Stochastic homogenization of maximal monotone relations and applications

We study the homogenization of a stationary random maximal monotone operator on a probability space equipped with an ergodic dynamical system. The proof relies on Fitzpatrick's variational formulation of monotone relations, on Visintin's scale integration/disintegration theory and on Tartar-Murat's compensated compactness. We provide applications to systems of PDEs with random coefficients arising in electromagnetism and in nonlinear elasticity.


Introduction
Stochastic homogenization is a broadly studied subject, starting from the seminal papers by Kozlov [11] and Papanicolaou-Varadhan [18], who studied boundary value problems for second order linear PDEs. We prove here an abstract homogenization result for the graph of a random maximal monotone operator v(x, ω) ∈ α ε (x, ω, u(x, ω)), where x ∈ R n and ω is a parameter in a probability space Ω. In the spirit of [10,Chapter 7], the random operator α ε is obtained from a stationary operator α via an ergodic dynamical system T x : Ω → Ω α ε (x, ω, ·) := α T x/ε ω, · . (1.1) The aim of this paper is to extend existing results where α is the subdifferential of a convex function [24,21] to general maximal monotone operators and to provide a simple proof based on Tartar's oscillating test function method. The crucial ingredient in our analysis is the scale integration/disintegration theory introduced by Visintin [25]. Moreover, relying on Fitzpatrick's variational formulation of monotone graphs [8], which perfectly suits the scale integration/disintegration setting [26], in the proof we can directly exploit the maximal monotonicity, without turning to (stochastic) two-scale convergence [2,20,9], Γ-convergence [6,5], G-convergence [16,17], nor epigraph convergence [13,14]. An advantage of our approach is that we don't need to assume an additional compact metric space structure on the probability space Ω. Moreover, the effective relation is obtained directly, i.e. without an intermediate two-scale problem, which often needs to be studied separately. We also obtain the existence of the oscillating test functions as a byproduct of scale disintegration, without having to study the auxiliary problem (see, e.g., [17,Section 3.2]. The outline of the proof is the following: Let X be a separable and reflexive Banach space, with dual X ′ , let A n : X → X ′ be a sequence of monotone operators, and let (x n , y n ) ∈ X ×X ′ be a sequence of points on the graphs of A n , i.e., such that y n = A n x n for all n ∈ N. Assuming that (x n , y n ) ⇀ (x, y) in X × X ′ and that we already know the limit maximal monotone operator A : X → X ′ , a classical question of functional analysis is: Under which assumptions can we conclude that y = Ax? A classical answer (see, e.g., [3]) is: If we can produce an auxiliary sequence of points on the graph of A n , and we know that they converge to a point on the graph of A, that is, if there exist (ξ n , η n ) ∈ X × X ′ such that η n = A n ξ n , (ξ n , η n ) ⇀ (ξ, η), and η = Aξ, (1.2) then, denoting by y, x the duality between x ∈ X and y ∈ X ′ , by monotonicity of A n y n − η n , x n − ξ n ≥ 0.
In order to pass to the limit as n → ∞, since the duality of weak converging sequences in general does not converge to the duality of the limit, we need the additional hypothesis lim sup n→∞ g n , f n ≤ g, f ∀ (f n , g n ) ⇀ (f, g) in X × X ′ , (1.3) which, together with the weak convergence of (x n , y n ) and (ξ n , η n ), yields y − η, x − ξ ≥ 0.
By maximal monotonicity of A, if the last inequality is satisfied for all (ξ, η) such that η = Aξ, then we can conclude that (x, y) is a point of the graph of A, i.e., y = Ax. Summarizing, this procedure is based on (1) Existence and weak compactness of solutions (x n , y n ) such that y n = A n x n and (x n , y n ) ⇀ (x, y); (2) A condition for the convergence of the duality product (1.3); (3) Existence of a recovery sequence (1.2) for all points in the limit graph.
