps in L 1 loc (R d ) d and such that for every bounded set B ⊂ R d the maps ω → ϕ(ω, ·) and ω → ∇ϕ(ω, ·) are measurable from Ω to L 1 (B) and to L 1 (B) d , respectively.
The above result allows us to introduce a corrector by solving a minimization problem on the probability space as explained in the lemma below.
We call φ ξ : Ω → W 1,1 loc (R d ) m given by Lemma 4.2 applied to the components of h ξ the corrector associated to the direction ξ. We assume in addition that − B1 φ ξ (ω, x) dx = 0. Then, as ε → 0, almost surely for any bounded, open set A ⊂ R d it holds that The function ξ → W hom (ξ) is convex, finite and superlinear at infinity. Moreover, the following is true: (i) Suppose that W satisfies in addition (A4). Then, there exists C 0 < +∞ such that for all ξ ∈ R m×d and all (ii) Suppose that W satisfies in addition (A5). Then for all ξ ∈ R m×d W hom (ξ) ≥ |ξ| p , and almost surely, as ε → 0, it holds that εφ Proof. The existence of minimizers in the weakly closed set (F 1 pot ) m (cf. Lemma 4.2) for the functional h → E[W (·, ξ + h)] follows from the direct method of the calculus of variations. Indeed, the convexity of W in the last variable turns the functional weakly lower semicontinuous for the L 1 (Ω)-topology, while the relative weak compactness of minimizing sequences can be shown using (A3) in the form of ω → sup |η|≤r W * (ω, η) ∈ L 1 (Ω) for all r > 0 as in the proof of Lemma 4.1, replacing the oscillating term x ε by 0 and the physical space by the probability space. Next, note that the constraint − B1 φ ξ (ω, x) dx = 0 does not affect the measurability property stated in Lemma 4.2 as this integral term is a measurable function of ω. We continue by showing the weak convergence to zero, dropping the dependence on ξ for the moment. For a.e. ω ∈ Ω we have ∇φ(ω, ·/ε) = h(τ ·/ε ω) and h ∈ L 1 (Ω). Hence the ergodic theorem in the form of Lemma 2.3 implies that for any bounded set B ⊂ R d we have where we used that h ∈ (F 1 pot ) m for the last equality. We will show that which yields the claim by Poincaré's inequality considering a ball B such that B 1 ∪ A ⊂ B. By a density argument one can show that for r ≥ 1 ∇φ(ω, ty)y dy dt.
By approximation with continuous functions one can show that the map t → − B1 ∇φ(ω, ty)y dy is continuous on (0, +∞) and by (4.3) it vanishes at infinity. Hence the right-hand side term in the above equality vanishes as r → +∞. This yields (4.4).
The function W hom is convex by the convexity of W in the last variable. It is finite since 0 ∈ (F 1 pot ) m is admissible in the minimization problem defining W hom , so that due to the integrability condition (A3) we have The superlinearity follows from the lower bound in (2.3). Indeed, for any C > 0 we have By Assumption (A3), the last expectation is finite for any fixed C > 0, so that dividing the above inequality by |ξ| and letting |ξ| → +∞, we infer that Superlinearity at infinity follows from the arbitrariness of C > 0. In order to prove the assertion in (i), fix ξ, ξ ∈ R m×d as in the statement and let h ξ ∈ (F 1 pot ) m be such Then, applying (A4) pointwise to the two matrix-valued functions ξ + h ξ and ξ + h ξ , we deduce that The additional statements in (ii) are well-known: the p-growth from below implies that h ξ satisfies in addition h ξ ∈ (L p (Ω; R d )) m and from this we deduce the weak convergence in W 1,p (A, R m ) for εφ ξ (ω, ·/ε). The coercivity for W hom follows by E[h ξ ] = 0, (A5) and Jensen's inequality as As a final result of this section, we show the almost sure existence of the limit in an asymptotic minimization formula in the physical space described in the following lemma.
There exists a convex function µ hom : R m×d → [0, +∞) and a set Ω ′ ⊂ Ω with P[Ω ′ ] = 1 such that the following is true: for every ω ∈ Ω ′ and every cube Q = x + (−η, η) d ⊂ R d and ξ ∈ R m×d it holds that Proof. We apply the subadditive ergodic theorem. According to Lemma B.1 the function ω → µ ξ (ω, O) is measurable. To show its integrability, we test the affine function u(x) = ξx as a candidate in the infimum problem. Since F 1 is nonnegative, we obtain Tonelli's theorem and stationarity of W yield that Hence µ ξ (·, O) ∈ L 1 (Ω). By stationarity of W also µ ξ is τ -stationary in the sense that Minimizing the right-hand side with respect to the variables v j , we deduce subadditivity in the form of By the subadditive ergodic theorem (see [1, Theorem 2.7]), a.s. there exists the a priori random limit for all cubes of the form Q = z + (−k, k) d with integer vertices k ∈ N and z ∈ Z d . The extension to arbitrary sequences t → +∞ and general cubes Q = x + (−η, η) d with x ∈ R d and η > 0 follows by approximation as in [30,Lemma 4.3], relying on the fact that W (·, ξ) ∈ L 1 (Ω), which allows us to apply the additive ergodic theorem in the form of Lemma 2.3 to the error terms that are due to integrating W (ω, x, ξ) over sets with small measure (relatively to the scale t d ). Similarly, one can prove that µ 0 is invariant under every group action τ z , so by ergodicity it is deterministic. We call this value µ hom (ξ).
To fix the issue that the exceptional set where convergence fails may depend on ξ, fix ξ 0 , ξ ∈ R m×d . For comparing µ ξ and µ ξ0 , we consider cubes of different size. For a cube Q = x + (−η, η) d and s > 0 set Q(s) = x + (−sη, sη) d and fix δ > 0. There exists a smooth cut-off function Given From the properties of ϕ, we infer that , the convexity of W in the last variable allows us to bound the error term by where C is independent of δ and t. We assume from now on that |ξ − ξ 0 | ≤ δ. Passing to the infimum over v, we deduce that Assume further that ξ 0 ∈ Q m×d and consider ω in the set of full probability such that the limit of t → |tQ| −1 µ ξ0 (ω, tQ) at +∞ exists for all rational matrices ξ 0 and all cubes. The additive ergodic theorem 2.3 applied to the last integral then yields that From an analogous construction with the cubes tQ and tQ(1 − δ) we further infer that which again by the ergodic theorem implies that lim sup Combining the two estimates (4.10) and (4.11) yields lim sup Considering a sequence of rational matrices that converges to ξ and then letting δ → 0, we deduce that lim sup so that the limit exists and hence the convergence holds for a uniform (with respect to ξ and Q) set of full probability. The convexity of µ hom is a consequence of the convexity of the map ξ → µ ξ (ω, O), which itself follows from the convexity of the map ξ → W (ω, x, ξ).

