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Research article Topical Sections

Effect of acid leaching conditions on impurity removal from silicon doped by magnesium

  • Received: 28 April 2017 Accepted: 11 July 2017 Published: 18 July 2017
  • The effect of magnesium addition into a commercial silicon and its leaching refining behavior is studied for producing solar grade silicon feedstock. Two different levels of Mg is added into a commercial silicon and the leaching of the produced alloys by 10% HCl solution at 60 ℃ for different durations is performed. It is shown that the microstructure of the alloy and in particular the distribution of eutectic phases is dependent on the amount of the added Mg. Moreover, the metallic impurities in silicon such as Fe, Al, Ca and Ti are mainly forming silicide particles with different compositions. These silicides are physically more detached from the primary silicon grains and their removal through chemical and physical separation in leaching is better for higher Mg additions. It is observed that the leaching is more effective for the purification of smaller silicon particles produced from each Mg-doped silicon alloy. It is shown that acid leaching by the applied method is effective to reach more than 70% of phosphorous removal. It is also shown that the purity of silicon is dependent on the total Mg removal and effectiveness of leaching on removing the Mg2Si phase.

    Citation: Stine Espelien, Jafar Safarian. Effect of acid leaching conditions on impurity removal from silicon doped by magnesium[J]. AIMS Energy, 2017, 5(4): 636-651. doi: 10.3934/energy.2017.4.636

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  • The effect of magnesium addition into a commercial silicon and its leaching refining behavior is studied for producing solar grade silicon feedstock. Two different levels of Mg is added into a commercial silicon and the leaching of the produced alloys by 10% HCl solution at 60 ℃ for different durations is performed. It is shown that the microstructure of the alloy and in particular the distribution of eutectic phases is dependent on the amount of the added Mg. Moreover, the metallic impurities in silicon such as Fe, Al, Ca and Ti are mainly forming silicide particles with different compositions. These silicides are physically more detached from the primary silicon grains and their removal through chemical and physical separation in leaching is better for higher Mg additions. It is observed that the leaching is more effective for the purification of smaller silicon particles produced from each Mg-doped silicon alloy. It is shown that acid leaching by the applied method is effective to reach more than 70% of phosphorous removal. It is also shown that the purity of silicon is dependent on the total Mg removal and effectiveness of leaching on removing the Mg2Si phase.


    Stochastic homogenization is a subject broadly studied starting from '80 since 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)αε(x,ω,u(x,ω)),

    where xRn and ω is a parameter in a probability space Ω. In the spirit of [10,Chapter 7], the random operator αε is obtained from a stationary operator α via an ergodic dynamical system Tx:ΩΩ

    αε(x,ω,):=α(Tx/εω,). (1)

    The aim of this paper is to extend existing results where α is the subdifferential of a convex function [21,24] 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 [2,9,20], Γ-convergence [5,6], G-convergence [16,17], nor epigraph convergence [13,14]. An advantage of our approach is that we do not 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 An:XX be a sequence of monotone operators, and for every nN let (xn,yn)X×X be a point in the graph of An, i.e., such that yn=Anxn for all nN. Assuming that (xn,yn)(x,y) in X×X and that we already know the limit maximal monotone operator A:XX, 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 An, and we know that they converge to a point on the graph of A, that is, if there exist

    (ξn,ηn)X×X such that ηn=Anξn(ξn,ηn)(ξ,η) and η=Aξ, (2)

    then, denoting by y,x the duality pairing between xX and yX, by monotonicity of An

    ynηn,xnξn0.

    In order to pass to the limit as n, since the duality pairing of weakly converging sequences in general does not converge to the duality pairing of the limit, we need the additional hypothesis

    lim supngn,fng,f(fn,gn)(f,g) in X×X, (3)

    which, together with the weak convergence of (xn,yn) and (ξn,ηn), yields

    yη,xξ0.

    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 (xn,yn) such that yn=Anxn and (xn,yn)(x,y);

    2. A condition for the convergence of the duality pairing (3);

    3. Existence of a recovery sequence (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 2.4) 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), the outline of this procedure is the following:

    α a) f b) f0 c) α0,

    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 ω-independent effective f0; c) a maximal monotone operator α0 is associated to f0. In Theorem 3.8 we prove that α0 is the homogenised limit of αε.

    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 3.8. The applications we provide in the last section are: Ohmic electric conduction with Hall effect (Section 4.1), and nonlinear elasticity, (Section 4.2).

    We use the notation ab for the standard scalar product for vectors in Rn. The arrows and denote weak and weak convergence, respectively. As usual, D(D) stands for the space of C-functions with compact support in DRn; its dual is denoted by D(D).

    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 pairing between xB and yB. Let α:BP(B) be a set-valued operator and let

    Gα:={(x,y)B×B:yα(x)}

    be its graph. (We will equivalently write yα(x) or (x,y)Gα.) The operator α is said to be monotone if

    (x,y)Gαyy0,xx00,(x0,y0)Gα (4)

    and strictly monotone if there is θ>0 such that

    (x,y)Gαyy0,xx0θ (5)

    We denote by \alpha^{-1} the inverse operator in the sense of graphs, that is

    x \in \alpha^{-1}(y)\;\;\;\;\; \Leftrightarrow \;\;\;\;\;y\in \alpha(x).

    The monotone operator \alpha is said to be maximal monotone if the reverse implication of (4) is also fulfilled, namely if

    \langle y-y_0, x-x_0\rangle \ge 0 \;\;\;\;\; \forall (x_0, y_0)\in \mathcal{G}_\alpha \;\;\;\;\; \Leftrightarrow \;\;\;\;\; (x, y) \in \mathcal{G}_\alpha.

    An operator \alpha is maximal monotone if and only if \alpha^{-1} is maximal monotone. For any operator \alpha \colon B \rightarrow \mathcal{P}(B'), which is not identically \emptyset, we introduce the Fitzpatrick function of \alpha as the function f_\alpha \colon B \times B' \to {\mathbb{R}} defined by

    \begin{align*} f_{\alpha}(x, y):& = \langle y, x \rangle + \sup\{\langle y-y_0, x_0-x \rangle :(x_0, y_0)\in\mathcal{G}_\alpha\}\\ & = \sup\left\{\langle y, x_0 \rangle + \langle y_0, x \rangle - \langle y_0, x_0 \rangle :(x_0, y_0)\in\mathcal{G}_\alpha\right\}. \end{align*}

    As a supremum of a family of linear functions, the Fitzpatrick function f_\alpha is convex and lower semicontinuous. Moreover, the following lemma holds true (see [8]).

    Lemma 2.1. An operator \alpha is monotone if and only if

    (x, y) \in \mathcal{G}_\alpha \;\;\;\;\;\Rightarrow \;\;\;\;\; f_{\alpha}(x, y) = \langle y, x \rangle,

    while \alpha is maximal monotone if and only if

    \left\{ \begin{array}{ll} f_{\alpha}(x, y) \ge \langle y, x \rangle\ \;\;\;\;\;\forall (x, y)\in B\times B'\\ f_{\alpha}(x, y) = \langle y, x \rangle \;\;\;\;\;\iff (x, y) \in \mathcal{G}_\alpha. \end{array} \right.

    In the case B = B' = {\mathbb{R}}, it is easy to compute some simple examples of Fitzpatrick function of a monotone operator:

    1. Let \alpha(x): = ax+b, with a>0, b\in {\mathbb{R}}. A straightforward computation shows that

    f_\alpha(x, y) = \frac{(y-b+ax)^2}{4a} +bx.

    2. Let

    \alpha(x) = \left\{ \begin{array}{cl} 1&\text{if }x > 0, \\ {[0, 1]}&\text{if }x = 0, \\ -1&\text{if }x < 0. \end{array} \right.

    Then

    f_\alpha(x, y) = \left\{ \begin{array}{cl} |x|&\;\;\;\;\;\text{if } |y| \leq 1, \\ +\infty&\;\;\;\;\;\text{if }|y| > 1. \end{array} \right.

    and in both cases f_\alpha coincides with g(x, y) = xy exactly on the graph of \alpha.

    We define \mathcal{F} = \mathcal{F}(B) to be the class of all proper, convex and lower semicontinuous functions f \colon B\times B'\rightarrow \mathbb{R}\cup \{+\infty \} such that

    f(x, y)\ge \langle y, x \rangle \;\;\;\;\;\forall (x, y)\in B\times B'.

    We call \mathcal F(B) the class of representative functions. The above discussion shows that given a monotone operator \alpha, one can construct its representative function in \mathcal F(B), and viceversa, given a function f\in \mathcal F(B), we define the operator represented by f, which we denote \alpha_f, by:

    \label{def:graph} (x, y) \in \mathcal G_{\alpha_f} \Leftrightarrow f(x, y) = \langle y, x \rangle. (6)

    A crucial point is whether \alpha_f is monotone (or maximal monotone, see also [26,Theorem 2.3]).

