Homogenization and correctors of Robin problem for linear stochastic equations in periodically perforated domains

Abstract

In this paper, we present homogenization and corrector results for stochastic linear parabolic equations in periodically perforated domains with non-homogeneous Robin conditions on the holes. We use the periodic unfolding method and probabilistic compactness results. Homogenization results presented in this paper are stochastic counterparts of some fundamental work given in [Cioranescu, Donato and Zaki in Port. Math. (N.S.) 63 (2006), 467–496]. We show that the sequence of solutions of the original problem converges in suitable topologies to the solution of a homogenized problem, which is a parabolic stochastic equation in fixed domain with Dirichlet condition on the boundary. In contrast to the two scale convergence method, the corrector results obtained in this paper are without any additional regularity assumptions on the solutions of the original problems.

Keywords

Periodic unfolding SPDEs Robin problem perforated domains probabilistic compactness results

1. Introduction

Homogenization is concerned with the description of macroscopic properties of heterogeneous materials immersed in various environments in terms of their microscopic properties. These heterogeneous materials have microscopic properties which are spatially oscillatory with the frequencies of oscillations depending on the distribution of the constituents of the underlying materials. This renders the direct numerical treatment of models of composites very difficult and in most cases impractical. Indeed, meshes in numerical schemes for approximating computational domains occupied by a composite must be very fine to capture microscopic behaviors. Homogenization seeks to mitigate this problem by approximating the underlying composite by a homogeneous one readily amenable to numerical computations. A typical homogenization problem involves the study of a physical process (heat conduction, deformation, etc.) in a composite. The first rigorous mathematical work in homogenization can be traced back to as late as the 1906s in the pioneering work of Khruslov and Marchenko [16], in which they dealt with the harder case of homogenization in general domains not subjected to any periodicity assumption. Also the work of Spagnolo [36], in which Spagnolo introduced the notion of G-convergence for certain types of linear operators and proved a G-compactness result which is the basis of the G-convergence approach to homogenization. One limitation of G-convergence is that the operator involved must be symmetric. One limitation of G-convergence is that the operator involved must be symmetric. In order to accommodate non-symmetric matrices, G-convergence is extended to H-convergence by Tartar and Murat [37] and [23], resulting in a sequential H-compactness theorem. It is imperative to stress that the techniques used to arrive at G-compactness and H-compactness are fundamentally different: the former uses semigroup theory for linear operators whereas the latter relies on compensated compactness which permeates nonlinearities. However, as in the case of G-convergences, H-convergence does not provide a mechanism for the identification of the H-limit. Another approach to homogenization that suffers from a similar problem is the Γ-convergence of Giorgi [11] which deals with the asymptotic analysis of optimization problems usually depending on some parameters destined to tend to zero or infinity. Compared to G-convergence and H-convergence, Γ-convergence imposes less regularity on the functions involved. Hence, it is able to deal with certain types of discontinuities. In 1989, Nguetseng [24] introduced a general convergence result to study the homogenization of boundary value problems with periodic rapidly oscillating coefficients. What makes the convergence of Nguetseng so revolutionary in the field of homogenization is, the weak limit he obtained which depends on two variables, the additional variable is a reflection of the micro oscillations in the sequence, and that is not captured in the classical weak limits. In 1992, Allaire [1] named the convergence of Nguetseng “two-scale convergence” and further developed and investigated its properties. He introduced several types of admissible oscillating test functions and he applied the two-scale convergence to the homogenization of linear and nonlinear boundary value problems. All these methods have been well established for homogenization of deterministic partial differential equations (PDEs), see for instance [1,5,9,24–26,33,34,37,38] and the references therein.

As far as the homogenization of stochastic partial differential equations (SPDEs) is concerned, very few results are known [2,15,17,19–22,31,32]. It should be noted that the first attempt to obtain homogenization results for SPDEs by means of the unfolding method was [18] where the author studied the homogenization of linear SPDEs of both parabolic and hyperbolic types in fixed domains.

In this paper, we adapt the periodic unfolding method in perforated domains, to treat linear stochastic partial differential equations with non-homogeneous boundary Robin conditions on the holes. Periodic unfolding method was introduced in 2002 by D. Cioranescu, A. Damlamian and G. Griso in [7], see also [8] to study homogenization of boundary value problems with periodically oscillating coefficients in fixed domains and was extended to tackle problems in periodically perforated domains by Cioranescu, Donato, Griso and Zaki, [10] and [6]. Later Gaveau in [13] uses the periodic unfolding method to obtain homogenization results for time-dependent problems in fixed domains, Donato and Zhanying in [12], see also [39] introduced the time-dependent unfolding operator to treat homogenization of the wave equation in perforated domains. We also note the newly extension of unfolding method from the periodic to the random setting by Heida, Neukamm and Varga [14].

We are concerned with the homogenization of the initial boundary value problem with oscillating data: $\begin{matrix} (P_{ϵ}) \{\begin{matrix} d u^{ϵ} - div A_{ϵ} \nabla u^{ϵ} d t = f^{ϵ} d t + g^{ϵ} d W in Q^{ϵ} \times (0, T) \\ u^{ϵ} (x, 0) = b^{ϵ} (x) in Q^{ϵ}, u^{ϵ} = 0 on (\partial Q^{ϵ} ∖ \partial S_{int}^{ϵ}) \times (0, T) \\ A_{ϵ} \nabla u^{ϵ} . n + α ϵ u^{ϵ} = ϵ h^{ϵ} on \partial S_{int}^{ϵ} \times (0, T); \end{matrix} \end{matrix}$ where $ϵ > 0$ is a sufficiently small parameter which ultimately tends to zero, $T > 0$ , Q is an open and bounded (at least Lipschitz) subset of $R^{n}$ , $Y = (0, l_{1}) \times \dots \times (0, l_{n})$ , S is an open set included in Y such that $\partial S$ does not meet the summits of Y and $Y^{*} = Y ∖ S$ . From which we have the perforated domain $\begin{matrix} Q^{ϵ} = Q ∖ S^{ϵ}, S^{ϵ} = ⋃_{ξ \in Z^{n}} ϵ (ξ + S), \end{matrix}$ $S_{int}^{ϵ}$ is the set of holes that do not intersect the boundary $\partial Q$ . We also assume that $Q^{ϵ}$ is connected, the holes can meet the $\partial Q$ and $| \partial Q | = 0$ . $W = (W (t))$ ( $t \in [0, T]$ ) an m-dimensional standard Wiener process defined on a given filtered complete probability space $(Ω, F, P, {(F_{t})}_{0 ⩽ t ⩽ T})$ ; $E$ denotes the corresponding mathematical expectation, $f^{ϵ} (x, t) = f (\frac{x}{ϵ}, t)$ , $g^{ϵ} (x, t) = g (\frac{x}{ϵ}, t)$ , $h^{ϵ} (x, t) = h (\frac{x}{ϵ}, t)$ , α is a constant, and $A_{ϵ} (x) = A (\frac{x}{ϵ}) = {(a_{i, j} (\frac{x}{ϵ}))}_{1 ⩽ i, j ⩽ n}$ is an $n \times n$ matrix, such that

There exists a constant $c > 0$ satisfying $\begin{array}{l} \sum_{i, j = 1}^{n} a_{i, j} ξ_{i} ξ_{j} ⩾ c \sum_{i = 1}^{n} ξ_{i}^{2} for ξ \in R^{n}, \end{array}$

$a_{i, j} \in L^{\infty} (R^{n})$ , $i, j = 1, \dots, n$ .