4.2.
Proof of the Γ-liminf inequality by truncation of W . For the Γ-liminf inequality, we approximate W from below by integrands with polynomial growth. This technique was already implemented for functionals satisfying p growth from below with p > 1, see e.g. [28,14]. Here, we generalize this method to cover the case of merely superlinear growth condition as in Assumption 1, where in contrast to the case with p-growth from below, the approximation only satisfies so-called non-standard or p − q-growth conditions.
Proof. Due to Remark 2.2, it is enough to perform the construction at the level of W , so that (A1) comes for free. We first define a suitable lower bound for W . Set ℓ(ω, ξ) = co(min{ℓ(ω, |ξ|), q −1 |ξ| q }), where co denotes the convex envelope and ℓ is the convex, monotone, superlinear function given by Remark 2.4 (ii). Then ℓ is jointly measurable 2 and ℓ(ω, ξ) ≤ W (ω, ξ). Next, since W ≥ 0, following the proof of [18,Theorem 6.36] there exist measurable and bounded functions a i : Ω → R and b i : Ω → R m such that for all ω ∈ Ω with W (ω, ·) being convex we have We define an approximation of W by setting Then W k is jointly measurable, increasing in k, convex in ξ and satisfies W k ↑ W pointwise in ξ for a typical element ω ∈ Ω (cf. the beginning of Section 4). Finally, we define Then W k is still jointly measurable, convex in ξ (as required in (A2)), increasing in k, and due to the lower bound ℓ(ω, ξ) ≤ W (ω, ξ) it also follows that W k ↑ W . Moreover, since the individual functions a i , b i are bounded, we have that which also implies the upper bound in (A3). Next, we show the bound on the conjugate function in (A3). To bound it from below, note that by definition of the conjugate For the upper bound, we will use several times that the convex envelope for finite, convex functions f : R m×d → R that are bounded below by an affine function (here zero) can be characterized by the biconjugate function, i.e., f * * = co(f ) (cf. [18,Remark 4.93 (iii)]) and that f * * * = f * (see [18,Proposition 4.88]), which in combination yields that co(f ) * = f * . (4.13) Since f ≤ g implies f * ≥ g * , it suffices to show that sup |η|≤r ( ℓ) * (·, η) ∈ L 1 (Ω).
According to (4.13) and the definition of ℓ, we know that where q ′ is the conjugate exponent to q. The function η → 1 q ′ |η| q ′ is deterministic and locally bounded, and therefore it suffices to show that sup |η|≤r ℓ * (ω, η) ∈ L 1 (Ω).
Recall the construction of ℓ in Remark 2.4 (ii) as the convex envelope of the radial minimum of W . According to (4.13) we have that Taking the supremum over |η| ≤ r and using the commutativity of suprema, we deduce that which concludes the proof of (A3).
In the next proposition we prove the Γ-liminf inequality for the truncated energies defined via the integrand W k given by the previous lemma.
Proof. To reduce notation, we drop k from the notation and assume in addition that W satisfies the growth condition W (ω, x, ξ) ≤ C(|ξ| q + 1) (4.14) for some 1 < q < 1 * and a.e. x ∈ R d . Without loss of generality, we assume that the limit inferior in the statement is finite and, passing to a non-relabeled subsequence, it is actually a limit. Following the classical blow-up method, define the absolutely continuous Radon-measure m ε on D by its action on Borel sets B ⊂ D via m ε (B) = B W (ω, x ε , ∇u ε (x)) dx. By our assumption, the sequence of measures m ε is equibounded, so that (up to passing to a further non-relabeled subsequence) m ε ⋆ ⇀ m for some nonnegative finite Radon measure m (possibly depending on ω). Using Lebesgue's decomposition theorem, we can write m = f (x)L d + ν, with ν a nonnegative measure ν that is singular with respect to the Lebesgue measure. Since D is open, the weak * convergence implies that Hence it suffices to show that f (x 0 ) ≥ µ hom (∇u(x 0 )) for a.e. x 0 ∈ D. The Besicovitch differentiation theorem [18,Theorem 1.153] and Portmanteau's theorem imply that for a.e. x 0 ∈ D we have W (ω, x ε , ∇u ε (x)) dx ≥ µ hom (∇u(x 0 )).
We claim further that To verify (4.16), note that due to the convexity of the map ξ → W (ω, y, ξ), for η ∈ (0, 1) it holds that . Since η ≤ 1, integrating this inequality over Q r (x 0 ) and taking the average, we obtain that and thus it suffices to show that the last integral vanishes. As ε → 0, we can apply the ergodic theorem and obtain For such x 0 we define the linearization of u at In what follows, we drop the dependence on x 0 from cubes and tacitly assume that they are centered at x 0 . We modify u ε close to ∂Q r such that the modification attains the boundary value L u,x0 : for 0 < η < 1 we pick a cut-off function ϕ η ∈ C ∞ c (Q r , [0, 1]) such that ϕ η (x) = 1 on Q ηr , which can be chosen such that ∇ϕ η ∞ ≤ C (1−η)r . Define then the function u ε,η : D → R m by u ε,η = ηϕ η u ε + η(1 − ϕ η )L u,x0 .
Since u ε,η = ηu ε on Q ηr , we can estimate the energy of u ε,η on Q r by We argue that the last term is asymptotically negligible relative to r d . To reduce notation, we set S r η = Q r \ Q ηr . Since u ε ∈ W 1,1 (D) m due to the global energy bound, the product rule yields that so that 0 ≤ η, ϕ η ≤ 1 and the convexity of ξ → W (ω, y, ξ) imply the estimate Let us first estimate the last term, using the polynomial bound (4.14). Inserting the uniform bound on ∇ϕ η , we find that Inserting this estimate into (4.20) and the resulting bound into (4.19), we infer that 1 We now let ε → 0. To estimate the left-hand side term from below, note that u ε,η = ηL u,x0 on ∂Q r , so that by a change of variables we have F ε (ω, u ε,η , Q r ) ≥ µ η∇u(x0) (ω, Q r /ε)ε d . In particular, from Lemma 4.4 we deduce that To treat the right-hand side terms in (4.21), we note that the second term is r −d m ε (Q r \ Q ηr ), so that we can estimate it using Portmanteau's theorem and switching to the closure of Q r \ Q ηr . For the third term we can apply the ergodic theorem to the integrand W (ω, x ε , ∇u(x 0 )). In order to pass to the limit in the fourth term, we note that u ε is bounded in W 1,1 (D) m and hence strongly converging in L q (D) due to the Sobolev embedding. In total, we obtain that Next, we let r → 0. On the one hand, by (4.15) and (4.18) we have that On the other hand, the L 1 * -differentiability of W 1,1 -functions (cf. Gathering these two pieces of information, we obtain that Finally, letting η → 1, the continuity of µ hom (which follows from convexity) yields which coincides with (4.17) and thus we conclude the proof.