The first step depends on the well-posedness of the application; the second step is ensured, e.g., by compensated compactness (in the sense of Murat-Tartar [15,23]), and, like the first one, it depends on the character of the differential operators that appear in the application, rather than on the homogenization procedure. In the present paper we focus on the third step: in the context of stochastic homogenization, we prove that the scale integration/disintegration idea introduced by Visintin [25], combined with Birkhoff's ergodic theorem (Theorem 2.5) yields the desired recovery sequence. We obtain an explicit formula for the limit operator A through the scale integration/disintegration procedure with Fitzpatrick's variational formulation. With the notation introduced in (1.1), the outline of this procedure is the following: where a) the random operator α(ω) is represented through a variational inequality involving Fitzpatrick's representation f (ω); b) the representation is "scale integrated" to a ωindependent effective f 0 ; c) a maximal monotone operator α 0 is associated to f 0 . In Theorem 3.10 we prove that α 0 is the correct homogenization of α ε .
In Section 2.1 we review the properties of maximal monotone operators and their variational formulation due to Fitzpatrick. In Section 2.2 we recall the basis of ergodic theory that we need in order to state our first main tool: Birkhoff's Ergodic Theorem. Section 3 is devoted to the translation to the stochastic setting of Visintin's scale integration-disintegration theory, which paves the way to our main result, Theorem 3.10. The applications we provide in the last section are: Ohmic electric conduction with Hall effect (Section 4.1), and nonlinear elasticity, (Section 4.2).

Notation and preliminaries
We use the notation a · b for the standard scalar product for vectors in R n . The arrows ⇀ and * ⇀ denote weak and weak * convergence, respectively. As usual, D(D) stands for the space of C ∞ -functions with compact support in D ⊂ R n ; its dual is denoted by D ′ (D).
2.1. Maximal monotone operators. In this section we summarize the variational representation of maximal monotone operators introduced in [8]. Further details and proofs of the statements can be found, e.g., in [27]. Let B be a reflexive, separable and real Banach space; we denote by B ′ its dual, by P(B ′ ) the power set of B ′ , and by y, x the duality between x ∈ B and y ∈ B ′ . Let α : B → P(B ′ ) be a set-valued operator and let be its graph. (We make use of the two equivalent notations y ∈ α(x) or (x, y) ∈ G α .) The operator α is said to be monotone if and strictly monotone if there is θ > 0 such that We denote by α −1 the inverse operator in the sense of graphs, that is The monotone operator α is said to be maximal monotone if the reverse implication of (2.1) is fulfilled, namely if An operator α is maximal monotone if and only if α −1 is maximal monotone. For any operator α : B → P(B ′ ), which is not identically ∅, we introduce the Fitzpatrick function of α as the function f α : As a supremum of a family of linear functions, the Fitzpatrick function f α is convex and lower semicontinuous. Moreover (see [8]) while α is maximal monotone if and only if In the case B = B ′ = R, it is easy to compute some simple examples of Fitzpatrick function of a monotone operator: 1. Let α(x) := ax + b, with a > 0, b ∈ R. A straightforward computation shows that

Let
and in both cases f α coincides with g(x, y) = xy exactly on the graph of α. We define F = F(B) to be the class of all proper, convex and lower semicontinuous functions f : We call F(B) the class of representative functions. The above discussion shows that given a monotone operator α, one can construct its representative function in F(B), and viceversa, given a function f ∈ F(B), we define the operator represented by f , which we denote α f , by: A crucial point is whether α f is monotone (or maximal monotone, see also [26,Theorem 2.3]). Proof. (i) If G α f is empty or reduced to a single element, then the statement is trivially satisfied. Let x 1 , x 2 ∈ B, y i ∈ α f (x i ), and assume, by contradiction, that y 2 −y 1 , x 2 −x 1 < 0. Define P i := (x i , y i ) ∈ B × B ′ and g(x, y) := y, x . We compute Since f ≥ g and f (x i , y i ) = g(x i , y i ), the last inequality implies which contradicts the convexity of f . (ii) Maximal monotone operators are representable by Lemma 2.1. To see that the inclusion is strict, assume that α f is maximal monotone. Let (x 0 , y 0 ) ∈ G α f and fix c ∈ R such that c > f (x 0 , y 0 ). The function and thus α h is not maximal.