    Lemma 2.2. Let f\in \mathcal F(B), then

    (i) the operator \alpha_f defined by (6) is monotone;

    (ii) the class of maximal monotone operators is strictly contained in the class of operators representable by functions in \mathcal F(B).

    Proof. (ⅰ) If \mathcal G_{\alpha_f} is empty or reduced to a single element, then the statement is trivially satisfied. Let x_1, x_2 \in B, y_i \in \alpha_f(x_i) for i = 1, 2, and assume, by contradiction, that \langle y_2 - y_1, x_2 -x_1 \rangle < 0. Define P_i: = (x_i, y_i)\in B \times B' and g(x, y): = \langle y, x \rangle. We compute

    \begin{align*} g\left( \frac{P_1+P_2}{2}\right) - \frac{g(P_1) +g(P_2)}{2} & = \frac14 \big( \langle y_1 +y_2, x_1+x_2\rangle \big) - \frac12 \big( \langle y_1, x_1\rangle + \langle y_2, x_2\rangle \big) \\ & = \frac14 \big( \langle y_1, x_2\rangle + \langle y_2, x_1\rangle - \langle y_1, x_1\rangle -\langle y_2, x_2\rangle\big) \\ & = -\frac14 \big( \langle y_2 -y_1, x_2 - x_1\rangle \big) > 0. \end{align*}

    Since f \geq g and f(x_i, y_i) = g(x_i, y_i) for i = 1, 2, the last inequality implies

    f\left( \frac{P_1+P_2}{2}\right) > \frac{f(P_1) +f(P_2)}{2},

    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 \alpha_f is maximal monotone. Let (x_0, y_0)\in \mathcal G_{\alpha_f}; since f is proper there exists c\in {\mathbb{R}} such that c > f(x_0, y_0). The function

    h(x, y) = \max\{c, f(x, y)\}

    clearly belongs to \mathcal F(B), but \mathcal G_{\alpha_h} is strictly contained in \mathcal G_{\alpha_f}. Indeed by the maximality of \alpha_f

    h(x_0, y_0) \geq c > f(x_0, y_0) = \langle y_0, x_0 \rangle,

    and thus \alpha_h is not maximal.

    Remark 1. When \alpha = \partial \varphi for some proper, convex, lower semicontinuous function \varphi:B \to {\mathbb{R}} \cup \{+\infty\}, the definition of Fenchel transform yields

    \varphi(x) + \varphi^*(y) \geq \langle y, x \rangle \;\;\;\;\;\forall\, (x, y)\in B \times B',
    y \in \alpha(x)\;\;\;\;\; \Leftrightarrow \;\;\;\;\; \varphi(x) + \varphi^*(y) = \langle y, x \rangle.

    Thus, Fitzpatrick's representative function f_\alpha generalizes \varphi + \varphi^* to maximal monotone operators which are not subdifferentials. Remark that, even if \alpha = \partial \varphi, in general f_\alpha \neq \varphi + \varphi^*: for example, let \alpha(x): = x in {\mathbb{R}}, then

    f_\alpha(x, y) = \frac{(x+y)^2}{4} \neq \frac{x^2}{2}+\frac{y^2}{2} = \varphi(x)+\varphi^*(y).

    We need to introduce also parameter-dependent operators. For any measurable space X we say that a set-valued mapping g\colon X \rightarrow \mathcal P(B') is measurable if for any open set R\subseteq B' the set

    g^{-1}(R) : = \{ x \in X : g(x) \cap R \neq \emptyset \}

    is measurable.

    Let \mathcal B(B) be the \sigma-algebra of the Borel subsets of the separable and reflexive Banach space B, let (\Omega, \mathcal{A}, \mu) be a probability space equipped with the \sigma-algebra \mathcal{A} and the probability measure \mu. We define a random (maximal) monotone operator as \alpha \colon B \times \Omega \rightarrow \mathcal{P}(B') such that

    \alpha \ \ \text{is }\ \mathcal{B}(\text{B})\otimes \mathcal{A}\text{-measurable}, (7)
    \alpha (x,\omega )\ \ \text{is}\ \ \text{closed}\ \ \text{for}\ \ \text{any }x\in B\ \ \text{and}\ \ \text{for }\mu \text{-a}.\text{e}.\ \ \omega \in \Omega , (8)
    \alpha (\cdot ,\omega )\ \ \text{is}\ (\text{maximal})\ \text{monotone}\ \ \text{for }\mu \text{-a}.\text{e}.\ \ \omega \in \Omega . (9)

    If \alpha fulfills (7) and (8) then for any \mathcal{A}-measurable mapping v:\Omega \rightarrow B, the multivalued mapping \omega \mapsto \alpha(v(\omega), \omega) is closed-valued and measurable. We denote by \mathcal{F}(\Omega;B) the set of all measurable representative functions f: B \times B' \times \Omega \to {\mathbb{R}} \cup \{+\infty\} such that

    (a) f\colon B\times B' \times \Omega \rightarrow \mathbb{R} \cup \{+\infty\} is \mathcal{B}(B \times B') \otimes \mathcal{A} -measurable,

    (b) f(\cdot, \cdot, \omega) is convex and lower semicontinuous for \mu-a.e. \omega \in \Omega,

    (c) f(x, y, \omega)\ge \langle y, x \rangle for all (x, y) \in B\times B' and for \mu-a.e. \omega \in \Omega.

    As above, f \in \mathcal{F}(\Omega;B) represents the operator \alpha = \alpha_f(\cdot, \omega) in the following sense:

    \label{eq:represent} y \in \alpha(x, \omega)\ \Leftrightarrow \ f(x, y, \omega) = \langle y, x\rangle \;\;\;\;\; \forall (x, y) \in B\times B', \, \text{for $\mu$-a.e. }\omega \in \Omega. (10)

    Precisely, any measurable representative function f\colon B \times B' \times\Omega \rightarrow \mathbb{R}\cup \{+\infty \} represents a closed-valued, measurable, monotone operator \alpha \colon B \times \Omega \rightarrow\mathcal{P}(B'), while any closed-valued, measurable, maximal monotone operator \alpha can be represented by a measurable representative function f, for instance, by its Fitzpatrick function [26,Proposition 3.2].

    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 (\Omega, \mathcal{A}, \mu) be a probability space, where \mathcal{A} is a \sigma-algebra of subsets of \Omega and \mu is a probability measure on \Omega. Let n \in {\mathbb N} with n \geq 1. An n-dimensional dynamical system T_x on \Omega is a family of mappings T_x \colon \Omega \to \Omega, with x \in {{\mathbb{R}}^{n}}, such that

    (a) T_0 is the identity and T_{x+y} = T_xT_y for any x, y \in {{\mathbb{R}}^{n}};

    (b) for every x \in {{\mathbb{R}}^{n}} and every set E \in \mathcal{A} we have T_xE \in \mathcal{A} and

    \mu(T_xE) = \mu(E) (11)

    (c) for any measurable function f \colon \Omega \to {\mathbb{R}}^m, the function \tilde f \colon {{\mathbb{R}}^{n}} \times \Omega \to {\mathbb{R}}^m given by

    \tilde f(x, \omega) = f(T_x\omega)

    is measurable.

    Given an n-dimensional dynamical system T on \Omega, a measurable function f defined on \Omega is said to be invariant if f(T_x\omega) = f(\omega) \mu-a.e. in \Omega, for each x \in {{\mathbb{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 \colon \Omega \to {{\mathbb{R}}^{n}} is defined as

    {\mathbb E}(f): = \int_\Omega f\, d\mu.

    In the context of stochastic homogenization, it is useful to provide an orthogonal decomposition of L^2(\Omega) into functions, whose realizations are curl-free and divergence-free, in the sense of distributions (see, e.g., [10,Section 7]). For p \in [1, +\infty[, Peter-Weyl's decomposition theorem [19] can be generalized to a relation of orthogonality between subspaces of L^p(\Omega) and L^{p'}(\Omega), where p' denotes the conjugate exponent of p. Let v\in L^p_{\rm loc}({{\mathbb{R}}^{n}};{{\mathbb{R}}^{n}}). We say that v is potential (or curl-free) in \mathbb{R}^{n} if

    \int \bigg( v^{i}\frac{\partial\varphi}{\partial x_{j}}-v^{j}\frac{\partial \varphi}{\partial x_{i}}\bigg)\, dx = 0, \ \ \ \ \forall i, j = 1, \dots, n, \, \;\;\;\;\;\forall \varphi \in \mathcal {D}(\mathbb{R}^{n})

    and we say that v is solenoidal (or divergence-free) in \mathbb{R}^{n} if

    \sum\limits_{i = 1}^{n}\int v^{i}\frac{\partial\varphi}{\partial x_{i}}\, dx = 0, \ \ \ \forall \varphi \in \mathcal {D}(\mathbb{R}^{n}).