The components $a_{i, j}$ , are Y-periodic

The differential $g^{ϵ} d W$ is understood in the sense of Itô.

In order to state some facts we need to introduce some spaces. We consider the well known spaces $L^{2} (Q)$ , $H^{1} (Q)$ , $H_{0}^{1} (Q)$ , $C_{per}^{\infty} (Y)$ the subspace of $C^{\infty} (R^{n})$ of Y-periodic functions. Let $H_{per}^{1} (Y)$ be the closure of $C_{per}^{\infty} (Y)$ in the $H^{1}$ -norm, and $H_{per} (Y)$ be the subspace of $H_{per}^{1} (Y)$ with zero mean on Y. $\begin{array}{l} V_{ϵ} = {ϕ \in H^{1} (Q^{ϵ}) | ϕ = 0 on \partial Q}, H_{per, S_{0}}^{1} (Y^{*}) = {ψ \in H_{per}^{1} (Y^{*}) | ψ = 0 in S} \\ W_{per} (Y^{*}) = {v \in H_{per}^{1} (Y^{*}) | M_{Y^{*}} (v) = 0} . \end{array}$ For a Banach space X, and $1 ⩽ p ⩽ \infty$ , we denote by $L^{p} (0, T; X)$ the space of measurable functions $ϕ : t \in [0, T] ⟶ ϕ (t) \in X$ and p-integrable with the norm $\begin{matrix} ‖ ϕ ‖_{L^{p} (0, T; X)} = {(\int_{0}^{T} ‖ ϕ ‖_{X}^{p} d t)}^{\frac{1}{p}}, 0 ⩽ p < \infty . \end{matrix}$ When $p = \infty$ , $L^{\infty} (0, T; X)$ is the space of all essentially bounded functions on the closed interval $[0, T]$ with values in X equipped with the norm $\begin{matrix} ‖ ϕ ‖_{L^{\infty} (0, T; X)} = ess sup_{[0, T]} ‖ ϕ ‖_{X} < \infty . \end{matrix}$ For $1 ⩽ q ⩽ \infty$ , the space $L^{q} (Ω, F, P; L^{p} (0, T; X))$ consists of all random functions $ϕ : (ω, t) \in Ω \times [0, T] ⟶ ϕ (ω, t, \cdot) \in X$ such that $ϕ (ω, t, x)$ is measurable with respect to $(ω, t)$ and all t ϕ is $F_{t}$ -measurable in ω. We endow this space with the norm $\begin{matrix} ‖ ϕ ‖_{L^{q} (Ω, F, P; L^{p} (0, T; X))} = {(E ‖ ϕ ‖_{L^{p} (0, T; X)}^{q})}^{1 / q} . \end{matrix}$ When $p = \infty$ , the norm in the space $L^{q} (Ω, F, P, L^{\infty} (0, T; X))$ is given by $\begin{matrix} ‖ ϕ ‖_{L^{q} (Ω, F, P; L^{\infty} (0, T; X))} = {(E ‖ ϕ ‖_{L^{\infty} (0, T; X)}^{q})}^{1 / q} . \end{matrix}$ It is well known that under the above norm, $L^{q} (Ω, F, P, L^{p} (0, T; X))$ is a Banach space.

When there is no ambiguity, we shall omit ω in the notation of $u^{ϵ}$ for the sake of brevity. In the following we introduce the notion of the strong probabilistic solution for our problem.

Definition 1.
We define the strong probabilistic solution of the problem $(P_{ϵ})$ on the prescribed filtered probability space $(Ω, F, P, {F_{t}}_{t \in [0, T]})$ as a process $\begin{matrix} u^{ϵ} : Ω \times [0, T] ⟶ V_{ϵ}, \end{matrix}$ such that
$u^{ϵ} \in L^{2} (Ω, F, P; C ([0, T]; V_{ϵ})) \cap L^{2} (Ω, F, P; L^{2} (0, T; L^{2} (Q^{ϵ})))$ ,

For any $(t, ω) \in (0, T) \times Ω$ , $u^{ϵ} (t, \cdot)$ satisfies $\begin{array}{l} \int_{0}^{t} {(d u^{ϵ} (t, \cdot), ϕ)}_{L^{2} (Q^{ϵ})} d s + \int_{0}^{t} {(A_{ϵ} \nabla u^{ϵ} (s, \cdot), \nabla ϕ)}_{{[L^{2} (Q^{ϵ})]}^{n}} d s + α ϵ \int_{0}^{t} {(u^{ϵ}, ϕ)}_{L^{2} (\partial S^{ϵ})} d s \\ = \int_{0}^{t} {(f^{ϵ} (s, \cdot) ϕ)}_{L^{2} (Q^{ϵ})} d s + ϵ \int_{0}^{t} {(h^{ϵ}, ϕ)}_{L^{2} (\partial S^{ϵ})} d s + {(\int_{0}^{t} g^{ϵ} (s, \cdot) d W (s), ϕ)}_{L^{2} (Q^{ϵ})}, \end{array}$ $\forall ϕ \in C_{0}^{\infty} (Q)$ .

For the existence and uniqueness of strong probabilistic solution of $(P_{ϵ})$ we refer to [28], see also [29]. The corresponding result follows.
Theorem 1.
Suppose that the assumptions (A1)–(A3) hold. Let
$b^{ϵ} \in L^{2} (Q)$ ,

$f^{ϵ} \in L^{2} (Q^{ϵ} \times (0, T))$ , $g^{ϵ} \in L^{2} (Q^{ϵ} \times (0, T))$ .

$h^{ϵ} (x, \cdot)$ is Y-periodic in the first argument in $L^{2} (\partial S^{ϵ} \times (0, T))$ .

Then for a fixed $ϵ > 0$ , the problem $(P_{ϵ})$ has a unique strong probabilistic solution $\begin{matrix} u^{ϵ} \in L^{2} (Ω, F, P; C ([0, T]; H_{0}^{1} (Q))) \cap L^{2} (Ω, F, P; L^{2} (0, T; L^{2} (Q)), \end{matrix}$ in the sense of definition 1 .