Note that h ξ,ε is well defined and measurable due to the joint measurability of ∇v ξ,ε and the joint measurability of the group action. To see that it is integrable, we can use Fubini's theorem and a change of variables in Ω to deduce that where we used the superlinearity (2.3) of W for a.e. x ∈ R d (cf. Remark 2.2). The last term is finite, since for a.e. ω ∈ Ω the function ∇v ξ,ε (ω, ·) is the gradient of an energy minimizer on Q/ε. We argue that h ξ,ε ∈ (F 1 pot ) m . Since for a.e. ω ∈ Ω the function ∇v ξ,ε is the weak gradient of u ξ,ε (ω) ∈ W 1,1 0 (Q/ε) m , it follows from Fubini's theorem and a change of variables in Ω that Hence, it suffices to show that all rows of x → h ξ,ε (τ x ω) satisfy the curl-free condition of Definition 4.2.
To this end, we derive a suitable formula for the distributional derivative of this map. Fix θ ∈ C ∞ c (R d ) and an index 1 ≤ j ≤ d. Since ∇v ξ,ε (ω, ·) = 0 on R d \ (Q/ε), we can write In order to conclude that h ξ,ε ∈ (F 1 pot ) m , it suffices to note that for a.e. ω ∈ Ω and almost every z ∈ R d the function y → ∇v ξ,ε (τ z ω, y) is the gradient of the Sobolev function u ξ,ε (τ z ω) ∈ W 1,1 0 (Q/ε) m , so that the curl-free conditions follows. Now we can conclude the proof. Since h ξ,ε ∈ (F 1 pot ) m , it follows from Jensen's inequality that where we used the stationarity of W , and that ∇v ξ,ε (ω, ·) = ∇u ξ,ε (ω) almost surely in the last step.
Next, we prove that lim k→+∞ W hom,k = W hom , which allows us to prove the Γ-liminf inequality by truncation.
Lemma 4.8. Assume that W satisfies (A1), (A2) and (A3). For k ∈ N let W k be the integrand given by Lemma 4.5 and denote by W hom,k and W hom the functions given by Lemma 4.3 applied to W k and W , respectively. Then for all ξ ∈ R m×d it holds that lim k→+∞ W hom,k (ξ) = W hom (ξ).
Proof. Since W k ≤ W , we clearly have W hom,k (ξ) ≤ W hom . For the reverse inequality, note that due to monotonicity it suffices to prove the claim up to subsequences. Let h ξ,k ∈ (F 1 pot ) m be a minimizer defining W hom,k (ξ) = E[W k (·, 0, ξ + h ξ,k )]. Due to the uniform superlinearity of W k (recall the monotonicity in k), we know that (up to a subsequence) h ξ,k ⇀ h ∈ (F 1 pot ) m in L 1 (Ω) m×d . Fix h ∈ (F 1 pot ) m . Then due to monotone convergence and lower semicontinuity, for every m ∈ N we have Letting m → +∞, we find that E[W (·, 0, ξ + h)] ≥ E[W (·, 0, ξ + h)]. Hence h is a minimizer and, as proven above Now we are in a position to prove the Γ-liminf inequality for the original functionals.
with W hom defined in Lemma 4.3.
Proof. Since W k ≤ W for all k ∈ N, it follows from the liminf inequality proven in Proposition 4.6 that Letting k → +∞, the claim follows from Lemma 4.8 and the monotone convergence theorem.

4.3.
Construction of a recovery sequence. In Proposition 4.9 we proved the Γ-lim inf inequality assuming only (A1), (A2) and (A3). In the proof of the Γ-lim sup inequality, see Proposition 4.11 below, we need to assume in addition either the mild monotonicity condition (A4) or the coercivity condition (A5). The main technical part in the proof of Proposition 4.11 is to construct a recovery sequence for affine functions that satisfy prescribed affine boundary values. In order to attain the boundary condition, we introduce a cut-off that causes an additional error term. Assuming the monotonicity condition (A4), we combine the cut-off with a truncation argument to control the additional error. Without a structure assumption such as (A4), truncation will not work and we rely on Sobolev embedding instead. Similar arguments were used e.g. in [14,28] assuming (A5) with p > d, exploiting the compact embedding of W 1,p into L ∞ . The improvement from p > d to p > d − 1 comes from suitably chosen cut-off functions in combination with the compact embedding of W 1,p (S 1 ) ⊂ L ∞ (S 1 ), where S 1 denotes the d−1-dimensional unit sphere, provided p > d − 1. The following lemma encodes the needed compactness property.
Step 1 We prove the statement for u 1 , . . . , u N ∈ C 1 (B R ) m . For i ∈ {1, . . . , N } and C := 4N , we set An elementary application of Fubinis Theorem and the definition of U i in the form An analogous computation yields that (4.32) Next, we define η ∈ W 1,∞ (B R ; [0, 1]) by By definition, we have that 0 ≤ η ≤ 1, η = 1 in B (1−δ)R , η ∈ W 1,∞ 0 (B R ) and for x = rz with r ∈ [0, R] and z ∈ S 1 Hence, recalling (4.32), the map η satisfies all the properties in (4.28). Next, we use p > d − 1 ≥ 1 in the form that the embedding W 1,p (S 1 ) m ⊂ L ∞ (S 1 ) m is compact. In particular, for every ρ > 0 there exists C ρ such that for all v ∈ C 1 (S 1 ) m it holds sup S1 |v| ≤ ρ D τ v L p (S1) + C δ v L p (S1) where D τ denotes the tangential derivative (see [26,Lemma 5.1]). Applying the above estimate to v r ∈ C 1 (S 1 , R m ) defined by v r (z) := u(rz) for all z ∈ S 1 with u ∈ C 1 (B R , R m ), we obtain with the chain rule sup z∈S1 |u(rz)| ≤ ρr ∇u(r·) L p (S1) + C ρ u(r·) L p (S1) . (4.34) Hence, for every ρ > 0 there exists C ρ < ∞ such that for all x = rz with r = |x| and z = x where we use the definition of U and 1 − δ ≥ 1 2 in the last inequality. The claimed estimate (4.29) follows by redefining the choice of ρ > 0 (depending on N ).