    Next we consider a vector field on (\Omega, \mathcal{A}, \mu). We say that f\in L^{p}(\Omega;{{\mathbb{R}}^{n}}) is potential if \mu-almost all its realizations x\mapsto f(T_{x}\omega) are potential. We denote by L_{\rm pot}^{p}(\Omega;{{\mathbb{R}}^{n}}) the space of all potential f\in L^{p}(\Omega;{{\mathbb{R}}^{n}}). In the same way, f\in L^{p}(\Omega;{{\mathbb{R}}^{n}}) is said to be solenoidal if \mu-almost all its realizations x\mapsto f(T_{x}\omega) are solenoidal and we denote by L_{\rm sol}^{p}(\Omega;{{\mathbb{R}}^{n}}) the space of all solenoidal f\in L^{p}(\Omega;{{\mathbb{R}}^{n}}). In the following Lemma we collect the main properties of potential and solenoidal L^p spaces (see [10,Section 15]).

    Lemma 2.3. Define the spaces

    \begin{align*} \mathcal{V}^{p}_{\rm pot}(\Omega;{{\mathbb{R}}^{n}})&: = \{ f\in L_{\rm pot}^{p}(\Omega;{{\mathbb{R}}^{n}}) : \mathbb E(f) = 0\}, \\ \mathcal{V}^{p}_{\rm sol}(\Omega;{{\mathbb{R}}^{n}})&: = \{ f\in L_{\rm sol}^{p}(\Omega;{{\mathbb{R}}^{n}}) : \mathbb E(f) = 0\}. \end{align*}

    The spaces L_{\rm pot}^{p}(\Omega;{{\mathbb{R}}^{n}}), L_{\rm sol}^{p}(\Omega;{{\mathbb{R}}^{n}}), \mathcal{V}^{p}_{\rm pot}(\Omega;{{\mathbb{R}}^{n}}), \mathcal{V}^{p}_{\rm sol}(\Omega;{{\mathbb{R}}^{n}}) are closed subspaces of L^{p}(\Omega;{{\mathbb{R}}^{n}}). If u \in L_{\rm sol}^p(\Omega;{{\mathbb{R}}^{n}}) and v \in L_{\rm pot}^{p'}(\Omega;{{\mathbb{R}}^{n}}) with 1/p+1/p' = 1, then

    \label{ort} \mathbb E(u \cdot v) = \mathbb E(u) \cdot \mathbb E(v) (12)

    and the relations

    (\mathcal{V}^{p}_{\rm sol}(\Omega;{{\mathbb{R}}^{n}}))^\perp = \mathcal{V}^{p'}_{\rm pot}(\Omega;{{\mathbb{R}}^{n}}) \oplus {{\mathbb{R}}^{n}}, \;\;\;\;\;(\mathcal{V}^{p}_{\rm pot}(\Omega;{{\mathbb{R}}^{n}}))^\perp = \mathcal{V}^{p'}_{\rm sol}(\Omega;{{\mathbb{R}}^{n}}) \oplus {{\mathbb{R}}^{n}}

    hold in the sense of duality pairing between the spaces L^p(\Omega) and L^{p'}(\Omega).

    One of the most important results regarding stochastic homogenization is Birk-hoff's Ergodic Theorem. We report the statement given in [10,Theorem 7.2].

    Theorem 2.4. (Birkhoff's Ergodic Theorem) Let f \in L^1(\Omega;{\mathbb{R}}^m) and let T be an n-dimensional ergodic dynamical system on \Omega. Then

    \mathbb E(f) = \lim\limits_{\varepsilon \to 0}\frac{1}{|K|}\int_K f\big(T_{x/\varepsilon }\omega\big)\, dx

    for \mu-a.e. \omega \in \Omega, for any K \subset {{\mathbb{R}}^{n}} bounded, measurable, with |K|>0.

    Remark 2. Birkhoff's theorem implies that \mu-almost every realization \tilde f_\varepsilon (x, \omega) = f(T_{x/\varepsilon }\omega) satisfies

    \lim\limits_{\varepsilon \to 0} \frac{1}{|K|}\int_K \tilde f_\varepsilon (x, \omega)\, dx = \mathbb{E}(f).

    Since this holds for every measurable bounded set K \subset {{\mathbb{R}}^{n}}, it entails in particular that if f\in L^p(\Omega), then

    {{\tilde{f}}_{\varepsilon }}\rightharpoonup \mathbb{E}(f)\ \ \text{weakly}\ \ \text{in }L_{\text{loc}}^{p}({{\mathbb{R}}^{\text{n}}};{{\mathbb{R}}^{\text{m}}})\ \text{for }\mu \text{-a}.\text{e}.\text{ }\ \ \omega \in \Omega . (13)

    In what follows, the dynamical system T_x is assumed to be ergodic and K\subset {{\mathbb{R}}^{n}} is bounded, measurable and |K|>0.

    Let be given a probability space (\Omega, \mathcal{A}, \mu) endowed with an n-dimensional ergodic dynamical system T_x \colon \Omega \to \Omega, x\in {{\mathbb{R}}^{n}}. Let p\in (1, +\infty), p' = \frac{p}{p-1} and let \alpha be a random maximal monotone operator, as in (7)-(9).

    We rephrase here Visintin's scale integration/disintegration [25,26] to the stochastic homogenization setting.

    Remark 3. 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 3.1 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 3.2 and Proposition 1 were taken for granted in [26], but we decided to write a proof for sake of clarity. Lemma 3.3 and Lemma 3.4 are essentially [26,Theorem 4.3] and [26,Theorem 4.4], cast in the framework of stochastic homogenization in the probability space (\Omega, \mathcal{A}, \mu), instead of periodic homogenization on the n-dimensional torus. Theorem 3.5 collects other results of [26]. Lemma 3.6 is an original remark.

    For every fixed \omega\in\Omega let f(\cdot, \cdot, \omega) \colon {{\mathbb{R}}^{n}} \times {{\mathbb{R}}^{n}} \to {\mathbb{R}} \cup \{+\infty\} be the Fitzpatrick representation of the operator \alpha(\cdot, \omega). We assume the following coercivity condition on f: there exist c>0 and k\in L^1(\Omega) such that for any \xi, \eta\in {{\mathbb{R}}^{n}}, for any \omega \in \Omega it holds

    f(\xi, \eta, \omega) \ge c\left(|\xi |^p+|\eta |^{p'}\right)+k(\omega). (14)

    We define the homogenised representation f_0 \colon {{\mathbb{R}}^{n}} \times {{\mathbb{R}}^{n}} \to {\mathbb{R}} \cup \{+\infty\} as

    f_0(\xi, \eta): = \inf \bigg\{ \int_\Omega f(\xi+v(\omega), \eta+u(\omega), \omega) \, d\mu :u \in \mathcal V^p_{\rm pot}(\Omega;{{\mathbb{R}}^{n}}), v\in \mathcal V^{p'}_{\rm sol}(\Omega;{{\mathbb{R}}^{n}}) \bigg\}. (15)

    Lemma 3.1. Let X be a reflexive Banach space, let K be a weakly closed subset of a reflexive Banach space. Let the function g \colon X \times K \to {\mathbb{R}} \cup \{+\infty\} be weakly lower semicontinuous and bounded from below. If g is coercive1, then the function h\colon X \to {\mathbb{R}} \cup \{+\infty\} given by

    1i.e., for all M>0 the set \{(x, y)\in X \times K : g(x, y) \leq M\} is bounded.

    h(x): = \inf\limits_{y\in K}g(x, y)

    is weakly lower semicontinuous and coercive. Moreover, if g and K are convex then h is convex.

    Proof. Let x_j \rightharpoonup x \in X, we must show that

    \label{eq2} \liminf\limits_{j \to +\infty} h(x_j) \geq h(x). (16)

    Let

    \ell: = \liminf\limits_{j \to +\infty} h(x_j).

    If \ell = +\infty, then (16) is trivially satisfied. On the other hand, since g is bounded from below, then \ell > -\infty, and we can assume that \ell \in {\mathbb{R}}. By definition of inferior limit, there exists a subsequence of \{x_j\} (not relabeled), such that \lim_{j\to \infty}h(x_j) = \ell. Up to extracting another subsequence, we can also assume that h(x_j)\leq 2\ell for all j\in \mathbb N. Let \varepsilon >0 be fixed, by definition of infimum, for all j\in \mathbb N, there exists y_j \in K such that

    \label{eq3} h(x_j) = \inf\limits_{y \in K} g(x_j, y) \geq g(x_j, y_j)-\varepsilon . (17)

    Therefore

    g(x_j, y_j) \leq 2\ell +\varepsilon \;\;\;\;\;\forall\, j\in \mathbb N.

