Maximum degree and minimum degree spectral radii of some graph operations

1.   Introduction

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

where $x \in {{\mathbb{R}}^{n}}$ and $\omega$ is a parameter in a probability space $\Omega$. In the spirit of [10,Chapter 7], the random operator $\alpha_\varepsilon$ is obtained from a stationary operator $\alpha$ via an ergodic dynamical system $T_x:\Omega \to \Omega$

The aim of this paper is to extend existing results where $\alpha$ 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], $\Gamma$-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 $\Omega$. 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 $A_n:X \to X'$ be a sequence of monotone operators, and for every $n\in{\mathbb N}$ let $(x_n, y_n)\in X \times X'$ be a point in the graph of $A_n$, i.e., such that $y_n = A_n x_n$ for all $n \in {\mathbb N}$. Assuming that $(x_n, y_n) \rightharpoonup (x, y)$ in $X \times X'$ and that we already know the limit maximal monotone operator $A:X \to X'$, a classical question of functional analysis is:

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

then, denoting by $\langle y, x \rangle$ the duality pairing between $x\in X$ and $y\in X'$, by monotonicity of $A_n$

In order to pass to the limit as $n \to \infty$, 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

which, together with the weak convergence of $(x_n, y_n)$ and $(\xi_n, \eta_n)$, yields

By maximal monotonicity of $A$, if the last inequality is satisfied for all $(\xi, \eta)$ such that $\eta = A\xi$, 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 $(x_n, y_n) \rightharpoonup (x, y)$;

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:

where a) the random operator $\alpha(\omega)$ is represented through a variational inequality involving Fitzpatrick's representation $f(\omega)$; b) the representation is "scale integrated" to a $\omega$-independent effective $f_0$; c) a maximal monotone operator $\alpha_0$ is associated to $f_0$. In Theorem 3.8 we prove that $\alpha_0$ is the homogenised limit of $\alpha_\varepsilon$.

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).

2.   Notation and preliminaries

We use the notation $a\cdot b$ for the standard scalar product for vectors in ${{\mathbb{R}}^{n}}$. The arrows $\rightharpoonup$ and $\stackrel{\ast}{\rightharpoonup}$ denote weak and weak$^*$ convergence, respectively. As usual, $\mathcal D(D)$ stands for the space of $C^\infty$-functions with compact support in $D\subset {{\mathbb{R}}^{n}}$; its dual is denoted by $\mathcal D'(D)$.

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 $\mathcal{P}(B')$ the power set of $B'$, and by $\langle y, x \rangle$ the duality pairing between $x\in B$ and $y \in B'$. Let $\alpha \colon B \rightarrow \mathcal{P}(B')$ be a set-valued operator and let

be its graph. (We will equivalently write $y \in \alpha(x)$ or $(x, y)\in \mathcal{G}_\alpha$.) The operator $\alpha$ is said to be monotone if

and strictly monotone if there is $\theta>0$ such that

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

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

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

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

while $\alpha$ is maximal monotone if and only if

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

2. Let

Then

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

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:

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

Since $f \geq g$ and $f(x_i, y_i) = g(x_i, y_i)$ for $i = 1, 2$, 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 $\alpha_f$ 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

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

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

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

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

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

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:

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].

2.2.   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 $(\Omega, \mathcal{A}, \mu)$ 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

(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

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

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

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

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

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

and the relations

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

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

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

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

3.   Stochastic homogenization

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).

3.1.   Stochastic scale integration/disintegration

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

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

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.

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

Let

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

Therefore

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 $g$

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 $g$

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

Regarding the coercivity of $h$, denote

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

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

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

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

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

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

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

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

Thus, for any $M\ge 0$, the set

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)

which yields the conclusion.

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

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

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

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

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

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

from which we obtain

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

then

Proof. By (25) and (12)

On the other hand, by definition of $f_0$,

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

$\bullet$ $f$ 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

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

3.2.   Main result

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

In addition, assume that

Then

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

such that

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

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

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

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

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

Equation (30) implies

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

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

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

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

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

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

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

such that

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

4.   Applications

4.1.   The Ohm-Hall model

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

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:

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

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

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. $x\in D, \, \forall v \in {\mathbb{R}}^3$

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

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

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

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

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

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

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

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

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

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,

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

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

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:

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

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

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 $\omega \in \Omega$.

4.2.   Nonlinear elasticity

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:

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:

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

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

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 $\varepsilon >0$

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

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,

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

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:

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

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)

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

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

Acknowledgments

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