The goals of this paper are to prove the following:
As $ϵ \to 0$ the solutions $u^{ϵ}$ of problem $(P_{ϵ})$ converges in suitable sense to a solution u of the following SPDE in fixed domain, referred to throughout the paper as problem $(P)$ $\begin{array}{l} θ d u - div A_{0} \nabla u d t + α \frac{| \partial S |}{| Y |} u d t \\ = θ f d t + \frac{| \partial S |}{| Y |} M_{(\partial S \times (0, T))} (h) d t + θ g d W in Q \times (0, T), \\ u (x, 0) = u_{0} (x) in L^{2} (Q), u = 0 on \partial Q \times (0, T), \end{array}$ where $A_{0}$ is a constant elliptic matrix defined by $\begin{array}{l} A_{0} = \frac{1}{| Y |} \int_{Y} (A (y) - A (y) χ (y)) d y, \end{array}$ and $χ \in H_{per} (Y^{})$ is the unique solution of the following boundary value problem. $\begin{matrix} (1) & \{\begin{matrix} {div}_{y} (A (y) \nabla_{y} χ (y)) = \nabla_{y} \cdot A (y) in Y^{} \\ χ is Y periodic, \end{matrix} \end{matrix}$ and $θ = \frac{| Y^{*} |}{| Y |}$ .

We prove some corrector results.

2. The time-dependent unfolding operator in perforated domains

The unfolding operator has been adapted to treat time-dependent problems in fixed domains in [13] and in perforated domains in [12]. In this section, we list some of the properties of the time-dependent unfolding operator in perforated domains, see [10]. We start by introducing the following notations. Let Q be a bounded and open subset of $R^{n}$ with Lipschitz boundary and $Y = (0, l_{1}) \times \dots \times (0, l_{n})$ , $(l_{1}, \dots, l_{n}) \in R^{n}$ be the reference cell. For $z \in R^{n}$ , let ${[z]}_{Y}$ to be the unique integer part of z such that $z - {[z]}_{Y} \in Y$ . Set ${z}_{Y} = z - {[z]}_{Y}$ to denote the fractional part. Then for $x \in R^{n}$ and $ϵ > 0$ , we have $\begin{matrix} x = ϵ ([\frac{x}{ϵ}] + {\frac{x}{ϵ}}) . \end{matrix}$ The following notations will be used frequently. $\begin{array}{l} Λ_{ϵ} = {ξ : ϵ (ξ + \bar{Y}) \cap Q \neq \emptyset}, \tilde{Q^{ϵ}} = Int {⋃_{ξ \in Λ_{ϵ}} ϵ (ξ + Y)}, \end{array}$ where $ξ = \sum_{i = 1}^{n} k_{i} l_{i}$ , $(k_{1}, \dots, k_{n}) \in Z^{n}$ . $\tilde{Q^{ϵ}}$ represents the smallest finite union of $ϵ Y$ cells containing Q. Now we recall the definition of the periodic unfolding operator for functions defined on the cylindrical $Q \times (0, T)$ .

Definition 2.
Let $ϕ \in L^{p} (0, T; L^{q} (Q^{ϵ}))$ , $p \in [1, \infty]$ , $q \in [1, \infty)$ . We define the unfolding operator $T_{}^{ϵ} : L^{p} (0, T; L^{q} (Q^{ϵ})) \to L^{p} (0, T; L^{q} (R^{n} \times Y^{}))$ as follows: $\begin{array}{l} T_{}^{ϵ} (ϕ) (x, y, t) = \tilde{ϕ} (ϵ [\frac{x}{ϵ}] + ϵ y, t) a.e. for (x, y, t) \in R^{n} \times Y^{} \times (0, T), \end{array}$ where $\tilde{v}$ is the extension by zero outside $Q^{ϵ}$ for any function defined on $L^{q} (Q^{ϵ})$ .

From the definition, it is easy to see that the unfolding operator $T_{}^{ϵ}$ satisfies the following
$T_{}^{ϵ} (ϕ ψ) = T_{}^{ϵ} (ϕ) T_{}^{ϵ} (ψ)$ $\forall ϕ$ , $ψ \in L^{p} (0, T; L^{q} (Q^{ϵ}))$ .

$T_{}^{ϵ} (ϕ ψ) = ϕ T_{}^{ϵ} (ψ) \forall ϕ \in L^{p} (0, T)$ and $ψ \in L^{q} (Q^{ϵ})$ .

If $ϕ \in L^{p} (Y^{})$ , is Y-periodic and has the form $ϕ^{ϵ} (x) = ϕ (\frac{x}{ϵ})$ , then $\begin{array}{l} (2) & T_{}^{ϵ} (ϕ) (x, y) = ϕ (y) . \end{array}$
The following proposition presents some of the properties of the unfolding operator. For the proof we refer to [10]. Proposition 1.
The unfolding operator $T_{}^{ϵ}$ satisfies the following properties:*
If $ϕ \in L^{p} (0, T; L^{1} (Q^{ϵ}))$ , $p \in [1, \infty]$ . Then the following integration formula holds $\begin{array}{l} \int_{0}^{T} \int_{Q^{ϵ}} ϕ (x, t) d x d t & = \frac{1}{| Y |} \int_{0}^{T} \int_{R^{n} \times Y^{}} T_{}^{ϵ} (ϕ) (x, y, t) d x d y d t \\ = \frac{1}{| Y |} \int_{0}^{T} \int_{\tilde{Q^{ϵ}} \times Y^{}} T_{}^{ϵ} (ϕ) (x, y, t) d x d y d t \end{array}$

Let $ϕ \in L^{p} (0, T; L^{q} (Q^{ϵ}))$ , $p \in [1, \infty]$ , $q \in [1, \infty)$ . Then $\begin{array}{l} (3) & {‖ T_{}^{ϵ} (ϕ) ‖}_{L^{p} (0, T; L^{q} (R^{n} \times Y^{}))} = ⩽ | Y |^{\frac{1}{q}} ‖ ϕ ‖_{L^{p} (0, T; L^{q} (Q^{ϵ}))} . \end{array}$

For $p \in [1, \infty]$ and $q \in [1, \infty)$ , let $ϕ^{ϵ}$ be a bounded sequence in $L^{p} (0, T; L^{q} (Q^{ϵ}))$ . Then $\begin{array}{l} (4) & \int_{0}^{T} \int_{Q^{ϵ}} ϕ^{ϵ} ψ d x d t = \frac{1}{| Y |} \int_{0}^{T} \int_{R^{n} \times Y^{}} T_{}^{ϵ} (ϕ^{ϵ}) T_{}^{ϵ} (ψ) d x d y d t, \end{array}$ for every* $ψ \in L^{p^{'}} (0, T; L^{q^{'}} (Q^{ϵ})) (\frac{1}{p} + \frac{1}{p^{'}} = 1)$ , $(\frac{1}{q} + \frac{1}{q^{'}} = 1)$ .

