We propose and analyze a mathematical model for wave propagation in infinite trees with self-similar structure at infinity. This emphasis is put on the construction and approximation of transparent boundary conditions. The performance of the constructed boundary conditions is then illustrated by numerical experiments.
Citation: Patrick Joly, Maryna Kachanovska, Adrien Semin. Wave propagation in fractal trees. Mathematical and numerical issues[J]. Networks and Heterogeneous Media, 2019, 14(2): 205-264. doi: 10.3934/nhm.2019010
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We propose and analyze a mathematical model for wave propagation in infinite trees with self-similar structure at infinity. This emphasis is put on the construction and approximation of transparent boundary conditions. The performance of the constructed boundary conditions is then illustrated by numerical experiments.
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, \omega) \in \alpha_\varepsilon (x, \omega, u(x, \omega)), $ |
where
$ \label{eq:alphaintro} \alpha_\varepsilon (x, \omega, \cdot): = \alpha\left(T_{x/\varepsilon }\omega, \cdot\right). $ | (1) |
The aim of this paper is to extend existing results where
The outline of the proof is the following: Let
$ 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
$ \label{eq:schemeaux} \text{$(\xi_n, \eta_n)\in X \times X'$ such that $\eta_n = A_n \xi_n$, $(\xi_n, \eta_n) \rightharpoonup (\xi, \eta)$ and $\eta = A \xi$}, $ | (2) |
then, denoting by
$ \langle y_n - \eta_n, x_n-\xi_n\rangle \geq 0. $ |
In order to pass to the limit as
$ \label{eq:schemecomp} \limsup\limits_{n \to \infty}\, \langle g_n, f_n\rangle \leq\langle g, f \rangle \;\;\;\;\; \forall\, (f_n, g_n)\rightharpoonup (f, g)\ \text{in }X \times X', $ | (3) |
which, together with the weak convergence of
$ \langle y - \eta, x-\xi\rangle \geq 0. $ |
By maximal monotonicity of
1. Existence and weak compactness of solutions
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
$ \alpha\ \stackrel{a)}{\longrightarrow}\ f\ \stackrel{b)}{\longrightarrow}\ f_0\ \stackrel{c)}{\longrightarrow}\ \alpha_0, $ |
where a) the random operator
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
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
$ \mathcal{G}_\alpha : = \{ (x, y) \in B\times B':y\in\alpha(x)\} $ |
be its graph. (We will equivalently write
$ \label{mon} (x, y) \in \mathcal{G}_\alpha \;\;\;\;\;\Rightarrow \;\;\;\;\; \langle y-y_0, x-x_0\rangle \ge 0,\;\;\;\;\; \forall (x_0, y_0)\in \mathcal{G}_\alpha $ | (4) |
and strictly monotone if there is
$ \label{mon-strict} (x, y) \in \mathcal{G}_\alpha \;\;\;\;\;\Rightarrow \;\;\;\;\;\langle y-y_0, x-x_0\rangle \ge \theta \|x-x_0\|^2, \;\;\;\;\;\forall (x_0, y_0)\in \mathcal{G}_\alpha. $ | (5) |
We denote by
$ x \in \alpha^{-1}(y)\;\;\;\;\; \Leftrightarrow \;\;\;\;\;y\in \alpha(x). $ |
The monotone operator
$ \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
$fα(x,y):=⟨y,x⟩+sup{⟨y−y0,x0−x⟩:(x0,y0)∈Gα}=sup{⟨y,x0⟩+⟨y0,x⟩−⟨y0,x0⟩:(x0,y0)∈Gα}.$ |
As a supremum of a family of linear functions, the Fitzpatrick function
Lemma 2.1. An operator
$ (x, y) \in \mathcal{G}_\alpha \;\;\;\;\;\Rightarrow \;\;\;\;\; f_{\alpha}(x, y) = \langle y, x \rangle, $ |
while
$ \left\{ fα(x,y)≥⟨y,x⟩ ∀(x,y)∈B×B′fα(x,y)=⟨y,x⟩⟺(x,y)∈Gα. \right. $ |
In the case
1. Let
$ f_\alpha(x, y) = \frac{(y-b+ax)^2}{4a} +bx. $ |
2. Let
$ \alpha(x) = \left\{ 1if x>0,[0,1]if x=0,−1if x<0. \right. $ |