    By the coercivity assumption on g, we deduce that y_j is bounded, we can therefore extract a subsequence \{y_{j_k}\} \subset K such that y_{j_k}\rightharpoonup y. Since K is weakly closed, then y \in K. We can now pass to the inferior limit in (17), using the lower semicontinuity of g

    \label{eq:4} \liminf\limits_{k \to +\infty} h(x_{j_k}) \geq \liminf\limits_{k \to +\infty} g(x_{j_k}, y_{j_k})-\varepsilon \geq g(x, y) -\varepsilon \geq h(x)-\varepsilon . (18)

    By arbitrariness of \varepsilon >0, this proves the weak lower semicontinuity of h. Assume now that K is convex. Take \lambda\in [0, 1], x_1, x_2\in X and y_1, y_2 \in K. By convexity of g

    h(\lambda x_1+(1-\lambda)x_2)\le g(\lambda x_1+(1-\lambda)x_2, \lambda y_1+(1-\lambda)y_2)\le \lambda g(x_1, y_1) +(1-\lambda)g(x_2, y_2).

    Passing to the infimum with respect to y_1, y_2\in K we conclude

    h(\lambda x_1+(1-\lambda)x_2)\le \lambda h(x_1) +(1-\lambda)h(x_2).

    Regarding the coercivity of h, denote

    B_t: = \{x \in X : h(x) \leq t\}, \;\;\;\;\;A_t: = \{x\in X : g(x, y)\leq t, \text{ for some }y\in K\}.

    Let M, \varepsilon >0, for all x\in B_M there exists y\in K such that g(x, y)\leq h(x)+\varepsilon \leq M+\varepsilon , therefore B_M \subseteq A_{M+\varepsilon }. Since g is coercive, A_{M+\varepsilon } is bounded and thus B_M is bounded, i.e., h is coercive.

    In the proof of Proposition 1 we need the following estimate

    Lemma 3.2. For all p\in[1, +\infty[ there exists C>0 such that

    \int_\Omega |\xi + u(\omega)|^p\, d\mu \geq C \int_\Omega |\xi|^p + |u(\omega)|^p\, d\mu

    for all \xi \in {{\mathbb{R}}^{n}}, for all u\in L^p(\Omega;{{\mathbb{R}}^{n}}) such that \mathbb E(u) = 0.

    Proof. Consider the operator

    \begin{align*} \Phi &:L^p(\Omega;{{\mathbb{R}}^{n}}) \to L^p(\Omega;{{\mathbb{R}}^{n}}) \times L^p(\Omega;{{\mathbb{R}}^{n}}) \\ & u \;\;\;\;\;\mapsto \left(\mathbb E(u), u-\mathbb E(u)\right). \end{align*}

    Clearly, \Phi is linear and continuous. Therefore, there exists C>0 such that

    \begin{align*} \int_\Omega |\mathbb E(u)|^p d\mu +\int_\Omega |u(\omega)- \mathbb E(u)|^p\, d\mu &\leq \left(\|\mathbb E(u)\|_{L^p} + \|u-\mathbb E(u)\|_{L^p}\right)^p\\ &\leq 2^{p/2}\left(\|\mathbb E(u)\|^2_{L^p} + \|u-\mathbb E(u)\|^2_{L^p}\right)^{p/2}\\ & = 2^{p/2}\|\Phi(u)\|^p_{L^p \times L^p} \leq C\|u\|^p_{L^p}\\ & = C\int_\Omega |u(\omega)|^p\, d\mu. \end{align*}

    Apply now the last inequality to u(\omega) = \xi + \tilde u(\omega), with \mathbb E(\tilde u) = 0:

    \int_\Omega |\xi|^p + |\tilde u(\omega)|^p\, d\mu \leq C\int_\Omega |\xi +\tilde u(\omega)|^p\, d\mu.

    Proposition 1. For all (\xi, \eta)\in {{\mathbb{R}}^{n}} \times {{\mathbb{R}}^{n}} there exists a couple (\widetilde{u}, \widetilde{v})\in \mathcal V^p_{\rm sol}(\Omega;{{\mathbb{R}}^{n}}) \times \mathcal V^{p'}_{\rm pot}(\Omega;{{\mathbb{R}}^{n}}) such that the infimum on the right-hand side of (15) is attained. Moreover, f_0 \in \mathcal F({{\mathbb{R}}^{n}}). In particular, it holds

    \label{ineq:f0} f_0(\xi, \eta)\ge \xi\cdot \eta \;\;\;\;\; \forall (\xi, \eta)\in {{\mathbb{R}}^{n}}\times {{\mathbb{R}}^{n}}. (19)

    Proof. Let K: = \mathcal V^p_{\rm sol}(\Omega;{{\mathbb{R}}^{n}})\times \mathcal V^{p'}_{\rm pot}(\Omega;{{\mathbb{R}}^{n}}). Then K is weakly closed in L^p(\Omega;{{\mathbb{R}}^{n}}) \times L^{p'}(\Omega;{{\mathbb{R}}^{n}}) since it is closed and convex. Let \xi, \eta\in {{\mathbb{R}}^{n}} be fixed, for any (u, v) \in K let

    F_{\xi, \eta}(u, v): = \int_\Omega f(\xi+v(\omega), \eta+u(\omega), \omega) \, d\mu.

    We prove that the problem \inf_KF_{\xi, \eta} has a solution applying the direct method of the Calculus of Variations. First, by (14), \inf_KF_{\xi, \eta}>-\infty. Then, if (u_h, v_h)\subset K is a minimizing sequence for F_{\xi, \eta}, by the coercivity assumption (14), up to subsequences, (u_h, v_h) \rightharpoonup (u, v) weakly in L^p(\Omega;{{\mathbb{R}}^{n}}) \times L^{p'}(\Omega;{{\mathbb{R}}^{n}}), therefore (u, v)\in K, since K is weakly closed. Finally, F_{\xi, \eta} is L^p\times L^{p'}-weakly lower semicontinuous since f(\cdot, \cdot, \omega) is convex, lower semicontinuous, and bounded from below by an integrable function (14), therefore

    F_{\xi, \eta}(u, v)\leq \liminf\limits_{h \to \infty}F_{\xi, \eta}(u_h, v_h) = \inf\limits_K F_{\xi, \eta}.

    This concludes the first part of the statement. We now want to show that f_0\in \mathcal F({{\mathbb{R}}^{n}}). Owing to (14) and Lemma 3.2, for all \xi, \eta, (u, v)\in {{\mathbb{R}}^{n}}\times {{\mathbb{R}}^{n}} \times K, there exists a constant C>0 such that

    \begin{align*} F_{\xi, \eta}(u, v)&\geq c\int_\Omega |\xi +v(\omega)|^p + |\eta +u(\omega)|^{p'} +k(\omega)\, d\mu \\ & \geq C\int_\Omega |\xi|^p +|u(\omega)|^p + |\eta|^{p'} +|v(\omega)|^{p'} d\mu+\mathbb E(k) \\ &\geq C\left( |\xi|^p + {\|u\|}^p_{L^p(\Omega)} + |\eta|^{p'} + {\|v\|}^{p'}_{L^{p'}(\Omega)} \right)-{\|k\|}_{L^1(\Omega)}. \end{align*}

    Thus, for any M\ge 0, the set

    \left\{(\xi, \eta, (u, v))\in R^n\times {{\mathbb{R}}^{n}} \times K : F_{\xi, \eta}(u, v) \leq M\right\}

    is bounded in {{\mathbb{R}}^{n}} \times {{\mathbb{R}}^{n}} \times L^p(\Omega;{{\mathbb{R}}^{n}}) \times L^{p'}(\Omega;{{\mathbb{R}}^{n}}). We are therefore in a position to apply Lemma 3.1 and to conclude that f_0 is convex and lower semicontinuous. Furthermore, let (\widetilde{u}, \widetilde{v})\in K be a minimizer of F_{\xi, \eta}, using (12)

    \begin{aligned} f_0(\xi, \eta)& = \int_\Omega f(\xi+\widetilde{u}(\omega), \eta+\widetilde{v}(\omega), \omega) \, d\mu\\ &\ge \int_\Omega (\xi+\widetilde{u}(\omega))\cdot (\eta+\widetilde{v}(\omega)) \, d\mu\\ & = \mathbb E(\xi+\widetilde{u}) \cdot \mathbb E(\eta + \widetilde{v})\\ & = \xi\cdot\eta, \end{aligned}

    which yields the conclusion.