Thus, Fitzpatrick's representative function f α generalizes ϕ + ϕ * to maximal monotone operators which are not subdifferentials. Remark that, even if α = ∂ϕ, in general f α = ϕ + ϕ * : for example, let α(x) := x in R, then We need to introduce also parameter-dependent operators. For any measurable space X we say that a set-valued mapping g : Let B(B) be the σ-algebra of the Borel subsets of the separable and reflexive Banach space B, let (Ω, A, µ) be a probability space equipped with the σ-algebra A and the probability measure µ. We define a random (maximal) monotone operator as α : α(x, ω) is closed for any x ∈ B and for a.e. ω ∈ Ω, If α fulfills (2.4) and (2.5) then for any A-measurable mapping v : Ω → B, the multivalued mapping ω → α(v(ω), ω) is closed-valued and measurable. We denote by F(Ω; B) the set of all measurable representative functions f : , ω) is convex and lower semicontinuous for a.e. ω ∈ Ω, (c) f (x, y, ω) ≥ y, x for all (x, y) ∈ B × B ′ and for a.e. ω ∈ Ω.

Stochastic analysis.
In this subsection we review the basic notions and results of stochastic analysis that we need in Section 3. For more details see [10,Chapter 7]. Let (Ω, A, µ) be a probability space, where A is a σ-algebra of subsets of Ω and µ is a probability measure on Ω. Let n ∈ N with n ≥ 1. A n-dimensional dynamical system T on Ω is a family of mappings T x : Ω → Ω, with x ∈ R n , such that (a) T 0 is the identity and T x+y = T x T y for any x, y ∈ R n ; (b) for every x ∈ R n and every set E ∈ A we have T x E ∈ A and is measurable.
Given a n-dimensional dynamical system T on Ω, a measurable function f defined on Ω is in Ω, for each x ∈ R n . A dynamical system is said to be ergodic if the only invariant functions are the constants. The expected value of a random variable f : Ω → R n is defined as In the context of stochastic homogenization, it is useful to provide an orthogonal decomposition of L 2 (Ω) into functions, the realizations of which are vortex-free and divergence-free, in the sense of distributions (see, e.g., [10,Section 7]). For p ∈ [1, +∞[, Peter-Weyl's decomposition theorem [19] can be generalized to a relation of orthogonality between subspaces of L p (Ω) and Next we consider a vector field on (Ω, A, µ). We say that f ∈ L p (Ω; R n ) is potential if µalmost all its realizations x → f (T x ω) are potential. We denote by L p pot (Ω; R n ) the space of all potential f ∈ L p (Ω; R n ). In the same way, f ∈ L p (Ω; R n ) is said to be solenoidal if µ-almost all its realizations x → f (T x ω) are solenoidal and we denote by L p sol (Ω; R n ) the space of all solenoidal f ∈ L p (Ω; R n ). In the following Lemma we collect the main properties of potential and solenoidal L p spaces (see [10,Section 15]).
and the relations in the sense of duality between the spaces L p (Ω) and L q (Ω).
One of the most important results regarding stochastic homogenization is Birkhoff's Ergodic Theorem. We report the statement given in [10, Theorem 7.2].
Since this holds for every measurable bounded set In what follows, the dynamical system T x is assumed to be ergodic and K ⊂ R n is bounded, measurable and |K| > 0.
Remark 3.1. While most of this subsection's statements are Visintin's results written in a different notation, some others contain a small, but original contribution. Namely: Lemma 3.2 can be found in [26,Lemma 4.1], where the assumption of boundedness for K is used to obtain the lower semicontinuity of the inf function. Since we prefer not to impose this condition, we independently proved the lower semicontinuity part, making use of the coercivity of g. Lemma 3.3 and Proposition 3.4 were given for granted in [26], but we decided to write a proof for sake of clarity. Lemma 3.5 and Lemma 3.6, are essentially [26,Theorem 4.3] and [26,Theorem 4.4], cast in the framework of stochastic homogenization in the probability space (Ω, A, µ), instead of periodic homogenization on the n-dimensional torus. Theorem 3.7 collects other results of [26]. Lemma 3.8 is an original remark.