For $p, q \in (1, \infty]$ let $ϕ^{ϵ} \in L^{p} (0, T; L^{q} (Q^{ϵ}))$ and $ψ^{ϵ} \in L^{p^{'}} (0, T; L^{q^{0}} (Q^{ϵ}))$ where $(\frac{1}{p} + \frac{1}{p^{'}} = 1)$ , $(\frac{1}{q} + \frac{1}{q^{0}} < 1)$ such that $\begin{array}{l} {‖ ϕ^{ϵ} ‖}_{L^{p} (0, T; L^{q} (Q^{ϵ}))} < \infty, {‖ ψ^{ϵ} ‖}_{L^{p^{'}} (0, T; L^{q^{0}} (Q^{ϵ}))} < \infty . \end{array}$ Then $\begin{array}{l} (5) & \int_{0}^{T} \int_{Q^{ϵ}} ϕ^{ϵ} ψ^{ϵ} d x d t = \frac{1}{| Y |} \int_{0}^{T} \int_{R^{n} \times Y^{}} T_{}^{ϵ} (ϕ^{ϵ}) T_{}^{ϵ} (ψ^{ϵ}) d x d y d t, \end{array}$

Let* $ϕ \in L^{p} (0, T; L^{q} (Q))$ , $p, q \in [1, \infty)$ . Then $\begin{array}{l} (6) & T_{}^{ϵ} (ϕ) \to \tilde{ϕ} strongly in L^{p} (0, T; L^{q} (R^{n} \times Y^{}) . \end{array}$

Let $ϕ^{ϵ} \in L^{p} (0, T; L^{q} (Q))$ , $p, q \in [1, \infty)$ such that $\begin{matrix} ϕ^{ϵ} \to ϕ strongly in L^{p} (0, T; L^{q} (Q), \end{matrix}$ then $\begin{array}{l} (7) & T_{}^{ϵ} (ϕ^{ϵ}) \to \tilde{ϕ} strongly in L^{p} (0, T; L^{q} (R^{n} \times Y^{}) . \end{array}$

Let $ϕ^{ϵ} \in L^{p} (0, T; L^{q} (Q^{ϵ}))$ , $p \in (1, \infty]$ and $q \in (1, \infty)$ such that $\begin{matrix} {‖ ϕ^{ϵ} ‖}_{L^{p} (0, T; L^{q} (Q^{ϵ}))} ⩽ C . \end{matrix}$ If $\begin{array}{l} T_{}^{ϵ} (ϕ^{ϵ}) ⇀ \hat{ϕ} weakly in L^{p} (0, T; L^{q} (R^{n} \times Y^{}), \end{array}$ then $\begin{array}{l} (8) & \tilde{ϕ^{ϵ}} ⇀ θ M_{Y^{}} (\hat{ϕ}) weakly in L^{p} (0, T; L^{q} (R^{n}), \end{array}$ where* $\tilde{v}$ is the extension by zero to Q for any function v defined on a subset of Q.

2.1. The time-dependent macro–micro decomposition in perforated domains

In this section, we present the notion of time-dependent Macro-Micro decomposition, we follow the presentation of [10]. We have the following notation $\begin{matrix} Q_{2 ϵ ρ (Y)}^{ϵ} = {x \in Q | d (x, \partial Q) > 2 ϵ ρ (Y)}, {\hat{Q^{ϵ}}}_{2 ϵ ρ (Y)} = int {⋃_{ξ \in Π_{ϵ}^{2 ϵ ρ (Y)}} ϵ (ξ + \bar{Y})}, \end{matrix}$ where $\begin{matrix} Π_{ϵ}^{2 ϵ ρ (Y)} = {ξ \in Z^{n} | ϵ (ξ + \bar{Y}) \subset Q_{2 ϵ ρ (Y)}^{ϵ}} \end{matrix}$ Now, for $p, q \in [1, \infty]$ and $ϕ \in L^{p} (0, T; L^{q} (Q^{ϵ}))$ we define the decomposition $ϕ = Q^{ϵ} (ϕ) + R^{ϵ}$ , where $Q^{ϵ} (ϕ)$ is defined on the sub-domain ${\hat{Q^{ϵ}}}_{2 ϵ ρ (Y)}$ as follows:

For every node $ϵ ξ$ in ${\hat{Q^{ϵ}}}_{2 ϵ ρ (Y)}$ we write $\begin{matrix} Q^{ϵ} (ϵ ξ, t) = \frac{1}{| B_{ϵ} |} \int_{B_{ϵ}} ϕ (ϵ ξ + ϵ z, t) d z . \end{matrix}$ Not that the ball $B_{ϵ}$ centred at any of the nodes of ${\hat{Q^{ϵ}}}_{2 ϵ ρ (Y)}$ is entirely contained $Q^{ϵ}$ and do not touch the hole $ϵ S$ .

The remainder $R^{ϵ}$ is designed to capture the oscillation.

The following compactness result is the time-dependent version of [10, Theorem 3.2] to functions defined on evolution spaces. This result plays a key role in obtaining the homogenization result as well as the corrector results.

Lemma 1.
Let $ϕ^{ϵ}$ be a sequence of functions in $L^{2} (0, T; H^{1} (Q^{ϵ}))$ , such that $\begin{array}{l} {‖ ϕ^{ϵ} ‖}_{L^{p} (0, T; H^{1} (Q^{ϵ}))} < \infty . \end{array}$ Then up to a subsequence there exist a couple of functions $(ϕ, \hat{ϕ})$ with $ϕ \in L^{2} (0, T; H^{1} (Q))$ and $\hat{ϕ} \in L^{2} (0, T; L^{2} (Q; H_{per}^{1} (Y)))$ such that $\begin{array}{l} (9) & Q^{ϵ} (ϕ^{ϵ}) ⇀ ϕ weakly in L^{2} (0, T; H_{l o c}^{1} (Q)), \\ (10) & T_{}^{ϵ} (ϕ^{ϵ}) ⇀ ϕ weakly in L^{2} (0, T; L_{l o c}^{2} (Q; H_{per}^{1} (Y))), \\ (11) & \frac{1}{ϵ} T_{}^{ϵ} (R^{ϵ} (ϕ^{ϵ})) ⇀ \hat{ϕ} weakly in L^{2} (0, T; L_{l o c}^{2} (Q; H_{per}^{1} (Y))), \\ (12) & T_{*}^{ϵ} (\nabla_{x} ϕ^{ϵ}) ⇀ \nabla_{x} ϕ + \nabla_{y} \hat{ϕ} weakly in L^{2} (0, T; L_{l o c}^{2} (Q \times Y)) . \end{array}$

2.2. The time-dependent averaging operator in perforated domains

In this subsection we introduce the time-dependent averaging operator in perforated domains and give some of its properties.