Step 2 Conclusion. Consider u 1 , . . . , u N ∈ W 1,p (B R ) m . By standard density results, we find ( Step 2, we find for every j ∈ N a cut-off function In view of the bounds in (4.35) and the Banach-Alaoglu Theorem, there exists η ∈ W 1,∞ 0 (B R ) such that up to subsequences (not relabeled) η j ⋆ ⇀ η in W 1,∞ (B R ). Moreover, η also satisfies the bounds in (4.28). Since ∇η j ⋆ ⇀ ∇η weakly * in L ∞ (B R ; R d ) and u i,j → u i (strongly) in L p (B R ) m , we deduce that ∇η j ⊗ u i,j converges weakly in L p (B R ) m×d to ∇η ⊗ u i and by the boundedness of the right-hand side in (4.36) also weakly * in L ∞ (B R ) m×d . Hence the claimed estimate (4.29) follows from (4.36) and weak * lower-semicontinuity of the norm. Now we are in a position to state and prove the Γ-lim sup inequality. Proof.
Step 1. Local recovery sequence for affine functions with rational gradient. We claim that for any bounded, open set A ⊂ R d , any affine function u : A → R m with ∇u = ξ ∈ Q m×d and u ε = u + εφ ξ (ω, ·/ε), where φ ξ is given as in Lemma 4.3, it holds that Indeed, the convergence u ε → u in L 1 (A) m is a direct consequence of Lemma 4.3, while the ergodic theorem in the form of Lemma 2.3 yields that Step 2. Recovery sequence with prescribed boundary values for affine functions -Assumption (A4).
Let A ⊂ R d be a bounded, open set and let u be an affine function with ∇u = ξ ∈ R m×d . We claim that there exists a sequence (v ε ) ε ⊂ W 1,1 (A) m satisfying Indeed, let (u j ) j be a sequence of affine functions satisfying u j → u in W 1,∞ (A) m and ∇u j ∈ Q m×d for all j ∈ N. In view of Step 1 there exists for every j ∈ N a sequence (u j,ε ) ε ⊂ W 1,1 (A) m satisfying u j,ε → u j in L 1 (A) m as ε ↓ 0 and (4.38). We glue u j,ε to u at the boundary of A and truncate peaks of u j,ε in A. More precisely, we consider for ε, δ, s > 0 and j ∈ N the function v ε,δ,j,s ∈ W 1,1 (A) m given by and T s (u j,ε ) ∈ L ∞ ∩ W 1,1 (A) m is obtained from u j,ε by 'component-wise' truncation, that is, T s (u j,ε ) · e k := max{min{u j,ε · e k , s}, −s} for k ∈ {1, . . . , m}. (4.42) By the product rule, we obtain for every t ∈ [0, 1) Hence, by convexity of W , For this, we start with the decomposition where we used (4.38) and the continuity of W hom . In order to estimate the term where truncation is active, we use the definition of T s in the form ∀k ∈ {1, . . . , m} : e T k ∇T s (u j,ε ) ∈ {0, e T k ∇u j,ε } a.e.
For s ≥ u L ∞ (D) + 2 and j sufficiently large such that it holds u j − u L ∞ (D) < 1, we have Hence, we obtain with help of (A4) and s and j as above that We claim that which together with (4.45) yields (4.44). In order to verify (4.46), recall that ∇u j,ε = ξ j + ∇φ ξj ( · ε ), so that the ergodic theorem in the form of Lemma 2.3 implies that W (ω, · ε , ∇u j,ε ) ⇀ W hom (ξ j ) and Λ(ω, · ε ) ⇀ E[Λ] weakly in L 1 (A) as ε ↓ 0. In particular, due to the Dunford-Pettis theorem (see e.g. [18,Theorem 2.54]) the sequences (W (ω, · ε , ∇u j,ε )) ε and (Λ(ω, · ε )) ε are equi-integrable and thus (4.46) follows from the convergence u j,ε → u j in Using once more the convexity of W , we can bound the integral by For the last term, recall that u j → u in L ∞ (A) m . Hence, given s, δ > 0 and t ∈ [0, 1), for j large enough we have 2t so that for such j Due to Assumption (A3) we can apply the ergodic theorem to the function on the right-hand side and deduce that lim sup To treat the other right-hand side term in (4.48), first recall that u j,ε → u j in L 1 (A) m . Using Egorov's theorem, for every sequence ε → 0 we find a subsequence such that for any η > 0 there exists a measurable set A η with measure less than η and such that u j,ε → u j uniformly on A \ A η . Consider j large enough such that u j − u L ∞ (A) < 1 and s ≥ u L ∞ (A) + 2. Then T s u j = u j a.e. on A. By construction, for any such s, and δ > 0 and t ∈ [0, 1) there exist r > 0 such that Since T s (u j,ε ) also converges uniformly to T s (u j ) = u j on A \ A η , for ε small enough we can bound the integrand by The second right-hand side term can be treated as before, while for the first one the equi-integrability of x → sup |ζ|≤r W (ω, x ε , ζ) (cf. Assumption (A3) and Lemma 2.3) implies that lim η→0 lim sup ε→0 Aη sup |ζ|≤r W (ω, x ε , ζ) dx = 0.
Since the limit does not depend on the subsequence we picked for Egorov's theorem, we proved that so that in combination with (4.49) and (4.48) we indeed obtain (4.47). Combining Passing to a suitable diagonal sequence and using the lim inf-inequality of Proposition 4.9 on the bounded, open set A, we obtain the desired recovery sequence satisfying (4.39).
Step 3. Recovery sequence with prescribed boundary values for affine functions -Assumption (A5). We show that for bounded, open set A ⊂ R d , every affine function u with ∇u = ξ ∈ R m×d there exists a sequence (v ε ) ε satisfying We first consider the case when A is a ball. Let B = B R (x 0 ) with R > 0, x 0 ∈ R d ,consider a sequence u j of affine functions satisfying ∇u j = ξ j ∈ Q m×d and u j → u in W 1,∞ (B) m . For δ ∈ (0, 1 2 ) and j ∈ N, let η = η ε,δ,j be as in Lemma 4.10 with N = 1 and u 1 = εφ ξj ( · ε ). We set v ε,δ,j := η ε,δ,j (u j + εφ ξj ( · ε )) + (1 − η ε,δ,j )u. Clearly, we have lim sup and by convexity, we have Similarly to Step 2, we obtain lim sup (4.53) Hence, it remains to estimate the last term in the above bound. By Lemma 4.10, we find for every ρ > 0 a constant C ρ < ∞ depending only on d, m, p and ρ > 0 such that By Lemma 4.3 the last integral vanishes as ε → 0, while the first right-hand side integral remains bounded as ε → 0. From the arbitrariness of ρ > 0 we thus infer that lim sup ε↓0 ∇η ε,δ,j ⊗ εφ ξj ( · ε ) L ∞ (B) = 0, which implies with help of the triangle inequality and (4.28) that for all t ∈ [0, 1), δ ∈ (0, 1 2 ] and j ∈ N it holds lim sup In particular, for all t ∈ (0, 1), δ ∈ (0, 1 2 ] and j sufficiently large (depending on t and δ) we have where we used Assumption (A3) and the ergodic theorem in the form of Lemma 2.3. Thus Combining (4.51), (4.52), (4.53) and (4.54), we obtain a diagonal sequence satisfying (4.50) in the case A = B. Next, we remove the restriction on A being a ball and consider a general bounded, open set A ⊂ R d and an affine function u with ∇u = ξ ∈ R m×d . By the Vitali covering theorem, we find a collection of disjoint balls The previous result for balls ensures that for all i ∈ {1, . . . , N }, we find a sequence (v we have that v ε ∈ u + W 1,p 0 (A) m and v ε → u in L p (A) m . Moreover, by the ergodic theorem it holds that where we used in the last step W hom ≥ 0. Letting N ↑ +∞, we conclude the proof.