Let f (·, ·, ω) : R n × R n → R ∪ {+∞} be the Fitzpatrick representation of the operator α(·, ω). We assume the following coercivity condition on f : there exist c > 0 and k ∈ L 1 (Ω) such that for any ξ, η ∈ R n , for any ω ∈ Ω it holds f (ξ, η, ω) ≥ c (|ξ| p + |η| q ) + k(ω). (3.1) We define the homogenized representation f 0 : Lemma 3.2. Let X be a reflexive Banach space, let K be a weakly closed subset of a reflexive Banach space. Let the function g : X × K → R ∪ {+∞} be weakly lower semicontinuous and bounded from below. If g is coercive, e.g. in the sense that for all M > 0 the set is weakly lower semicontinuous and coercive. Moreover, if g and K are convex then h is convex. If ℓ = +∞, then (3.3) is trivially satisfied. On the other hand, since g is bounded from below, then ℓ > −∞, and we can assume that ℓ ∈ R. By definition of inferior limit, there exists a subsequence of {x j } (not relabeled), such that lim j→∞ h(x j ) = ℓ. Up to extracting another subsequence, we can also assume that h(x j ) ≤ 2ℓ for all j ∈ N. Let ε > 0 be fixed, by definition of infimum, for all j ∈ N, there exists y j ∈ K such that By the coercivity assumption on g, we deduce that y j is bounded, we can therefore extract a subsequence {y j k } ⊂ K such that y j k ⇀ y. Since K is weakly closed, then y ∈ K. We can now pass to the inferior limit in (3.4), using the lower semicontinuity of g By arbitrariness of ε > 0, this proves the weak lower semicontinuity of h. Assume now that K is convex. Take λ ∈ [0, 1], x 1 , x 2 ∈ X and y 1 , y 2 ∈ K. By convexity of g Passing to the infimum with respect to y 1 , y 2 ∈ K we conclude Regarding the coercivity of h, denote Since g is coercive, A M +ε is bounded and thus B M is bounded, i.e., h is coercive.
In the proof of Proposition 3.4 we need the following estimate Proof. Consider the operator On the image space L p (Ω; R n ) × L p (Ω; R n ), choose the equivalent norm Clearly, Φ is linear and continuous. Therefore, there exists C > 0 such that Apply now the last inequality to u(ω) = ξ +ũ(ω), with E(ũ) = 0: Proposition 3.4. For all (ξ, η) ∈ R n × R n there exists a couple ( u, v) ∈ V p sol (Ω; R n ) × V q pot (Ω; R n ) such that the infimum on the right-hand side of (3.2) is attained. Moreover, f 0 ∈ F(R n ). In particular, it holds f 0 (ξ, η) ≥ ξ · η ∀(ξ, η) ∈ R n × R n . (3.6) Proof. Let K := V p sol (Ω; R n ) × V q pot (Ω; R n ). Then K is weakly closed in L p (Ω; R n ) × L q (Ω; R n ) since it is closed and convex. Let ξ, η ∈ R n be fixed, for any (u, v) ∈ K let We prove that the problem inf K F ξ,η has a solution applying the direct method of the Calculus of Variations. First, by (3.1), inf K F ξ,η > −∞. Then, if (u h , v h ) ∈ K is a minimizing sequence for F ξ,η , by the coercivity assumption (3.1), up to subsequences, (u h , v h ) ⇀ (u, v) weakly in L p (Ω; R n ) × L q (Ω; R n ), therefore (u, v) ∈ K, since K is weakly closed. Finally, F ξ,η is L p × L qweakly lower semicontinuous since f (·, ·, ω) is convex, lower semicontinuous, and bounded from below by an integrable function (3.1), therefore This concludes the first part of the statement. We now want to show that f 0 ∈ F(R n ). Owing to (3.1) and Lemma 3.3, for all ξ, η, (u, v) ∈ R n × R n × K, there exists a constant C > 0 such that . Thus, for any M ≥ 0, the set is bounded in R n × R n × L p (Ω; R n ) × L q (Ω; R n ). We are therefore in a position to apply Lemma 3.2 and to conclude that f 0 is convex and lower semicontinuous. Furthermore, let ( u, v) ∈ K be a minimizer of F ξ,η , using (2.9) which yields the conclusion.