Definition 3.
Let $ϕ \in L^{2} (0, T; L^{2} (R^{n} \times Y))$ . We define $\begin{array}{l} U^{ϵ} (ϕ) (x, t) = \frac{1}{| Y^{} |} \int_{Y^{}} ϕ (ϵ {[\frac{x}{ϵ}]}_{Y} + ϵ z, ϵ {\frac{x}{ϵ}}_{Y}, t) d z, a.e. for (x, t) \in R^{n} \times (0, T) . \end{array}$

In what follows we give some of the properties of the averaging operator $U^{ϵ}$ , see [12] and [10] for more details. Lemma 2.

Let $ϕ \in L^{2} (0, T; L^{2} (R^{n}))$ . Then $\begin{array}{l} U_{}^{ϵ} (ϕ) \to ϕ strongly in L^{2} (0, T; L^{2} (R^{n})) . \end{array}$

Let* $ϕ \in L^{2} (0, T; L^{2} (R^{n} \times Y^{}))$ . Then* $\begin{array}{l} T_{}^{ϵ} (χ_{Q^{ϵ}} (U_{}^{ϵ} (ϕ))) \to ϕ strongly in L^{p} (0, T; L^{q} (R^{n} \times Y^{})) \end{array}$ and* $\begin{array}{l} U_{}^{ϵ} (ϕ) ⇀ \frac{1}{| Y^{} |} \int_{Y} ϕ (\cdot, \cdot y) d y weakly in L^{2} (0, T; L^{2} (R^{n})) . \end{array}$

Let $ϕ^{ϵ}$ be a sequence in $L^{2} (0, T; L^{2} (Q^{ϵ}))$ and ϕ in $L^{2} (0, T; L^{2} (R^{n} \times Y^{}))$ . Then the following assertions are equivalent*

$‖ T_{}^{ϵ} (ϕ^{ϵ}) - ϕ ‖_{L^{2} (0, T; L^{2} (R^{n} \times Y^{}))} \to 0$ .

$‖ \tilde{ϕ^{ϵ}} - U_{}^{ϵ} (ϕ) ‖_{L^{2} (0, T; L^{2} (R^{n}))} \to 0$ .

Let* $ϕ^{ϵ}$ be a sequence in $L^{2} (0, T; L^{2} (Q^{ϵ}))$ and ϕ in $L^{2} (0, T; L^{2} (R^{n}))$ . Then the following assertions are equivalent

$‖ T_{}^{ϵ} (ϕ^{ϵ}) - ϕ ‖_{L^{2} (0, T; L_{loc}^{2} (R^{n} \times Y^{}))} \to 0$

$‖ \tilde{ϕ^{ϵ}} - ϕ ‖_{L^{2} (0, T; L_{loc}^{2} (R^{n}))} \to 0$ .

2.3. The time-dependent boundary unfolding operator in perforated domains

This subsection, is devoted to the adaptation of the boundary unfolding operator in perforated domains in [12] and [10] to treat evolution problems.

Definition 4.
Let $ϕ \in L^{2} (0, T; L^{2} (\partial S^{ϵ}))$ . We define the boundary unfolding operator $T_{b}^{ϵ} : L^{2} (0, T; L^{2} (\partial S^{ϵ})) \to L^{2} (0, T; L^{2} (R^{n} \times \partial S))$ as follows: $\begin{matrix} T_{b}^{ϵ} (ϕ) (x, y, t) = ϕ (ϵ [\frac{x}{ϵ}] + ϵ y, t) a.e. for (x, y, t) \in R^{n} \times \partial S \times (0, T) . \end{matrix}$

We have the following proposition
Proposition 2.

For $ψ \in L^{2} (0, T; L^{2} (\partial S^{ϵ}))$ , we have $\begin{array}{l} \int_{0}^{T} \int_{\partial S^{ϵ}} ψ (x, t) d σ_{x} d t = \frac{1}{ϵ | Y |} \int_{0}^{T} \int_{R^{n} \times \partial S} T_{b}^{ϵ} (ψ) (x, y, t) d x d σ_{y} d t \\ = \frac{1}{ϵ | Y |} \int_{0}^{T} \int_{\tilde{Q^{ϵ}} \times \partial S} T_{b}^{ϵ} (ψ) (x, y, t) d x d σ_{y} d t . \end{array}$

Let $ψ \in L^{2} (0, T; L^{2} (Q))$ . Then $\begin{matrix} T_{b}^{ϵ} (ψ) \to \tilde{ψ} strongly in L^{2} (0, T; L^{2} (R^{n} \times \partial S) \end{matrix}$

Let $ψ^{ϵ} \in L^{2} (0, T; L^{2} (\partial S^{ϵ}))$ , such that $\begin{matrix} T_{b}^{ϵ} (ψ^{ϵ}) ⇀ \tilde{ψ} weakly in L^{2} (0, T; L^{2} (R^{n} \times \partial S)) . \end{matrix}$ Then $\begin{array}{l} ϵ \int_{0}^{T} \int_{\partial S^{ϵ}} ψ^{ϵ} ϕ d σ_{x} d t ⟶ \frac{1}{| Y |} \int_{0}^{T} \int_{R^{n} \times \partial S} \tilde{ψ} (x, y, t) ϕ (x, t) d x d σ_{y} d t, \end{array}$ for all $ϕ \in L^{2} (0, T; H^{1} (Q))$ .
Remark 1.
From Remark 2.4 of [6], for every $ψ \in L^{2} (0, T; W^{1, 2} (Q^{ϵ}))$ , one has $\begin{matrix} {‖ T_{b}^{ϵ} (ψ) ‖}_{L^{2} (0, T; L^{2} (Q \times \partial S))} ⩽ C (‖ ψ ‖_{L^{2} (0, T; L^{2} (Q^{ϵ}))} + ϵ ‖ \nabla ψ ‖_{L^{2} (0, T; {[L^{2} (Q^{ϵ})]}^{n})}) \end{matrix}$