Step 4. The general case. In this step, we pass from the limsup-inequality for affine functions to the limsup-inequality for general functions u ∈ W 1,1 (D) m . Let us first consider u ∈ W 1,∞ (D) m which is piecewise affine, i.e., there exist finitely many disjoint open sets A i ⊂ D, i = 1, . . . , K such that |D \ ∪ i A i | = 0 and u| Ai (x) = ξ i x + b i for some ξ i ∈ R m×d and b i ∈ R m . In view of Step 2 and Step 3, we find for every A i a sequence (u i,ε ) ε ∈ u + W 1,1 0 (A i ) m satisfying either (4.39) or (4.50). Since all u i,ε , i = 1, . . . , K coincide with u at the boundary of A i , we can glue them together and obtain a recovery sequence for u on D.
The limsup-inequality for general u ∈ W 1,1 (D) m follows as in [14,28]. By the 'locality of the recovery sequence' (see [14,Corollary 3.3]), we can assume that D ⊂ R n is an open ball. Indeed, for a bounded, open set with Lipschitz boundary D we find a ball B such that D ⊂⊂ B. Assume that we have a recovery sequence (u ε ) ε satisfying u ε → u in L 1 (B) m and F ε (ω, u ε , B) → B W hom (∇u) dx as ε ↓ 0. Then, where we used in the second inequality that u ε is a recovery sequence on B and the lim inf inequality Proposition 4.9 Hence, it suffices to assume that D ⊂ R n is an open ball and thus smooth and star-shaped. Fix u ∈ W 1,1 (D) m such that D W hom (∇u) dx < +∞. Since W hom : R m×n → [0, +∞) is convex (cf. Lemma 4.3), we can apply [28, Lemma 3.6] and find a sequence (u j ) j of piecewise affine functions satisfying u j → u in W 1,1 (D) m and D W hom (∇u j ) dx → D W hom (∇u) dx as j → ∞. Hence, by the previous argument, we find (u j,ε ) ε with u j,ε → u j in L 1 (D) m and F ε (ω, u j,ε , D) → D W hom (∇u j ) dx and thus The claim follows again by a diagonal argument.

4.4.
Convergence with boundary value problems and external forces. In this section we prove the Γ-convergence result under Dirichlet boundary conditions and with external forces, i.e., Theorem 3.3.
Proof of Theorem 3.3. In a first step we consider the case f ε = f = 0. Weak W 1,1 -compactness (or W 1,pcompactness assuming (A5)) of energy bounded sequences follows from the unconstrained case (cf. Lemma 4.1) and the fact that the Dirichlet boundary condition allows us to apply a Poincaré inequality to deduce L 1 -boundedness. The Γ-liminf inequality is also a consequence of the unconstrained case (cf. Lemma 4.9) since the boundary condition is stable under weak convergence in W 1,1 (D) m . Hence it remains to show the Γ-limsup inequality, which requires more work. Fix u ∈ W 1,1 (D) m such that u = g on ∂D in the sense of traces and F hom (u) < +∞. Let t ∈ (0, 1). Then u t := g + t(u − g) ∈ g + W 1,1 0 (D) m and by convexity we have and F hom (u t ) ≤ (1 − t)F hom (g) + tF hom (u) < +∞. In particular, and since u t → u in L 1 (D) m when t ↑ 1 and 1+t 2t > 1, a diagonal argument allows us to show the Γ-limsup inequality for functions u such that additionally F hom (s(u − g)) < +∞ for some s > 1. By Lemma C.1 and the properties of W hom (cf. (4.55) By choosing a suitable subsequence (not relabeled) we can assume in the following that (v n ) n satisfies in addition v n → u − g and ∇v n → ∇v − ∇g a.e. in D. Next, we show Indeed, by Fatous lemma and the non-negativity of and W hom , we have To show the corresponding inequality for the lim sup, we first observe that for all ξ ∈ R m×d Hence, the desired inequality follows with help of estimate (4.56), Fatous lemma and (4.55): Since g is Lipschitz-continuous, we thus deduce that there exists a sequence u n ∈ Lip(D) m such that u n = g on ∂D with u n → u in Since u n is piecewise affine in D n , there exist disjoint open sets D 1 n , . . . , D Nn n such that D n = ∪ j D j n and ∇u = ξ j n on D j n . By Step 2 and Step 3 of the proof of Proposition 4.11, we find for every j ∈ {1, . . . , N n } a recovery sequence u j ε,n ∈ u n + W 1,1 0 (D j n ) m satisfying u j ε,n → u n in L 1 (D j n ) m and lim ε↓0 F ε (ω, u j ε,n , D j n ) = D j n W hom (∇u n ) dx.
Since u j ε,n ∈ u n + W 1,1 0 (D j n ) m , we have that u ε,n : D → R m defined by Finally, u j ε,n being a recovery sequence on D j n , it holds that u ε,n → u n in L 1 (D) m as ε → 0 and We estimate the last term as follows: the function ∇u n L ∞ (D) is uniformly bounded by (4.58), so that for some r > 0 Assumption (A3) and the ergodic theorem in the form of Lemma 2.3 imply that Inserting this bound into (4.60), we infer from (4.57) that Since u n → u in L 1 (D) m , using another diagonal argument we conclude the proof without external forces.
Next, we consider the case of non-trivial forcing terms f ε and f . Here, we only prove the case that f ε , f ∈ L d (D) m are such that f ε ⇀ f in L d (D) m , the refined results if W satisfies (A5) are simpler and left to the reader. We first show relative compactness of energy bounded sequence. Due to Hölder's inequality, the Sobolev embedding in W 1,1 (D) m , and Poincaré's inequality in W 1,1 0 (D) m , for any admissible u we deduce that and Remark 2.2 shows that Due to the ergodic theorem the sequence a ε (ω) is bounded when ε → 0. Hence boundedness of F ε,fε,g (ω, u ε , D) implies that also F ε (ω, u ε , D) is bounded, so the weak relative compactness of u ε in W 1,1 (D) m is a consequence of the case f ε = 0. Moreover, as shown in [30,Theorem B.1], the weak convergence in W 1,1 (D) m implies the strong convergence in L d/(d−1) (D) m . Hence along any sequence with equibounded energy and with u ε → u in L 1 (D) m , the term D f ε · u ε dx converges to D f · u dx. Thus also the Γ-convergence is a consequence of the case f ε = 0.
where h ξ is given by Lemma 4.3.