Then
$ f_\alpha(x, y) = \left\{ |x|if |y|≤1,+∞if |y|>1. \right. $ |
and in both cases
We define
$ f(x, y)\ge \langle y, x \rangle \;\;\;\;\;\forall (x, y)\in B\times B'. $ |
We call
$ \label{def:graph} (x, y) \in \mathcal G_{\alpha_f} \Leftrightarrow f(x, y) = \langle y, x \rangle. $ | (6) |
A crucial point is whether
Lemma 2.2. Let
(i) the operator
(ii) the class of maximal monotone operators is strictly contained in the class of operators representable by functions in
Proof. (ⅰ) If
$g(P1+P22)−g(P1)+g(P2)2=14(⟨y1+y2,x1+x2⟩)−12(⟨y1,x1⟩+⟨y2,x2⟩)=14(⟨y1,x2⟩+⟨y2,x1⟩−⟨y1,x1⟩−⟨y2,x2⟩)=−14(⟨y2−y1,x2−x1⟩)>0.$ |
Since
$ f\left( \frac{P_1+P_2}{2}\right) > \frac{f(P_1) +f(P_2)}{2}, $ |
which contradicts the convexity of
(ⅱ) Maximal monotone operators are representable by Lemma 2.1. To see that the inclusion is strict, assume that
$ h(x, y) = \max\{c, f(x, y)\} $ |
clearly belongs to
$ h(x_0, y_0) \geq c > f(x_0, y_0) = \langle y_0, x_0 \rangle, $ |
and thus
Remark 1. When
$ \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(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
$ g^{-1}(R) : = \{ x \in X : g(x) \cap R \neq \emptyset \} $ |
is measurable.
Let
$ \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
(a)
(b)
(c)
As above,
$ \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
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
(a)
(b) for every
$\mu(T_xE) = \mu(E)$ | (11) |
(c) for any measurable function
$ \tilde f(x, \omega) = f(T_x\omega) $ |
is measurable.
Given an
$ {\mathbb E}(f): = \int_\Omega f\, d\mu. $ |
In the context of stochastic homogenization, it is useful to provide an orthogonal decomposition of
$ \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
$ \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
Lemma 2.3. Define the spaces
$Vppot(Ω;Rn):={f∈Lppot(Ω;Rn):E(f)=0},Vpsol(Ω;Rn):={f∈Lpsol(Ω;Rn):E(f)=0}.$ |
The spaces
$ \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
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
$ \mathbb E(f) = \lim\limits_{\varepsilon \to 0}\frac{1}{|K|}\int_K f\big(T_{x/\varepsilon }\omega\big)\, dx $ |
for
Remark 2. Birkhoff's theorem implies that
$ \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
$ {{\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
Let be given a probability space
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
For every fixed
$ f(\xi, \eta, \omega) \ge c\left(|\xi |^p+|\eta |^{p'}\right)+k(\omega). $ | (14) |
We define the homogenised representation
$ 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
1i.e., for all
$ h(x): = \inf\limits_{y\in K}g(x, y) $ |
is weakly lower semicontinuous and coercive. Moreover, if
Proof. Let
$ \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
$ \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
$ \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
$ 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
$ h(\lambda x_1+(1-\lambda)x_2)\le \lambda h(x_1) +(1-\lambda)h(x_2). $ |
Regarding the coercivity of
$ 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
In the proof of Proposition 1 we need the following estimate
Lemma 3.2. For all
$ \int_\Omega |\xi + u(\omega)|^p\, d\mu \geq C \int_\Omega |\xi|^p + |u(\omega)|^p\, d\mu $ |
for all
Proof. Consider the operator
$Φ:Lp(Ω;Rn)→Lp(Ω;Rn)×Lp(Ω;Rn)u↦(E(u),u−E(u)).$ |
Clearly,
$∫Ω|E(u)|pdμ+∫Ω|u(ω)−E(u)|pdμ≤(‖$ |
Apply now the last inequality to