The following result is an adaptation of proposition 5 in [10] and proposition 5.3 in [12] to evolution problems. Proposition 3.
Let $h \in L^{2} (0, T; L^{2} (\partial S))$ and Y-periodic. Set $h^{ϵ} (x, t) = h (\frac{x}{ϵ}, t)$ for all $x \in R^{n} ∖ ⋃_{ξ \in Z^{n}} ϵ (ξ + S)$ and $t \in (0, T)$ . Let $ψ \in L^{2} (Ω, P, F; L^{2} (0, T; H^{1} (Q)))$ . Then we have $\begin{array}{l} ϵ \int_{0}^{T} \int_{\partial S^{ϵ}} h^{ϵ} (x, t) ψ (x, t) d σ_{x} d t ⟶ \frac{| \partial S | T M_{(\partial S \times (0, T))} (h)}{| Y |} \int_{0}^{T} \int_{Q} ψ (x, t) d x d t, P -a.s. \end{array}$ Furthermore, $\begin{array}{l} E | \int_{0}^{T} \int_{\partial S^{ϵ}} h^{ϵ} (x, t) ψ (x, t) d σ_{x} d t | \\ ⩽ \frac{C}{ϵ} [| M_{(\partial S \times (0, T))} (h) | E ‖ ϕ ‖_{L^{2} (0, T; L^{2} (Q^{ϵ}))} + ϵ E ‖ \nabla ϕ ‖_{L^{2} (0, T; L^{2} ({[Q^{ϵ}]}^{n}))}] . \end{array}$ Finally, if $M_{(\partial S \times (0, T))} (h) = 0$ ,then $\begin{array}{l} \int_{0}^{T} \int_{\partial S^{ϵ}} h^{ϵ} (x, t) ψ (x, t) d σ_{x} d t ⟶ 0 P -a.s. \end{array}$
Proof.
We prove this results for $ψ \in L^{2} (Ω, P, F; D (0, T \times R^{n}))$ and then density gives the result for $ψ \in L^{2} (Ω, P, F; L^{2} (0, T; H^{1} (Q)))$ . From Proposition 1 and Definition 4, we have $\begin{array}{l} \int_{0}^{T} \int_{\partial S^{ϵ}} h (\frac{x}{ϵ}, t) ψ (x, t) d σ_{x} d t \\ = \frac{1}{ϵ | Y |} \int_{0}^{T} \int_{R^{n} \times \partial S} T_{b}^{ϵ} (h) (y, t) T_{b}^{ϵ} (ψ) (x, y, t) d x d σ_{y} d t \\ = \frac{1}{ϵ | Y |} \int_{0}^{T} \int_{\tilde{Q^{ϵ}} \times \partial S} T_{b}^{ϵ} (h) (y, t) T_{b}^{ϵ} (ψ) (x, y, t) d x d σ_{y} d t \\ = \frac{1}{ϵ | Y |} \int_{0}^{T} \int_{\tilde{Q^{ϵ}} \times \partial S} h (y, t) ψ (ϵ [\frac{x}{ϵ}] + ϵ y, t) d x d σ_{y} d t \\ = \frac{1}{ϵ | Y |} \int_{0}^{T} \int_{\tilde{Q^{ϵ}} \times \partial S} h (y, t) ψ (ϵ [\frac{x}{ϵ}], t) d x d σ_{y} d t \\ (13) & + \frac{1}{ϵ | Y |} \int_{0}^{T} \int_{\tilde{Q^{ϵ}} \times \partial S} h (y, t) [ψ (ϵ [\frac{x}{ϵ}] + ϵ y, t) - ψ (ϵ [\frac{x}{ϵ}], t)] d x d σ_{y} d t \end{array}$ Now, passing to the limit as ϵ goes to zero, we have $\begin{array}{l} lim_{ϵ ⟶ 0} ϵ \int_{0}^{T} \int_{\partial S^{ϵ}} h (\frac{x}{ϵ}, t) ψ (x, t) d σ_{x} d t & = \frac{1}{| Y |} \int_{0}^{T} \int_{\tilde{Q^{ϵ}} \times \partial S} h (y, t) ψ (x, t) d x d σ_{y} d t P -a.s. \\ = \frac{| \partial S | T M_{(\partial S \times (0, T))} (h)}{| Y |} \int_{0}^{T} \int_{Q} ψ (x, t) d x d t, P -a.s. \end{array}$ Simple calculations in (13), gives $\begin{array}{l} E | \int_{0}^{T} \int_{\partial S^{ϵ}} h^{ϵ} (x, t) ψ (x, t) d σ_{x} d t | \\ ⩽ \frac{C}{ϵ} [| M_{(\partial S \times (0, T))} (h) | E ‖ ϕ ‖_{L^{2} (0, T; L^{2} (Q^{ϵ}))} + ϵ E ‖ \nabla ϕ ‖_{L^{2} (0, T; L^{2} ({[Q^{ϵ}]}^{n}))}] . \end{array}$ □

3. Energy estimates

In this section, we derive crucial energy estimates of the solution of problem $(P_{ϵ})$ that will be used in the subsequent sections.