    We denote by \alpha_0 the operator on {{\mathbb{R}}^{n}} represented by f_0 through the usual relation

    \eta \in \alpha_0(\xi) \;\;\;\;\; \Leftrightarrow\;\;\;\;\; f_0(\xi, \eta) = \xi \cdot \eta.

    We refer to \alpha_0 as the scale integration of \alpha, since it is obtained through f_0, which is the scale integration of the Fitzpatrick representative f of \alpha.

    Lemma 3.3. Let \xi, \eta\in {{\mathbb{R}}^{n}} be such that \eta \in \alpha_0(\xi). Then, there exist u\in L^p_{\rm sol}(\Omega;{{\mathbb{R}}^{n}}) and v\in L^{p'}_{\rm pot}(\Omega;{{\mathbb{R}}^{n}}) such that

    \label{auxiliary} v(\omega)\in \alpha(u(\omega), \omega), \;\;\;\;\; for \;\;\;\;\; \mu -a.e. \omega\in\Omega. (20)

    Moreover, \mathbb{E}(u) = \xi and \mathbb{E}(v) = \eta, that is

    \mathbb E( v) \in \alpha_0(\mathbb E( u)). (21)

    Proof. Since f_0 represents \alpha_0, \eta \in \alpha_0(\xi) implies

    \label{eq:f0repr} f_0(\xi, \eta) = \xi \cdot \eta. (22)

    Take now \widetilde{u} \in \mathcal V^p_{\rm sol}(\Omega;{{\mathbb{R}}^{n}}) and \widetilde{v} \in \mathcal V^{p'}_{\rm pot}(\Omega;{{\mathbb{R}}^{n}}) such that

    \label{eq:hyp1} f_0(\xi, \eta) = \int_\Omega f(\xi+\widetilde{u}(\omega), \eta+\widetilde{v}(\omega), \omega) \, d\mu. (23)

    Since \mathbb E(\widetilde{u}) = \mathbb E(\widetilde{v}) = 0,

    \begin{align*} \xi \cdot \eta& = \mathbb E(\xi+\widetilde{u}) \cdot \mathbb E(\eta+\widetilde{v})\\ & \stackrel{(12)}{ = }\int_\Omega (\xi+\widetilde{u}(\omega))\cdot (\eta+\widetilde{v}(\omega))\, d\mu\\ &\stackrel{f \in \mathcal F({{\mathbb{R}}^{n}})}{\le} \int_\Omega f(\xi+\widetilde{u}(\omega), \eta+\widetilde{v}(\omega), \omega) \, d\mu\\ & \stackrel{(23)}{ = } f_0(\xi, \eta)\\ & \stackrel{(22)}{ = }\xi \cdot \eta \end{align*}

    from which we obtain

    \label{eq:312} (\xi+\widetilde{u}(\omega))\cdot (\eta+\widetilde{v}(\omega)) = f(\xi+\widetilde{u}(\omega), \eta+\widetilde{v}(\omega), \omega), \;\;\;\;\; \text{$\mu$-a.e. }\omega\in\Omega. (24)

    Let u(\omega): = \xi+\widetilde{u}(\omega) and v(\omega): = \eta+\widetilde{v}(\omega). Since f represents \alpha, (24) is equivalent to (20). Moreover, since \mathbb E(u) = \xi and \mathbb E( v) = \eta, \eta \in \alpha_0(\xi) implies also (21).

    Lemma 3.3 is also referred to as scale disintegration (see [26,Theorem 4.4]), as it shows that given a solution (\xi, \eta) to the integrated problem \eta \in \alpha_0(\xi), it is possible to build a solution to the original problem v(\omega) \in \alpha(u(\omega), \omega). The converse, known as scale integration (see [26,Theorem 4.3]) is provided by the next Lemma.

    Lemma 3.4. Let u\in L^p_{\rm sol}(\Omega;{{\mathbb{R}}^{n}}) and v\in L^{p'}_{\rm pot}(\Omega;{{\mathbb{R}}^{n}}) satisfy

    \label{auxiliary:int} v(\omega)\in \alpha(u(\omega), \omega), \;\;\;\;\;for\;\;\;\;\; \mu-a.e. \omega\in\Omega, (25)

    then

    \label{eq:integrated} \mathbb E(v) \in \alpha_0(\mathbb E(u)). (26)

    Proof. By (25) and (12)

    \int_\Omega f(u(\omega), v(\omega), \omega)\, d\mu = \int_\Omega u(\omega)\cdot v(\omega)\, d\mu = \mathbb E(u)\cdot \mathbb E(v).

    On the other hand, by definition of f_0,

    \int_\Omega f(u(\omega), v(\omega), \omega)\, d\mu \geq f_0(\mathbb E(u), \mathbb E(v)) \geq \mathbb E(u)\cdot \mathbb E(v).

    We conclude that f_0(\mathbb E(u), \mathbb E(v)) = \mathbb E(u)\cdot \mathbb E(v), which yields (26).

    How the properties of \alpha and f reflect on \alpha_0 and f_0 was thoroughly studied in [26]:

    Theorem 3.5. If

    \bullet f\in \mathcal F(\Omega, {{\mathbb{R}}^{n}}) is uniformly bounded from below,

    \bullet there exists (u, v)\in L^p(\Omega;{{\mathbb{R}}^{n}}) \times L^{p'}(\Omega;{{\mathbb{R}}^{n}}) such that

    \int_\Omega f(u(\omega), v(\omega), \omega)\, d\mu < +\infty,

    \bullet f represents a maximal monotone operator for \mu-a.e. \omega \in \Omega then f_0 represents a (proper) maximal monotone operator [26,Theorem 5.3]. Moreover, if f is strictly convex, then

    \bullet f_0 is strictly convex [26,Lemma 5.4],

    \bullet the operators \alpha_0 and \alpha_0^{-1} are both strictly monotone [26,Proposition 5.5] and if Dom(\alpha_0) and Dom(\alpha_0^{-1}) are unbounded, then \alpha_0 and \alpha_0^{-1} are coercive [26,Proposition 5.6].

    In order to obtain strict monotonicity of \alpha_0 and \alpha_0^{-1}, by the next Lemma we provide an alternative to strict convexity of the Fitzpatrick function.

    Lemma 3.6. Let \alpha(\cdot, \omega):B \to B' be maximal and strictly monotone, and assume that its Fitzpatrick representation f is coercive, in the sense of (14). Then its scale integration \alpha_0 is strictly monotone.

    Proof. For all \eta_i\in \alpha_0(\xi_i), i = 1, 2, by Lemma 3.3, there exist u_i\in L^p_{\rm sol}(\Omega;{{\mathbb{R}}^{n}}) and v_i\in L^{p'}_{\rm pot}(\Omega;{{\mathbb{R}}^{n}}) such that

    \label{smon1} v_i(\omega)\in \alpha(u_i(\omega), \omega), \;\;\;\;\; \text{for $\mu$-a.e. }\omega\in\Omega (27)

    and \mathbb{E}(u_i) = \xi_i, \mathbb{E}(v_i) = \eta_i. By (12), strict monotonicity of \alpha, and Jensen's inequality

    \begin{align*} (\eta_2-\eta_1)\cdot(\xi_2-\xi_1) & = \int_\Omega (v_2(\omega)-v_1(\omega))\cdot(u_2(\omega)-u_1(\omega)) d\mu \\ & \geq \theta \int_\Omega |u_2(\omega)-u_1(\omega)|^2d\mu \\ & \geq \theta \left|\int_\Omega u_2(\omega)-u_1(\omega)d\mu \right|^2\\ & = \theta \left| \xi_2 -\xi_1 \right|^2. \end{align*}

    Let D\subset {{\mathbb{R}}^{n}} be a Lipschitz and bounded domain with |D|>0. We recall the following classical result.

    Lemma 3.7 (Div-Curl lemma, [15]). Let p\in]1, +\infty[, let v^n, v \in L^{p'}(D;{\mathbb{R}}^m) and u^n, u \in L^p(D;{\mathbb{R}}^m) be such that

    v^n \rightharpoonup v \;\;\;\;\;weakly\;\;\;\;\; in L^{p'}(D;{\mathbb{R}}^m), \;\;\;\;\; u^n \rightharpoonup u \;\;\;\;\;weakly\;\;\;\;\; in \;\;\;\;\;L^p(D;{\mathbb{R}}^m).

    In addition, assume that

    \{\text{curl}{{v}^{n}}\}\text{ }is\text{ }compact\text{ }in\text{ }{{W}^{-1,{p}'}}(D;{{\mathbb{R}}^{m\times m}}),\ \ \ \ \ \{\text{div}\ {{u}^{n}}\}\text{ }is\text{ }compact\text{ }in\text{ }{{W}^{-1,p}}(D).

    Then

    v^n \cdot u^n \stackrel{*}{\rightharpoonup} v \cdot u \;\;\;\;\;\mbox{in }\mathcal D'(D).