Proof. It suffices to show that W hom is differentiable with the claimed derivative. The continuity of the derivative follows from general convex analysis [18,Theorem 4.65]. Recall that We already know that W hom is a real-valued, convex function on R m×d . In particular, its subdifferential is always non-empty and W hom is differentiable at ξ ∈ R m×d if and only if the subdifferential at ξ contains exactly one element. We aim to express the subdifferential via a suitable chain rule. To this end, define the set-valued mapping F : R m×d ⇒ L ϕ (Ω) m×d by Due to the estimate ϕ(ω, ·) ≤ CW (ω, ·) + Λ(ω) (cf. (2.8)), it follows indeed that dom(G) ⊂ L ϕ (Ω) m×d . Since G is convex, the graph of F defined by gph(F ) = {(ξ, h) ∈ R m×d × L ϕ (Ω) m×d , h ∈ F (ξ)} is a convex subset of R m×d × L ϕ (Ω) m×d . Clearly, we can rewrite the first identity in (4.62) by W hom (ξ) = min{G(y) : y ∈ F (ξ)}. (4.63) We aim to use the representation result for subdifferentials of optimal-value functions as in (4.63) given in [27,Corollary 7.3]. For this it is left to check that G is finite and continuous at a point in F (ξ) with respect to convergence in L ϕ (Ω) m×d . We choose the constant function ξ ∈ F (ξ) and let h n → 0 in L ϕ (Ω) m×d .
Set t n = 1 − h n ϕ . Then the convexity of W and of ϕ together with (2.8) and ϕ(·, 0) = 0 yield that for n large enough such that t n ∈ (0, 1) Hence, h n ϕ → 0 as n → ∞ yields lim sup The reverse inequality follows from Fatous's lemma since h n → 0 also in L 1 (Ω) m×d by the continuous embedding. Now we are in a position to apply [27,Corollary 7.3] to conclude that where in the above equation ℓ(h − (ξ + h ξ )) stands for the duality pairing and h ξ ∈ (F 1 pot ) m is a minimizer in (4.62). Our goal is to show that the sets above are all singletons containing the same element, which then concludes the proof that W hom is differentiable with the claimed derivative. For this, we first observe that for every z ∈ R m×d and t ∈ (0, 1), convexity of W and Assumption (A3) imply so that z + th ξ ∈ dom(G) and thus z + th ξ ∈ F (z). Hence, for η ∈ ∂W hom (ξ) we deduce from (4.64) This holds for all z ∈ R m×d and therefore η, z = ℓ(z) for all z ∈ R m×d .
We claim that the above expression does not depend on the element ℓ ∈ ∂G(ξ + h ξ ). To this end, let us recall the expression for the subdifferential of G at ξ + h ξ in Section 2.3: for a function h ∈ L ϕ (Ω) m×d we have In particular, there is only one possibility for the component ℓ a . Moreover, as noted in Section 2.3 it holds that ℓ s (z) = 0 for all constant functions z ∈ R m×d and all ℓ s ∈ S ϕ (Ω). This in turn yields that for all z ∈ R m×d . Hence ∂W hom (ξ) is a singleton containing the claimed element and we conclude the proof.
4.6. Stochastic homogenization of the Euler-Lagrange equations. In this last section we provide the arguments for our main result on the Euler-Lagrange equations. The notation relies heavily on the generalized Sobolev-Orlicz spaces introduced in Remark 2.6 (see also Section 2.3).
Proof of Theorem 3.5. i): We start showing existence of minimizers for the problem min Recall that in (4.61), for all u ∈ g + W 1,1 0 (D) m we proved the estimate where a ε (ω) is finite and converging as ε → 0. This implies the compactness of minimizing sequences since for fixed ε > 0 the coercivity of F ε (ω, ·, D) with respect to weak convergence in W 1,1 can be proven as in Lemma 4.1 and moreover due to the Lipschitz regularity of g and Assumption (A3). Moreover, F ε (ω, ·, D) is weakly lower semicontinuous due to convexity and strong lower semicontinuity, while the term involving f ε is continuous with respect to weak convergence in W 1,1 (D) m due to the Sobolev embedding. We thus proved the existence of a minimizer. Next, let us show that minimizers are characterized by 0 (D) m and F ε (ω, u + φ, D) < +∞, which proves i). Assume first that u ε satisfies the above system. Then, due to convexity of W , for any which shows minimality since F ε and F ε,fε,g have the same domain on g + W 1,1 0 . For the reverse implication, we omit the dependence on ω to reduce notation. Define the two proper convex functionals G : Note that F is real-valued and continuous since W 1,ϕε 0 (D) m embeds into W 1,1 0 (D) m . Moreover, 0 ∈ dom(G), so that we can apply the sum rule for the subdifferential of G + F [27, Theorem 6.1] to obtain To find a suitable formula for the subdifferential of G, we first write G = G 0 • ∇, where the gradient ∇ : W 1,ϕε 0 (D) m → L ϕε (D) m×d is a bounded linear map, and G 0 : L ϕε (D) m×d → [0, +∞] is defined by Arguing as in the proof of Proposition 4.12, the boundedness of ∇g implies that G 0 is finite and continuous in 0. Hence the chain rule for subdifferentials implies that ∂G(u) = ∇ * ∂G 0 (∇u), where ∇ * denotes the adjoint operator of the gradient map. In particular, for any v, φ ∈ W 1,ϕε 0 (D) m it holds that ∂G(v)φ = ∂G 0 (∇v)∇φ, while by the results in Section 2.3 the subdifferential of G 0 at a function h ∈ L ϕε (D) m×d is given by The function u = g + v being a solution of the minimization problem (4.65) is equivalent to the fact that which is equivalent to the facts that ∂ ξ W ( · ε , ∇u) ∈ L ϕ * ε (D) m×d and that there exists ℓ s ∈ S ϕε (D) m×d with ℓ s (· − ∇v) ≤ 0 on dom(G 0 ), which combined satisfy If φ ∈ W 1,ϕε 0 (D) m×d is such that F ε (ω, u + φ, D) < +∞, then ∇v + ∇φ ∈ dom(G 0 ) and therefore ℓ s (∇φ) ≤ 0, which then yields Moreover, as noted in Section 2.3 we have that ℓ s (∇φ) = 0 whenever φ ∈ W 1,∞ 0 (D) m . This shows that any minimizer satisfies the system in Theorem 3.5 i).