Lemma 3.
Under the assumptions (A1)–(A6), the solution $u^{ϵ}$ of $(P_{ϵ})$ satisfies the following estimates $\begin{array}{l} (14) & E sup_{0 ⩽ t ⩽ T} {‖ u^{ϵ} (t) ‖}_{L^{2} (Q^{ϵ})}^{2} ⩽ C_{1}, \\ (15) & E \int_{0}^{T} {‖ \nabla u^{ϵ} (t) ‖}_{{[L^{2} (Q^{ϵ})]}^{n}}^{2} d t ⩽ C_{2}, \\ (16) & E \int_{0}^{T} {‖ u^{ϵ} (t) ‖}_{L^{2} (\partial S^{ϵ})}^{2} d t ⩽ C_{3}, \end{array}$ where $C_{1}$ , $C_{2}$ and $C_{3}$ are positive constants independent of ϵ. Proof.
We apply Itô’s formula to obtain $\begin{array}{l} d {‖ u^{ϵ} ‖}_{L^{2} (Q^{ϵ})}^{2} + 2 {(A_{ϵ} \nabla u^{ϵ}, \nabla u^{ϵ})}_{{[L^{2} (Q^{ϵ})]}^{n}} d t + ϵ α {(u^{ϵ}, u^{ϵ})}_{L^{2} (\partial S^{ϵ})} \\ = ϵ {(h^{ϵ}, u^{ϵ})}_{L^{2} (\partial S^{ϵ})} + 2 {(f^{ϵ}, u^{ϵ})}_{L^{2} (Q^{ϵ})} d t \\ + 2 {(g^{ϵ}, u^{ϵ})}_{L^{2} (Q^{ϵ})} d W + {‖ g^{ϵ} ‖}_{L^{2} (Q^{ϵ})}^{2} d t . \end{array}$ Integrating over $(0, t)$ , $t ⩽ T$ , we get $\begin{array}{l} {‖ u^{ϵ} (t) ‖}_{L^{2} (Q^{ϵ})}^{2} + 2 \int_{0}^{t} {(A_{ϵ} \nabla u^{ϵ} (τ), \nabla u^{ϵ} (τ))}_{L^{2} (Q^{ϵ})} d τ + ϵ α \int_{0}^{t} (u^{ϵ} (τ), u^{ϵ} {(τ)}_{L^{2} (\partial S^{ϵ})} d τ \\ = {‖ b^{ϵ} ‖}_{L^{2} (Q^{ϵ})}^{2} + ϵ \int_{0}^{t} (h^{ϵ} (τ), u^{ϵ} {(τ)}_{L^{2} (\partial S^{ϵ})} d τ + 2 \int_{0}^{t} (f^{ϵ} (τ), u^{ϵ} {(τ)}_{L^{2} (Q^{ϵ})} d τ \\ + 2 \int_{0}^{t} (g^{ϵ} (τ), u^{ϵ} {(τ)}_{L^{2} (Q^{ϵ})} d W + \int_{0}^{t} {‖ g^{ϵ} ‖}_{L^{2} (Q^{ϵ})}^{2} d τ . \end{array}$ Using the assumptions on the matrix A and taking the supremum over $t \in [0, T]$ , we have $\begin{array}{l} sup_{0 ⩽ t ⩽ T} {‖ u^{ϵ} (t) ‖}_{L^{2} (Q^{ϵ})}^{2} + \int_{0}^{T} {‖ \nabla u^{ϵ} (t) ‖}_{L^{2} (Q^{ϵ})}^{2} d t + ϵ α \int_{0}^{T} {‖ u^{ϵ} (t) ‖}_{L^{2} (\partial S^{ϵ})} d t \\ ⩽ C {‖ b^{ϵ} ‖}_{L^{2} (Q^{ϵ})}^{2} + ϵ C | \int_{0}^{T} {(h^{ϵ}, u^{ϵ})}_{L^{2} (\partial S^{ϵ})} d t | + 2 C | \int_{0}^{T} {(f^{ϵ}, u^{ϵ})}_{L^{2} (Q^{ϵ})} d t | \\ + 2 C sup_{0 ⩽ t ⩽ T} | \int_{0}^{t} {(g^{ϵ}, u^{ϵ})}_{L^{2} (Q^{ϵ})} d W | + C \int_{0}^{T} {‖ g^{ϵ} ‖}_{L^{2} (Q)}^{2} d t . \end{array}$ Taking the expectation on both sides, we have $\begin{array}{l} E [sup_{0 ⩽ t ⩽ T} {‖ u^{ϵ} (t) ‖}_{L^{2} (Q^{ϵ})}^{2} + \int_{0}^{T} {‖ \nabla u^{ϵ} (t) ‖}_{L^{2} (Q^{ϵ})}^{2} d t + ϵ α \int_{0}^{T} {‖ u^{ϵ} (t) ‖}_{L^{2} (\partial S^{ϵ})} d t] \\ ⩽ C E [C_{1} + ϵ | \int_{0}^{T} {(h^{ϵ}, u^{ϵ})}_{L^{2} (\partial S^{ϵ})} d t | \\ (17) & + | \int_{0}^{T} {(f^{ϵ}, u^{ϵ})}_{L^{2} (Q^{ϵ})} d t | + sup_{0 ⩽ t ⩽ T} | \int_{0}^{t} {(g^{ϵ}, u^{ϵ})}_{L^{2} (Q^{ϵ})} d W |], \end{array}$ where $\begin{matrix} C_{1} = {‖ b^{ϵ} ‖}_{L^{2} (Q)}^{2} + \int_{0}^{T} {‖ g^{ϵ} ‖}_{L^{2} (Q)}^{2} d s . \end{matrix}$

Using Cauchy–Schwarz’s and Young’s inequalites, we have $\begin{array}{l} E | \int_{0}^{T} {(f^{ϵ}, u^{ϵ})}_{L^{2} (Q^{ϵ})} d t | & ⩽ E \int_{0}^{T} {‖ f^{ϵ} ‖}_{L^{2} (Q^{ϵ})} {‖ u^{ϵ} ‖}_{L^{2} (Q^{ϵ})} d t \\ ⩽ E sup_{0 ⩽ t ⩽ T} {‖ u^{ϵ} (t) ‖}_{L^{2} (Q^{ϵ})} \int_{0}^{T} {‖ f^{ϵ} ‖}_{L^{2} (Q^{ϵ})} d t \\ (18) & ⩽ ε E sup_{0 ⩽ t ⩽ T} {‖ u^{ϵ} (t) ‖}_{L^{2} (Q^{ϵ})}^{2} + C (ε) {(\int_{0}^{T} {‖ f^{ϵ} ‖}_{L^{2} (Q^{ϵ})}^{2} d t)}^{2} \end{array}$ where $ε > 0$ is sufficiently small. Thanks to Burkholder–Davis–Gundy’s inequality and Cauchy Schwarz’s inequality. From Proposition 3 , we see that $\begin{array}{l} E ϵ | \int_{0}^{T} {(h^{ϵ}, u^{ϵ})}_{L^{2} (\partial S^{ϵ})} d t | \\ (19) & ⩽ C [| M_{(\partial S \times (0, T))} (h) | E \int_{0}^{T} {‖ u^{ϵ} ‖}_{L^{2} (Q^{ϵ})} d t + ϵ E \int_{0}^{T} {‖ \nabla u^{ϵ} ‖}_{L^{2} ({[Q^{ϵ}]}^{n})} d t] . \end{array}$ The last term in the right-hand side of ( 17 ) can be estimated as $\begin{array}{l} E sup_{0 ⩽ t ⩽ T} | \int_{0}^{t} {(g^{ϵ}, u^{ϵ})}_{L^{2} (Q^{ϵ})} d W | & ⩽ C E {(\int_{0}^{T} {(g^{ϵ}, u^{ϵ})}^{2} d t)}_{L^{2} (Q^{ϵ})}^{\frac{1}{2}} \\ ⩽ C E {(\int_{0}^{T} {‖ g^{ϵ} ‖}_{L^{2} (Q^{ϵ})}^{2} {‖ u^{ϵ} ‖}_{L^{2} (Q^{ϵ})}^{2} d t)}^{\frac{1}{2}} . \end{array}$ Again using Young’s inequality, we get $\begin{array}{l} C E {(\int_{0}^{T} {‖ g^{ϵ} ‖}_{L^{2} (Q^{ϵ})} {‖ u^{ϵ} ‖}_{L^{2} (Q^{ϵ})} d t)}^{\frac{1}{2}} & ⩽ C E sup_{0 ⩽ t ⩽ T} {‖ u^{ϵ} (t) ‖}_{L^{2} (Q^{ϵ})} {(\int_{0}^{T} {‖ g^{ϵ} ‖}_{L^{2} (Q^{ϵ})}^{2} d t)}^{\frac{1}{2}} \\ ⩽ C (ε) E sup_{0 ⩽ t ⩽ T} {‖ u^{ϵ} (t) ‖}_{L^{2} (Q^{ϵ})}^{2} \\ (20) & + ε C \int_{0}^{T} {‖ g^{ϵ} ‖}_{L^{2} (Q^{ϵ})}^{2} d t, \end{array}$ where $ε > 0$ is small enough. Using ( 18 ), ( 19 ) and ( 20 ) into ( 17 ) together with Growall’s inequality give the estimates ( 14 ), ( 15 )and ( 16 ). Thus the proof is complete. □