    We are now ready to prove our main result concerning the stochastic homogenization of a maximal monotone relation.

    Theorem 3.8. Let (\Omega, \mathcal{A}, \mu) be a probability space with an n-dimensional ergodic dynamical system T_x:\Omega \to \Omega, x\in{{\mathbb{R}}^{n}}. Let D\subset {{\mathbb{R}}^{n}} be a bounded domain, let p \in]1, +\infty[. Let \alpha: {{\mathbb{R}}^{n}} \times \Omega \to \mathcal P({{\mathbb{R}}^{n}}) be a closed-valued, measurable, maximal monotone random operator, in the sense of (7)-(9).

    Let f be a Fitzpatrick representation of \alpha, as in (10). Let f satisfy (14) and assume that for \mu-a.e. \omega \in \Omega and \varepsilon \geq 0 there exists a couple

    (J_\omega^\varepsilon, E_\omega^\varepsilon)\in L^p(D;{{\mathbb{R}}^{n}})\times L^{p'}(D;{{\mathbb{R}}^{n}})

    such that

    {{\{\text{div}J_{\omega }^{\varepsilon }\}}_{\varepsilon \ge 0}}\text{ }is\text{ }compact\text{ }in\text{ }{{W}^{-1,p}}(D),{{\{\text{curl}E_{\omega }^{\varepsilon }\}}_{\varepsilon \ge 0}}\text{ }is\text{ }compact\text{ }in\text{ }{{W}^{-1,{p}'}}(D;{{\mathbb{R}}^{n\times n}}), (28a)
    \label{e:convergence} \lim\limits_{\varepsilon \to 0} J_\omega^\varepsilon = J_\omega^0 \;\;\;\;\; weakly \;\;\;\;\;in \;\;\;\;\;L^p(D;{{\mathbb{R}}^{n}}), \;\;\;\;\; \lim\limits_{\varepsilon \to 0} E_\omega^\varepsilon = E_\omega^0 \;\;\;\;\; weakly \;\;\;\;\;in \;\;\;\;\;L^{p'}(D;{{\mathbb{R}}^{n}}), (28b)
    \label{e:inclusion} E_\omega^\varepsilon(x) \in \alpha(J_\omega^\varepsilon(x), T_{x/\varepsilon }\omega) \;\;\;\;\; a.e. \;\;\;in \;\;\;\;\;D. (28c)

    Then, for \mu-a.e. \omega \in \Omega

    \label{hom1} E_\omega^0(x) \in \alpha_0(J_\omega^0(x)) \;\;\;\;\; a.e. \;\;\;in \;\;\;\;\;D, (29)

    where \alpha_0 is the maximal monotone operator represented by the homogenized representation f_0 \colon {{\mathbb{R}}^{n}} \times {{\mathbb{R}}^{n}} \to {\mathbb{R}} \cup \{+\infty\} defined by

    f_0(\xi, \eta): = \inf \bigg\{ \int_\Omega f(\xi+u(\omega), \eta+v(\omega), \omega) \, d\mu :u \in \mathcal V^p_{\rm sol}(\Omega;{{\mathbb{R}}^{n}}), v\in \mathcal V^{p'}_{\rm pot}(\Omega;{{\mathbb{R}}^{n}}) \bigg\}.

    Proof. By Lemma 3.3 for all \xi, \eta\in {{\mathbb{R}}^{n}} such that \eta \in \alpha_0(\xi) there exist u\in L^p_{\rm sol}(\Omega;{{\mathbb{R}}^{n}}) and v\in L^{p'}_{\rm pot}(\Omega;{{\mathbb{R}}^{n}}) such that \mathbb{E}(u) = \xi, \mathbb{E}(v) = \eta, and

    \label{auxiliary2} v(\omega)\in \alpha(u(\omega), \omega), \;\;\;\;\; \text{for $\mu$-a.e. }\omega\in\Omega. (30)

    Define the stationary random fields u_\varepsilon, v_\varepsilon \colon D\times \Omega \to {{\mathbb{R}}^{n}} as

    u_\varepsilon(x, \omega): = u(T_{x/\varepsilon }\omega), \;\;\;\;\; v_\varepsilon(x, \omega): = v(T_{x/\varepsilon }\omega).

    For \mu-a.e. \omega \in \Omega

    x \mapsto u_\varepsilon(x, \omega) \in L^p_{\rm loc}({{\mathbb{R}}^{n}};{{\mathbb{R}}^{n}}), \;\;\;\;\; x \mapsto v_\varepsilon(x, \omega) \in L^{p'}_{\rm loc}({{\mathbb{R}}^{n}};{{\mathbb{R}}^{n}}).

    Equation (30) implies

    \label{eq:318} v_\varepsilon(x, \omega) \in \alpha(u_\varepsilon(x, \omega), T_{x/\varepsilon }\omega), \;\;\;\;\; \textrm{for a.e. }x\in D, \ \mu\textrm{-a.e. }\omega\in \Omega. (31)

    By Birkhoff's Theorem (and (13), in particular), for \mu-a.e. \omega \in \Omega we have

    \label{weak} u_\varepsilon(\cdot, \omega) \rightharpoonup \mathbb E(u) \;\;\;\;\;\textrm{weakly in }L^p(D;{{\mathbb{R}}^{n}}), \;\;\;\;\;v_\varepsilon(\cdot, \omega) \rightharpoonup \mathbb E(v) \;\;\;\;\; \textrm{weakly in }L^{p'}(D;{{\mathbb{R}}^{n}}). (32)

    Since \alpha is monotone, by (28c) and (31), for \mu-a.e. \omega \in \Omega

    \label{ineq} \int_D (E_\omega^\varepsilon(x)- v_\varepsilon(x, \omega))\cdot (J_\omega^\varepsilon(x)- u_\varepsilon(x, \omega))\phi(x)\, dx \ge 0, (33)

    for any \phi\in C_c^\infty(D) with \phi\ge 0. Since u_\varepsilon is solenoidal and v_\varepsilon is potential, by (28a)

    \begin{align} & {{\{\text{curl}(E_{\omega }^{\varepsilon }-{{v}_{\varepsilon }}(\cdot ,\omega ))\}}_{\varepsilon }}\text{ is compact in }{{W}^{-1,{p}'}}(D;{{\mathbb{R}}^{n\times n}}), \\ & {{\{\text{div}(J_{\omega }^{\varepsilon }-{{u}_{\varepsilon }}(\cdot ,\omega ))\}}_{\varepsilon }}\text{ is compact in }{{W}^{-1,p}}(D). \\ \end{align}

    By (28b), (32), and Lemma 3.7, we can thus pass to the limit as \varepsilon \to 0 in (33):

    \int_D (E_\omega^0(x) -\mathbb E( v))\cdot (J_\omega^0(x) -\mathbb E( u))\phi(x)\, dx \ge 0, \;\;\;\;\; \textrm{for $\mu$-a.e. }\omega \in \Omega.

    Since the last inequality holds for all nonnegative \phi\in C_c^\infty(D), it holds also pointwise, for almost every x \in D:

    (E_\omega^0(x) -\mathbb E( v))\cdot (J_\omega^0(x) -\mathbb E( u)) \ge 0, \;\;\;\;\;\textrm{for $\mu$-a.e. }\omega \in \Omega.

    To conclude, since \mathbb{E}(u) = \xi, \mathbb{E}(v) = \eta are arbitrary vectors in \mathcal G_{\alpha_0}, the maximal monotonicity of \alpha_0 implies that

    E_\omega^0(x) \in \alpha_0(J_\omega^0(x))

    for a.e. x\in D, \mu-a.e. \omega\in \Omega.

    Remark 4. In this section's results, the function spaces L^p_{\rm sol}(\Omega) and L^{p'}_{\rm pot}(\Omega) can be generalized to a couple of nonempty, closed and convex sets

    \mathcal U\subset L^p(\Omega;{{\mathbb{R}}^{n}}), \;\;\;\;\; \mathcal V\subset L^{p'}(\Omega;{{\mathbb{R}}^{n}})

    such that

    \mathbb E(u \cdot v) = \mathbb E(u)\cdot \mathbb E(v), \;\;\;\;\; \forall (u, v)\in \mathcal U \times \mathcal V.

    Furthermore, Proposition 1 and Lemma 3.3 remain valid if the previous equality is replaced by the inequality

    \mathbb E(u \cdot v) \geq \mathbb E(u)\cdot \mathbb E(v), \;\;\;\;\; \forall (u, v)\in \mathcal U \times \mathcal V.