$ \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
$ \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
$ F_{\xi, \eta}(u, v): = \int_\Omega f(\xi+v(\omega), \eta+u(\omega), \omega) \, d\mu. $ |
We prove that the problem
$ 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
$\begin{align*} F_{\xi, \eta}(u, v)&\geq c\int_\Omega |\xi +v(\omega)|^p + |\eta +u(\omega)|^{p'} +k(\omega)\, d\mu \\ & \geq C\int_\Omega |\xi|^p +|u(\omega)|^p + |\eta|^{p'} +|v(\omega)|^{p'} d\mu+\mathbb E(k) \\ &\geq C\left( |\xi|^p + {\|u\|}^p_{L^p(\Omega)} + |\eta|^{p'} + {\|v\|}^{p'}_{L^{p'}(\Omega)} \right)-{\|k\|}_{L^1(\Omega)}. \end{align*}$ |
Thus, for any
$ \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
$ \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
$ \eta \in \alpha_0(\xi) \;\;\;\;\; \Leftrightarrow\;\;\;\;\; f_0(\xi, \eta) = \xi \cdot \eta. $ |
We refer to
Lemma 3.3. Let
$ \label{auxiliary} v(\omega)\in \alpha(u(\omega), \omega), \;\;\;\;\; for \;\;\;\;\; \mu -a.e. \omega\in\Omega. $ | (20) |
Moreover,
$ \mathbb E( v) \in \alpha_0(\mathbb E( u)). $ | (21) |
Proof. Since
$ \label{eq:f0repr} f_0(\xi, \eta) = \xi \cdot \eta. $ | (22) |
Take now
$ \label{eq:hyp1} f_0(\xi, \eta) = \int_\Omega f(\xi+\widetilde{u}(\omega), \eta+\widetilde{v}(\omega), \omega) \, d\mu. $ | (23) |
Since
$\begin{align*} \xi \cdot \eta& = \mathbb E(\xi+\widetilde{u}) \cdot \mathbb E(\eta+\widetilde{v})\\ & \stackrel{(12)}{ = }\int_\Omega (\xi+\widetilde{u}(\omega))\cdot (\eta+\widetilde{v}(\omega))\, d\mu\\ &\stackrel{f \in \mathcal F({{\mathbb{R}}^{n}})}{\le} \int_\Omega f(\xi+\widetilde{u}(\omega), \eta+\widetilde{v}(\omega), \omega) \, d\mu\\ & \stackrel{(23)}{ = } f_0(\xi, \eta)\\ & \stackrel{(22)}{ = }\xi \cdot \eta \end{align*}$ |
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
Lemma 3.3 is also referred to as scale disintegration (see [26,Theorem 4.4]), as it shows that given a solution
Lemma 3.4. Let
$ \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
$ \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
How the properties of
Theorem 3.5. If
$ \int_\Omega f(u(\omega), v(\omega), \omega)\, d\mu < +\infty, $ |
In order to obtain strict monotonicity of
Lemma 3.6. Let
Proof. For all
$ \label{smon1} v_i(\omega)\in \alpha(u_i(\omega), \omega), \;\;\;\;\; \text{for $\mu$-a.e. }\omega\in\Omega $ | (27) |
and
$\begin{align*} (\eta_2-\eta_1)\cdot(\xi_2-\xi_1) & = \int_\Omega (v_2(\omega)-v_1(\omega))\cdot(u_2(\omega)-u_1(\omega)) d\mu \\ & \geq \theta \int_\Omega |u_2(\omega)-u_1(\omega)|^2d\mu \\ & \geq \theta \left|\int_\Omega u_2(\omega)-u_1(\omega)d\mu \right|^2\\ & = \theta \left| \xi_2 -\xi_1 \right|^2. \end{align*}$ |
Let
Lemma 3.7 (Div-Curl lemma, [15]). Let
$ 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
Let
$ (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
$ \label{hom1} E_\omega^0(x) \in \alpha_0(J_\omega^0(x)) \;\;\;\;\; a.e. \;\;\;in \;\;\;\;\;D, $ | (29) |
where
$ 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
$ \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(x, \omega): = u(T_{x/\varepsilon }\omega), \;\;\;\;\; v_\varepsilon(x, \omega): = v(T_{x/\varepsilon }\omega). $ |
For
$ 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
$ \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
$ \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