How the properties of α and f reflect on α 0 and f 0 was thoroughly studied in [26]: • f represents a maximal monotone operator for µ-a.e. ω ∈ Ω then f 0 represents a (proper) maximal monotone operator [26,Theorem 5.3]. Moreover, if f is strictly convex, then In order to obtain strict monotonicity of α 0 and α −1 0 , by the next Lemma we provide an alternative to strict convexity of the Fitzpatrick function.
Lemma 3.8. Let α(·, ω) : B → B ′ be maximal and strictly monotone, and assume that its Fitzpatrick representation f is coercive, in the sense of (3.1). Then its scale integration α 0 is strictly monotone.

Main result.
Let D ⊂ R n be a Lipschitz and bounded domain with |D| > 0. We recall the following classical result. In addition, assume that Then v n · u n * ⇀ v · u in D ′ (D).
We are now ready to prove our main result concerning the stochastic homogenization of a maximal monotone relation. Theorem 3.10. Let (Ω, A, µ) be a probability space with an n-dimensional ergodic dynamical system T x : Ω → Ω, x ∈ R n . Let D ⊂ R n be a bounded domain, let p ∈]1, +∞[ and q = p/(p − 1). Let α : R n × Ω → P(R n ) be a closed-valued, measurable, maximal monotone random operator, in the sense of (2.4)-(2.6).

4.1.
The Ohm-Hall model. In this paragraph we address the homogenization problem for the Ohm-Hall model for an electric conductor. For further information about the Ohm-Hall effect we refer the reader to [1, pp. 11-15], [12,Section 22]. We consider a non homogeneous electric conductor, that occupies a bounded Lipschitz domain D ⊂ R 3 and is subjected to a magnetic field. We assume that the electric field E and the current density J fulfill the constitutive law where α(·, x) : R 3 → R 3 is a (single-valued) maximal monotone mapping for a.e. x ∈ D, B is the magnetic induction field, h is the (material dependent) Hall coefficient, and E a is an applied electromotive force. We couple (4.1) with the Faraday law and with the stationary law of charge-conservation: Following [26], we assume that h, B, E a are given, we deal with the stationary system, thus dropping the time variable, and we define the maximal monotone operator β(·, x) : R 3 → R 3 and the vector field g : A single-valued parameter-dependent operator β is strictly monotone uniformly in x, if there exists θ > 0 such that for a.e. x ∈ D The following existence and uniqueness result is a classical consequence of the maximal monotonicity of α (see, e.g., [22,26]). Theorem 4.1. Let D ⊂ R 3 be a bounded Lipschitz domain. Let {β(·, x)} x∈D be a family of single-valued maximal monotone operators on R 3 . Assume moreover that there exist constants a, c > 0 and b ≥ 0 such that for a.e. x ∈ D, ∀v ∈ R 3 Let g ∈ L 2 (D; R 3 ) be given, such that ∇ · g = 0, distributionally. Then, there exists E, J ∈ L 2 (D; R 3 ) such that and, denoting by ν the outward unit normal to ∂D, Moreover, if β is strictly monotone uniformly in x ∈ D, then the field J is uniquely determined, while if β −1 is strictly monotone uniformly in x ∈ D, then the field E is uniquely determined.
We couple (4.25) with the conservation of linear momentum: where ρ is the density and f represents the external forces. For sake of simplicity, we choose to deal with the stationary system only and we set ρ∂ 2 t u = 0. The following existence and uniqueness result is a classical consequence of the maximal monotonicity of β (see, e.g., [7,22]).  Moreover, if β is strictly monotone uniformly in x ∈ D, then u is uniquely determined, while if β −1 is strictly monotone uniformly in x ∈ D, then σ is uniquely determined.
Finally, the strict monotonicity of the limit operators β 0 and β −1 0 yields uniqueness and therefore independence of ω for the solution (u, σ).