    In this subsection 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] and we also follow [26] for the suitable mathematical formulation in terms of maximal monotone operators. We consider a non-homogeneous electric conductor, that occupies a bounded Lipschitz domain D \subset {\mathbb{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

    \label{eq:ohmhall} E(x) \in \alpha(J(x), x) +h(x)J(x) \times B(x) + E_a(x) \;\;\;\;\; \textrm{in }D, (34)

    where \alpha(\cdot, x):{\mathbb{R}}^3 \mapsto {\mathbb{R}}^3 is a maximal monotone mapping for a.e. x\in D, B is the magnetic induction field, h is the (material-dependent) Hall coefficient, and E_a is an applied electromotive force. We deal with a stationary system and thus we couple (34) with the Faraday law and with the stationary law of charge-conservation:

    \begin{align} & \text{curl}E=g, \\ & \text{div}J=0, \\ \end{align}

    where the vector field g:D \to {\mathbb{R}}^3 is prescribed. Following [26], we assume that h, B, E_a are given and we define the maximal monotone operator \beta(\cdot, x):{\mathbb{R}}^3 \mapsto {\mathbb{R}}^3 as

    \beta(J, x): = \alpha(J, x) +h(x)J \times B(x) + E_a(x).

    A single-valued parameter-dependent operator \beta is strictly monotone uniformly in x, if there exists \theta>0 such that for a.e. x\in D

    \label{eq:smon} (\beta(v_1, x) - \beta(v_2, x))\cdot (v_1-v_2) \geq \theta{\| v_1-v_2\|}^2 \;\;\;\;\;\forall\, v_1, v_2\in {\mathbb{R}}^3. (35)

    The following existence and uniqueness result is a classical consequence of the maximal monotonicity of \alpha (see, e.g., [22,26]).

    Theorem 4.1. Let D \subset {\mathbb{R}}^3 be a bounded Lipschitz domain. Let \{\beta(\cdot, x)\}_{x\in D} be a family of single-valued maximal monotone operators on {\mathbb{R}}^3. Assume moreover that there exist constants a, c>0 and b\geq 0 such that for a.e. x\in D, \, \forall v \in {\mathbb{R}}^3

    \begin{align} |\beta(x, v)| &\leq c(1+|v|), \label{eq:bounded} \end{align} (36)
    \begin{align} \beta(x, v)\cdot v &\geq a|v|^2 -b. \label{eq:coercivity} \end{align} (37)

    Let g \in L^2(D;{\mathbb{R}}^3) be given, such that div\, g = 0, distributionally. Then, there exists E, J\in L^2(D;{\mathbb{R}}^3) such that

    \label{eq:estimates} {\|E\|}_{L^2} +{\|J\|}_{L^2}\leq C\left(1+{\|g\|}_{L^2}\right) (38)

    and, denoting by \nu the outward unit normal to \partial D,

    \begin{align} E(x) & = \beta(J(x), x) \;\;\;\;\; \ in\;\; D, \label{P:incl}\end{align} (39)
    \begin{align} curl\, E(x) & = g(x) \;\;\;\;\;\ in \;\;D, \label{P:ele}\end{align} (40)
    \begin{align} div\, J(x)& = 0 \;\;\;\;\; \ in\;\; D, \label{P:magn}\end{align} (41)
    \begin{align} E(x) \times \nu(x) & = 0 \;\;\;\;\;\ on\;\; \partial D. \label{P:bound} \end{align} (42)

    Moreover, if \beta is strictly monotone uniformly in x\in D, then the field J is uniquely determined, while if \beta^{-1} is strictly monotone uniformly in x\in D, then the field E is uniquely determined.

    Remark 5. Conditions (40)-(41) have to be intended in the weak sense -see below -while (42) holds in H^{-1/2}(\partial D;{\mathbb{R}}^3). Note that, according to (40), the divergence of g also vanishes.

    Let (\Omega, \mathcal{A}, \mu) be a probability space endowed with a 3-dimensional ergodic dynamical system T_x \colon \Omega \to \Omega, with x\in {\mathbb{R}}^3. Let \{\alpha(\cdot, \omega)\}_{\omega\in \Omega} be a family of maximal monotone operators on {\mathbb{R}}^3, and let

    \label{hyp:data} h \in L^\infty(\Omega), \;\;\;\;\; B \in L^\infty(\Omega;{\mathbb{R}}^3), \;\;\;\;\; E_a\in L^2(\Omega;{\mathbb{R}}^3). (43)

    For any J \in {\mathbb{R}}^3 and for any (x, \omega)\in D\times\Omega let

    \label{hyp:beta} \beta(J, \omega): = \alpha(J, \omega)+h(\omega)J \times B(\omega)+E_a(\omega). (44)

    In order to apply the scale integration procedure, we assume that

    \label{hyp:fsc} \text{the representative function $f$ of $\beta$ is coercive, in the sense of (14), } (45)

    moreover, to ensure uniqueness of a solution (E, J), we assume that

    \label{hyp:smon} \beta\text{ and }\beta^{-1} \text{ are strictly monotone, uniformly with respect to }x\in D. (46)

    As in the previous section \beta_0 stands for the maximal monotone operator represented by f_0 given by (15). For any \varepsilon >0 define

    \beta_\varepsilon (\cdot, x, \omega): = \beta(\cdot, T_{x/\varepsilon }\omega).

    Then \{\beta_\varepsilon (\cdot, x, \omega)\}_{(x, \omega)\in D\times \Omega} is a family of maximal monotone operators on {\mathbb{R}}^3. Let g_\varepsilon \in L^2(D\times \Omega;{\mathbb{R}}^3) with g_\varepsilon \rightharpoonup g in L^2(D;{\mathbb{R}}^3) for some g \in L^2(D;{\mathbb{R}}^3), for \mu-a.e. \omega\in\Omega; assume that

    \label{hyp:divge} div\, g_\varepsilon = 0, \;\;\;\;\; \text{in }\mathcal D'(D), \text{ for $\mu$-a.e. }\omega\in\Omega. (47)

    We are ready to state and prove the homogenization result for the Ohm-Hall model.

    Theorem 4.2. Assume that (43)-(47) are fulfilled. Then

    1. For \mu-a.e. \omega\in\Omega, for any \varepsilon >0 there exists (E_\omega^\varepsilon, J_\omega^\varepsilon)\in L^2(D;{\mathbb{R}}^3) \times L^2(D;{\mathbb{R}}^3) such that

    \begin{align} & E_\omega^\varepsilon(x) = \beta_\varepsilon (J_\omega^\varepsilon(x), x, \omega) & &in\;\;\;D, \label{P:incl-eps}\end{align} (48)
    \begin{align}& {\rm{curl}}\, E_\omega^\varepsilon(x) = g_\varepsilon (x, \omega) & &in\;\;\;D, \label{P:ele-eps}\end{align} (49)
    \begin{align}& {\rm{div}}\, J_\omega^\varepsilon(x) = 0 & &in\;\;\;D, \label{P:magn-eps}\end{align} (50)
    \begin{align}&E_\omega^\varepsilon(x) \times \nu(x) = 0 & &on \;\;\;\partial D. \label{P:bound-eps} \end{align} (51)

    2. There exists (E, J)\in L^2(D;{\mathbb{R}}^3) \times L^2(D;{\mathbb{R}}^3) such that, up to a subsequence,

    \label{eq:conv} E_\omega^\varepsilon \rightharpoonup E \;\;\;\;\;and\;\;\;\;\; J_\omega^\varepsilon \rightharpoonup J (52)

    as \varepsilon \to0, weakly in L^2(D;{\mathbb{R}}^3).

    3. The limit couple (E, J) is a weak solution of

    \begin{align} & E(x) = \beta_0(J(x)) \;\;\;\;\; & &in\;\;\; D, \label{P:incl-hom} \end{align} (53)
    \begin{align}& {\rm{curl}}\, E(x) = g(x)\;\;\;\;\; & &in\;\;\; D, \label{P:ele-hom} \end{align} (54)
    \begin{align}& {\rm{div}}\, J(x) = 0 \;\;\;\;\; & &in \;\;\; D, \label{P:magn-hom} \end{align} (55)
    \begin{align}& E(x) \times \nu(x) = 0 \;\;\;\;\; & &on\;\;\; \partial D. \label{P:bound-hom} \end{align} (56)

    Proof. 1. Assumption (46) implies that \beta is single valued and that almost every realization (x, v)\mapsto \beta(v, T_x\omega) satisfies the boundedness and coercivity assumptions (36) and (37). Therefore, by Theorem 4.1 for almost any \omega\in\Omega and for any \varepsilon >0 problem (48)-(51) has a unique solution.

    2. Let \omega\in \Omega be fixed. By (38) the families \{E_\omega^\varepsilon\}_\varepsilon and \{J_\omega^\varepsilon\}_\varepsilon are weakly relatively compact in L^2(D;{\mathbb{R}}^3), therefore, there exist a subsequence \varepsilon_n \to 0 and a couple (E_\omega, J_\omega)\in L^2(D;{\mathbb{R}}^3) \times L^2(D;{\mathbb{R}}^3) satisfying (52). A priori, (E_\omega, J_\omega) depends on \omega\in \Omega.