ii) The fact that Assumption 2 implies W hom ∈ C 1 was shown in Proposition 4.12 iii) The proof for the homogenized equation is analogous once one notes that W hom satisfies the same assumptions as W on a deterministic level. Indeed, we already know that W hom is differentiable (Proposition 4.12), while Assumption 2 implies that so that also W hom is almost even, which allows us to define the associated Sobolev-Orlicz space. Moreover, Assumption (A3) holds since W hom is convex, finite and superlinear at +∞ (the last property ensures that W * hom is also convex and finite), while (A4) or (A5) (which were not needed in the proof of i) anyway) were proven in Lemma 4.3.
iv) Under the additional assumption that there exists s > 1 such that su ε ∈ dom(F ε (ω, ·, D)) with u ε = g + v we know that s(∇v + ∇g) − ∇g ∈ dom(G 0 ). Then for any φ ∈ W 1,ϕε 0 (D) m and δ > 0, convexity implies that Since G 0 is continuous in −∇g with G 0 (−∇g) = D W ( x ε , 0) dx, for δ small enough the right-hand side is finite and we can therefore conclude that ℓ s (δ∇φ) ≤ 0 (ℓ s as in the subdifferential representation in i)), which yields that ℓ s (∇φ) ≤ 0. Since this also holds for −ϕ, we conclude that ℓ s (∇φ) = 0 and therefore The proof for the homogenized functional is the same.
v) The convergence assertion in the strict convex case is a consequence of i) and iii) combined with Remark 3.4 (ii).

Appendix B. Measurability
Here we prove the following general measurability result that in particular covers our integrand W satisfying Assumption 1.
Proof. Redefining W (ω, x, ξ) = |ξ| 2 on the set of ω, where µ ξ (ω, O) is not finite (and when no minimizer exists in the second case), we can assume without loss of generality that all properties holds for all ω ∈ Ω. Note that this is possible since the modified integrand is still jointly measurable due to the completeness of the probability space. We first prove that the functional (ω, u) → O W (ω, x, ξ +∇u) dx is F ⊗B(W 1,1 0 (O) m )-measurable. By truncation, we can assume without loss of generality that W is bounded. If W is additionally continuous in the third variable, then the joint measurability is a consequence of Fubini's theorem (which shows measurability in ω) and continuity of the functional with respect to strong convergence in W 1,1 0 (O) m . Indeed, joint measurability then holds due to the separability of W 1,1 0 (O) m , which ensures joint measurability of Carathéodory-functions.
To remove the continuity assumption on W , we use a Monotone Class Theorem for functions. To this end consider the classes of functions defined as Note that if h ∈ C, then h is F ⊗ L d ⊗ B m×d -measurable, thus the argument above shows that C ⊂ R. Moreover, R contains the constant functions, is a vector space of bounded functions, and is closed under uniformly bounded, increasing limits. Finally, the set C is closed under multiplication. Thus [11, Chapter I, Theorem 21] ensures that R contains all bounded functions that are measurable with respect to the σ-algebra generated by C. By definition of C this σ-algebra coincides with F ⊗ L d ⊗ B m×d .
Having the joint measurability of (ω, u) → O W (ω, x, ξ + ∇u) dx at hand, the measurability of the optimal value function µ ξ (·, O) follows from the measurable projection theorem. Indeed, for every t ∈ R we know from joint measurability that By assumption (Ω, F , P) is a complete probability space. Since W 1,1 0 (O, R m ) is a complete, separable, metric space, the projection theorem [18, Theorem 1.136] yields the F -measurability of the projection of (B.1) onto Ω. Therefore we have ω ∈ Ω : inf which proves the F -measurability of µ ξ (·, A). To show the existence of a measurable selection of minimizers, define the multi-valued map Γ : By the measurability of µ ξ (·, O) and the joint measurability of the functional, we see that the graph of the this multi-function is F ⊗ B(W 1,1 0 (O) m )-measurable. Due to the completeness of (Ω, F , P) and the separability and completeness of W 1,1 0 (O) m , we can now apply Aumann's measurable selection theorem [18, Theorem 6.10] and conclude the proof.

Appendix C. Approximation results in the vectorial case
In this section we extend [16, Proposition 2.6, Chapter X] to the vectorial setting, a result that has already been used in the literature, but the proof in [16] is only valid for scalar functions. For (probably) technical reasons, we need to assume in addition the analogue of (A4) in the homogeneous setting or the lower bound W (·) ≥ | · | p for some p > d − 1. Let us emphasize that in the literature the result was used under the stronger assumption p > d.
Lemma C.1. Let W : R m×d → [0, +∞) be convex and D ⊂ R d be a bounded, open set with Lipschitz boundary. Assume that one of the following assumptions holds true: (1) there exists C < +∞ such that for all ξ ∈ R m×d and all ξ ∈ R m×d with e T j (ξ − ξ) ∈ {0, e T j ξ} for all 1 ≤ j ≤ m it holds that W ( ξ) ≤ C(W (ξ) + 1).
Then for any u ∈ W 1,1 0 (D) m there exists a sequence u n ∈ C ∞ c (D) m such that u n → u strongly in Proof. It suffices to treat the case when D W (∇u) dx < +∞, since otherwise the statement reduces to well-known density results in W 1,1 0 . We only prove that it is sufficient to consider u with compact support in D. Once, this is established the statement follows by a routine regularization by convolution (see [16] or [28,Lemma 3.6]).