$\begin{align} & {{\{\text{curl}(E_{\omega }^{\varepsilon }-{{v}_{\varepsilon }}(\cdot ,\omega ))\}}_{\varepsilon }}\text{ is compact in }{{W}^{-1,{p}'}}(D;{{\mathbb{R}}^{n\times n}}), \\ & {{\{\text{div}(J_{\omega }^{\varepsilon }-{{u}_{\varepsilon }}(\cdot ,\omega ))\}}_{\varepsilon }}\text{ is compact in }{{W}^{-1,p}}(D). \\ \end{align}$ |
By (28b), (32), and Lemma 3.7, we can thus pass to the limit as
$ \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
$ (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
$ E_\omega^0(x) \in \alpha_0(J_\omega^0(x)) $ |
for a.e.
Remark 4. In this section's results, the function spaces
$ \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
$ \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
$\begin{align} & \text{curl}E=g, \\ & \text{div}J=0, \\ \end{align}$ |
where the vector field
$ \beta(J, x): = \alpha(J, x) +h(x)J \times B(x) + E_a(x). $ |
A single-valued parameter-dependent operator
$ \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
Theorem 4.1. Let
$ \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
$ \label{eq:estimates} {\|E\|}_{L^2} +{\|J\|}_{L^2}\leq C\left(1+{\|g\|}_{L^2}\right) $ | (38) |
and, denoting by
$ \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
Remark 5. Conditions (40)-(41) have to be intended in the weak sense -see below -while (42) holds in
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
$ \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
$ \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_\varepsilon (\cdot, x, \omega): = \beta(\cdot, T_{x/\varepsilon }\omega). $ |
Then
$ \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
$ \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
$ \label{eq:conv} E_\omega^\varepsilon \rightharpoonup E \;\;\;\;\;and\;\;\;\;\; J_\omega^\varepsilon \rightharpoonup J $ | (52) |
as
3. The limit couple
$ \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
2. Let
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
$ \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(x) = \beta_0(J_\omega(x)). $ |
We have thus proved that
4. By Lemma 3.6 and assumption (46),
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
$ \label{eq:nlelastic} \sigma(x, t) = \beta(\nabla u(x, t), x), $ | (58) |
where
$ \rho \partial _{t}^{2}u-\text{div}\sigma =F, $ |
where
The following existence and uniqueness result is a classical consequence of the maximal monotonicity of
Theorem 4.3. Let
$ \label{Q:estimates} {\|u\|}_{H^1} +{\|\sigma\|}_{L^2}\leq C\left(1+{\|F\|}_{L^2}\right) $ | (59) |
and, denoting by
$ \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
As above, we consider a family of maximal monotone operators
$ \beta_\varepsilon (\cdot, x, \omega): = \beta(\cdot, T_{x/\varepsilon }\omega) $ |
defines a family of maximal monotone operators on
Theorem 4.4. Assume that (45) and (46) are fulfilled. Then
1. For
$ \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
$ \label{Q:conv} u_\omega^\varepsilon \rightharpoonup u \;\;\;\;\;and\;\;\;\;\; \sigma_\omega^\varepsilon \rightharpoonup \sigma $ | (66) |
as
3. The limit couple
$ \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
$ \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
$ {{\{\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_\omega(x) = \beta_0(\nabla u_\omega(x)). $ |
Finally, the strict monotonicity of the limit operators
We would like to thank the anonymous referees for their valuable comments and remarks.
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