    3. The weak formulation of (49)-(51) is:

    \label{eq:weak} \int_D E_\omega^\varepsilon \cdot \text{curl}\, \phi + J_\omega^\varepsilon \cdot \nabla \psi\, dx = \int_D g_\varepsilon \cdot \phi\, dx, (57)

    for all \phi\in \{H^1(D;{\mathbb{R}}^3): \phi \times \nu = 0 \text{ on }\partial D\}, for all \psi\in H^1_0(D). Passing to the limit in (57), one gets

    \int_D E_\omega \cdot \text{curl}\, \phi + J_\omega \cdot \nabla \psi\, dx = \int_D g \cdot \phi\, dx,

    which is exactly the weak formulation of (54)-(56). Equations (49) and (50) imply that \{E_\omega^\varepsilon\}_\varepsilon and \{J_\omega^\varepsilon\}_\varepsilon satisfy also the div-curl compactness condition (28a). Therefore, we can apply the abstract stochastic homogenization Theorem 3.8, which yields

    E_\omega(x) = \beta_0(J_\omega(x)).

    We have thus proved that (E_\omega, J_\omega) is a weak solution of (53)-(56). In order to conclude we have to eliminate the dependence on \omega\in \Omega.

    4. By Lemma 3.6 and assumption (46), \beta_0 and \beta_0^{-1} are strictly monotone, and therefore (53)-(56) admits a unique solution. Thus, (E, J): = (E_\omega, J_\omega) is independent of \omega \in \Omega.

    Another straightforward application of the homogenization theorem 3.8 is given in the framework of deformations in continuum mechanics (see, e.g., [4,Chapter 3]). Elastic materials are usually described through the deformation vector u:D\times(0, T) \to {\mathbb{R}}^3 and the stress tensor \sigma:D\times(0, T) \to {\mathbb{R}}^{3 \times 3}_s. Here D\subset {\mathbb{R}}^3 is the spatial domain and {\mathbb{R}}^{3 \times 3}_s the space of symmetric 3\times 3 matrices. We assume the following constitutive relation relating stress and deformation:

    \label{eq:nlelastic} \sigma(x, t) = \beta(\nabla u(x, t), x), (58)

    where \beta(\cdot, x):{\mathbb{R}}^{3\times 3} \mapsto {\mathbb{R}}^{3\times 3} is a (single-valued) maximal monotone mapping for a.e. x\in D. We couple (58) with the conservation of linear momentum:

    \rho \partial _{t}^{2}u-\text{div}\sigma =F,

    where \rho 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 \rho\partial_t^2 u = 0.

    The following existence and uniqueness result is a classical consequence of the maximal monotonicity of \beta (see, e.g., [7,22]).

    Theorem 4.3. Let D \subset {\mathbb{R}}^3 be a bounded Lipschitz domain. Let \{\beta(\cdot, x)\}_{x\in D} be a family of single-valued maximal monotone operators on {\mathbb{R}}^{3\times 3} that satisfy (36) and (37). Let F \in L^2(D;{\mathbb{R}}^3) be given. Then, there exists \sigma\in L^2(D;{\mathbb{R}}^{3\times 3}) and u\in H^1_0(D;{\mathbb{R}}^3) such that

    \label{Q:estimates} {\|u\|}_{H^1} +{\|\sigma\|}_{L^2}\leq C\left(1+{\|F\|}_{L^2}\right) (59)

    and, denoting by \nu the outward unit normal to \partial D,

    \begin{align} \sigma(x) & = \beta(\nabla u(x), x)\;\;\;\;\; in\;\;\; D, \label{Q:incl}\end{align} (60)
    \begin{align} -div\, \sigma(x) & = F(x) \;\;\;\;\; in\;\;\; D, \label{Q:ele}\end{align} (61)
    \begin{align} u(x) & = 0 \;\;\;\;\; on\;\;\; \partial D. \label{Q:bound} \end{align} (62)

    Moreover, if \beta is strictly monotone uniformly in x\in D, then u is uniquely determined, while if \beta^{-1} is strictly monotone uniformly in x\in D, then \sigma is uniquely determined.

    As above, we consider a family of maximal monotone operators \{\beta(\cdot, \omega)\}_{\omega\in \Omega} on {\mathbb{R}}^{3\times 3}, \beta_0 stands for the maximal monotone operator represented by f_0, and for any \varepsilon >0

    \beta_\varepsilon (\cdot, x, \omega): = \beta(\cdot, T_{x/\varepsilon }\omega)

    defines a family of maximal monotone operators on {\mathbb{R}}^{3\times 3}. Let F_\varepsilon \in L^2(D\times \Omega;{\mathbb{R}}^3) with F_\varepsilon \rightharpoonup F in L^2(D;{\mathbb{R}}^3) for some F \in L^2(D;{\mathbb{R}}^3), for \mu-a.e. \omega\in\Omega. The correspondent homogenization theorem is the following.

    Theorem 4.4. Assume that (45) and (46) are fulfilled. Then

    1. For \mu-a.e. \omega\in\Omega, for any \varepsilon >0 there exist (u_\omega^\varepsilon, \sigma_\omega^\varepsilon)\in H^1_0(D;{\mathbb{R}}^3) \times L^2(D;{\mathbb{R}}^3) such that

    \begin{align} & \sigma_\omega^\varepsilon(x) = \beta_\varepsilon (\nabla u_\omega^\varepsilon(x), x, \omega) & &in\;\;\; D, \label{Q:incl-eps}\end{align} (63)
    \begin{align}& -{\rm{div}}\, \sigma_\omega^\varepsilon(x) = F_\varepsilon (x, \omega) & &in \;\;\;D, \label{Q:ele-eps}\end{align} (64)
    \begin{align}&u_\omega^\varepsilon(x) = 0 & &on\;\;\; \partial D. \label{Q:bound-eps} \end{align} (65)

    2. There exist (u, \sigma)\in H^1_0(D;{\mathbb{R}}^3) \times L^2(D;{\mathbb{R}}^3) such that, up to a subsequence,

    \label{Q:conv} u_\omega^\varepsilon \rightharpoonup u \;\;\;\;\;and\;\;\;\;\; \sigma_\omega^\varepsilon \rightharpoonup \sigma (66)

    as \varepsilon \to0, weakly in H^1(D;{\mathbb{R}}^3) and L^2(D;{\mathbb{R}}^3), respectively.

    3. The limit couple (u, \sigma) is a weak solution of

    \begin{align} & \sigma(x) = \beta_0(\nabla u(x)) & &in\;\;\; D, \label{Q:incl-hom}\end{align} (67)
    \begin{align}& -{\rm{div}}\, \sigma(x) = F(x) & &in \;\;D, \label{Q:ele-hom}\end{align} (68)
    \begin{align}& u(x) = 0 & &on\;\;\; \partial D. \label{Q:bound-hom} \end{align} (69)

    Proof. Steps 1. and 2. follow exactly as in the proof of Theorem 4.2.

    3. The weak formulation of (64)-(65) is the following:

    \label{Q:weak} \int_D \sigma_\omega^\varepsilon \cdot \nabla \phi\, dx = \int_D F_\varepsilon \phi\, dx, (70)

    for all \phi\in H^1_0(D). Passing to the limit as \varepsilon \to 0, one gets

    \int_D \sigma_\omega \cdot \nabla \phi\, dx = \int_D F \phi\, dx,

    which is exactly the weak formulation of (68)-(69). Equation (64) and estimate (59) imply that \{\sigma_\omega^\varepsilon\}_\varepsilon and \{\nabla u_\omega^\varepsilon\}_\varepsilon satisfy also the div-curl compactness condition (28a)

    {{\{\text{div}\sigma _{\omega }^{\varepsilon }\}}_{\varepsilon \ge 0}}\text{ is compact in }{{W}^{-1,2}}(D;{{\mathbb{R}}^{3}}),
    {{\{\text{curl}\nabla u_{\omega }^{\varepsilon }\}}_{\varepsilon \ge 0}}\text{ is compact in }{{W}^{-1,2}}(D;{{\mathbb{R}}^{3\times 3}}).

    Therefore, we can apply the abstract stochastic homogenization Theorem 3.8, (with \sigma in place of J and \nabla u in place of E), which yields

    \sigma_\omega(x) = \beta_0(\nabla u_\omega(x)).

    Finally, the strict monotonicity of the limit operators \beta_0 and \beta_0^{-1} yields uniqueness and therefore independence of \omega for the solution (u, \sigma).

    We would like to thank the anonymous referees for their valuable comments and remarks.

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