We first treat the more involved case (C.2) and explain the necessary modifications/simplifications for (C.1) afterwards. Let u ∈ W 1,p 0 (D) m be given. First extend u to be zero outside D, so that u ∈ W 1,p (R d ) m . For every x ∈ D we consider a ball B rx (x) ⊂⊂ D, while for x ∈ ∂D the Lipschitz regularity of ∂D implies that (up to an Euclidean motion) there exists a cylinder C x = B d− Choose then r x < min{r ′ x , h x } such that B rx (x) ⊂⊂ C x and such that the Lipschitz-constant L x of ψ x satisfies 0 < 2L x r x ≤ h x + inf |y ′ |≤r ′ x ψ x (y ′ ), (C.5) which is possible due to (C.4). Due to the compactness of D, we find a finite family of above balls B i = B rx i (x i ) (1 ≤ i ≤ N ) that cover D. Let us emphasize that these balls will be fixed throughout the rest of the proof, so we omit the dependence on the radii r xi or the number N of certain quantities. For interior points x i ∈ D, we define z i = x i , while for points x i ∈ ∂D we choose z i ∈ R d such that in the local coordinates we have z i = (0, −h xi ) (i.e., at the bottom of the local graph representation). Now let 0 < ρ k < 1 be such that lim k ρ k = 1 and for any 1 ≤ i ≤ N we define u k,i (x) = ρ k u(z i + 1 ρ k (x − z i )). Since ρ k → 1, it holds that u k,i → u in W 1,p (R d ) m as k → +∞. Next, let (φ i ) N i=0 be a smooth partition of unity subordinated to the cover {R d \ D, (B i ) N i=1 } of R d (note that we work on the 'manifold' R d and therefore supp(φ i ) is compactly contained in B i and φ 0 vanishes on D). We build an ad hoc Lipschitz partition of unity as follows: choose δ 0 > 0 such that for each 1 ≤ i ≤ N we have supp(φ i ) ⊂⊂ B (1−δ0)rx i (x i ) and then use Lemma 4.10 (with ρ = 1 and δ = δ 0 that are considered as fixed for the rest of the proof) for the finite family of functions {u k,j − u} N j=1 ⊂ W 1,p (B i , R m ) to obtain η k,i ∈ W 1,∞ 0 (B i ) such that 0 ≤ η k,i ≤ 1, η k,i = 1 on supp(ϕ i ), ∇η k,i L ∞ (Bi) ≤ C. and for all 1 ≤ j ≤ N ∇η k,i ⊗ (u k,j − u) L ∞ (Bi) ≤ C u k,j − u W 1,p (Bi) k→+∞ −→ 0. (C. 6) Similar to [20,Theorem 2], we now define the Lipschitz partition of unity as follows: for k ∈ N we set Here the denominator is always ≥ 1 since η k,j = 1 on supp(φ j ). Therefore the gradient of ϕ k,i (1 ≤ i ≤ N ) satisfies for any vector a ∈ R m the pointwise estimate |∇ϕ k,i ⊗a| ≤ |∇η k,i ⊗ a| We then set for which the gradient on D (here ϕ k,0 vanishes) can be expressed as As a convex combination, we still have that u k → u in L 1 (D) m , while for gradients the estimate (C.7) and the uniform gradient bound for η k,i imply that |u k,i − u| + |∇u k,i − ∇u| dx → 0 as k → +∞ .
Finally, due to the convexity of W , for any t ∈ (0, 1) it holds that Let us discuss the two right-hand side integrals separately. By a change of variables we have (χ D ϕ k,i )(z i + ρ k (y − z i ))W (∇u(y)) dy.
Due to (C.7) and the uniform gradient bound for η k,i , we know that the functions ϕ k,i are equi-Lipschitz with respect to k and i, so that for z i + ρ k (y − z i ) ∈ D we have the estimate Inserting the bound into the previous equality, we deduce that χ D (z i + ρ k (y − z i ))W (∇u(y)) dy.
The pre-factor of the right-hand side integral converges to 1, while for the integrand we can use the dominated convergence theorem to infer that max 1≤i≤N χ D (z i + ρ k (· − z i ))W (∇u) → χ D W (∇u) in It remains to show that the second integral in (C.8) is negligible. Applying (C.7) with a = u k,i (x) − u(x) and inserting (C.6) in the corresponding right-hand side, due to the fact that ∇φ 0 = 0 on D we find that The right-hand side converges to 0 as k → +∞, so we find a constant c = c t,D,u such that pointwise on D where we used that W is bounded on bounded sets by continuity. The dominated convergence theorem yields that In total, starting from (C.8) we deduce the estimate lim sup Since tu k → tu in W 1,1 (D) m , we can use a diagonal argument with respect to k and t to find a sequence u k such that u k → u in W 1,1 (D) m and, combined with Fatou's lemma, Moreover, it holds that supp( u k ) = supp(u k ′ ) for some k ′ ∈ N, so it remains to show that all u k have compact support in D. Let δ k ≪ 1 and consider x ∈ D such that dist(x, ∂D) ≤ δ k and a ball B i with x ∈ B i . We show that ϕ k,i (x)u k,i (x) = 0 for such i and δ k small enough, which then implies that u k (x) = 0, so that u k has compact support. Since there are only finitely many sets in the covering and 'interior' balls are compactly contained in D, for δ k small enough we know from our construction of the covering and (C.3) that, up to a Euclidean motion, with the corresponding cylinders C i = C xi and radii r ′ i = r ′ xi we can write D ∩ C i = {(y ′ , y d ) ∈ C i : y d < ψ i (y ′ )} for a Lipschitz function ψ i : Since supp(ϕ k,i ) = supp(η k,i ) ⊂ B i ⊂⊂ C i , there exists 0 < η < min i r i such that ϕ k,i (x) = 0 whenever dist(x, ∂C i ) ≤ η. Hence we can assume that dist(x, ∂C i ) > η. We will show that for k large enough (independent of x = (x ′ , x d )) we have z i + 1 ρ k (x − z i ) ∈ C i \ D, which then implies that u k,i (x) = 0. Since x ∈ C i and dist(x, ∂C i ) > η and the z i are fixed, it follows that for ρ k is sufficiently close to 1 we have z i + 1 ρ k (x − z i ) ∈ C i . Hence, in order to show that this point does not belong to D, it suffices to show that in the local coordinates (where z i = (0, −h i )) we have Let d x ∈ ∂D be such that |x − d x | = dist(x, ∂D). Note that d x ∈ C i (otherwise the line from x to d x intersects ∂C i at a distance less than δ k ≪ η), so that we can write d x = (y ′ , ψ i (y ′ )) for some y ′ ∈ B d−1 To show (C.9), let L i be the Lipschitz constant of ψ i . Then we can estimate where we used (C.5) in the last estimate. Choosing δ k ≪ 1 ρ k − 1 , it clearly follows that the right-hand side is negative and we conclude the proof for the case (C.2).
If we assume (C.1) instead, we can show that the analysis reduces to the case that u ∈ L ∞ (D) m . Indeed, consider for s ≫ 1 the componentwise truncation T s u at level s (cf. (4.42)). Then T s u ∈ W 1,1 0 (D) m and T s u → u in W 1,1 (D) m as s → +∞ by the dominated convergence theorem (both for the function and the gradient). Moreover, by the non-negativity of W we have and by a diagonal argument it suffices to show the claim for u ∈ L ∞ (D) m . In this case we can argue is in the first part (directly working with the k-independent partition of unity {φ i } N i=1 ). The only difference occurs when proving that the second integral in (C.8) is negligible. Since now u ∈ L ∞ (R d ) m and u k,i L ∞ (R d ) ≤ u L ∞ (R d ) , we find a constant c = c t,D,u such that Email address: mathias.schaeffner@mathematik.uni-halle.de