With no major difficulties, one can use the estimates (14) and (15) and [10, Proposition 3.1], to prove the following estimates for the macro and micro representations of $u^{ϵ}$ the solution of problem $(P_{ϵ})$ , see also [12, Proposition 2.15] and [6]. We have the following Lemma
Lemma 4.
The macro $Q^{ϵ} (u^{ϵ})$ and micro $R^{ϵ} (u^{ϵ})$ representations of $u^{ϵ}$ , satisfy the following estimates: $\begin{array}{l} (21) & E {‖ Q^{ϵ} (u^{ϵ}) ‖}_{L^{2} (0, T; L^{2} ({\hat{Q^{ϵ}}}_{2 ϵ ρ (Y)}))} ⩽ C, \\ (22) & E {‖ \nabla Q^{ϵ} (u^{ϵ}) ‖}_{L^{2} (0, T; {[L^{2} ({\hat{Q^{ϵ}}}_{2 ϵ ρ (Y)})]}^{n})} ⩽ C, \\ (23) & E {‖ R^{ϵ} (u^{ϵ}) ‖}_{L^{2} (0, T; L^{2} ({\hat{Q^{ϵ}}}_{2 ϵ ρ (Y)}))} ⩽ ϵ C, \\ (24) & E {‖ \nabla R^{ϵ} (u^{ϵ}) ‖}_{L^{2} (0, T; {[L^{2} ({\hat{Q^{ϵ}}}_{2 ϵ ρ (Y)})]}^{n})} ⩽ C, \end{array}$ where C is a constant independent of ϵ.

Note that, the extensions by zero $\tilde{Q^{ϵ}} (u^{ϵ} (t))$ and $\tilde{\nabla Q^{ϵ}} (u^{ϵ} (t))$ (resp.) of $Q^{ϵ} (u^{ϵ} (t))$ and $\nabla Q^{ϵ} (u^{ϵ} (t))$ (resp.) from ${\hat{Q^{ϵ}}}_{2 ϵ ρ (Y)}$ to Q satisfy estimates (21) and (22). Similarly, the extensions by zero $\tilde{R^{ϵ}} (u^{ϵ} (t))$ and $\tilde{\nabla R^{ϵ}} (u^{ϵ} (t))$ (resp.) of $R^{ϵ} (u^{ϵ} (t))$ and $\nabla R^{ϵ} (u^{ϵ} (t))$ (resp.) from ${\hat{Q^{ϵ}}}_{2 ϵ ρ (Y)}$ to $Q^{ϵ}$ satisfies estimate (23) and (24). It easy seeing from (23), (24) and Proposition 1, that $\begin{array}{l} (25) & \frac{1}{ϵ} T_{}^{ϵ} (\tilde{R^{ϵ}} (u^{ϵ})) ⇀ \hat{u}, weakly in L^{2} (0, T; L^{2} (Q \times Y^{})) P -a.s., \\ (26) & T_{}^{ϵ} (\tilde{\nabla R^{ϵ}} (u^{ϵ})) \to \nabla_{y} \hat{u}, weakly in L^{2} (0, T; L^{2} (Q \times Y^{})) P -a.s., \\ (27) & T_{}^{ϵ} (\tilde{R^{ϵ}} (u^{ϵ})) \to 0, strongly in L^{2} (0, T; L^{2} (Q \times Y^{})) P -a.s. \end{array}$ Now, in order to prove strong convergence of $\tilde{Q^{ϵ}} (u^{ϵ})$ , one needs to use some compactness results. But $\tilde{Q^{ϵ}} (u^{ϵ})$ is a random, therefore the classical compactness [12] will not work. For this we prove in the next section some tightness results for the probability measures generated by $\tilde{Q^{ϵ}} (u^{ϵ})$ . We introduce the assumption (A7) below which is required for the proof of Lemma 5 below.
${\tilde{g}}^{ϵ} \in L^{4} ((0, T); H^{- 1} (Q))$ .
Lemma 5.
Under the assumptions (A1)–(A7), $\tilde{Q^{ϵ}} (u^{ϵ})$ satisfies the following $\begin{matrix} (28) & E sup_{θ ⩽ δ} \int_{0}^{T} {‖ \tilde{Q^{ϵ}} (u^{ϵ}) (t + θ) - \tilde{Q^{ϵ}} (u^{ϵ}) (t) ‖}_{H^{- 1} (Q)}^{2} d t < C δ, \end{matrix}$ for any $ϵ > 0$ and sufficiently small $δ > 0$ .
Proof.
By linearity, we see that $\tilde{Q^{ϵ}} (u^{ϵ})$ satisfies the first equation in problem $(P_{ϵ})$ , when the Macro representation of the data $f^{ϵ}$ and $g^{ϵ}$ extended by zero outside ${\hat{Q^{ϵ}}}_{2 ϵ ρ (Y)} \times (0, T)$ . Therefore, the proof is obtained by arguing exactly as in [19, Lemma 2]. □

4. Tightness property of probability measures

A crucial step in the homogenization of SPDEs is the compactness results for the family of probability measures generated by the sequence of solutions of the original (in-homogenized) problems. In this section, we establish the tightness of probability measures generated by $\tilde{Q_{*}^{ϵ}} (u^{ϵ})$ and the Wiener process $W (t)$ . We start by introducing analytic and probabilistic compactness results. We have from [35]

Lemma 6.
Let $B_{0}$ , B and $B_{1}$ be some Banach spaces such that $B_{0} \subset B \subset B_{1}$ and the injection $B_{0} \subset B$ is compact. For any $1 ⩽ p$ , $q ⩽ \infty$ , and $0 < s ⩽ 1$ let E be a set bounded in $L^{q} (0, T; B_{0}) \cap N^{s, p} (0, T; B_{1})$ , where $\begin{matrix} N^{s, p} (0, T; B_{1}) = {v \in L^{p} (0, T; B_{1}) : sup_{h > 0} h^{- s} {‖ v (t + ρ) - v (t) ‖}_{L^{p} (0, T - ρ, B_{1})} < \infty} . \end{matrix}$ Then E is relatively compact in $L^{p} (0, T; B)$

The following two lemmas are taken from [4]. Let $S$ be a separable Banach space and consider its Borel σ-field to be $B (S)$ . We have
Lemma 7 (Prokhorov).

A sequence of probability measures ${(Π_{n})}_{n \in N}$ on $(S, B (S))$ is tight if and only if it is relatively compact.

Lemma 8 (Skorokhod).

Suppose that the probability measures ${(μ_{n})}_{n \in N}$ on $(S, B (S))$ weakly converge to a probability measure μ. Then there exist random variables $ξ, ξ_{1}, \dots ξ_{n}, \dots$ , defined on a common probability space $(Ω, F, P)$ , such that $L (ξ_{n}) = μ_{n}$ and $L (ξ) = μ$ and $\begin{matrix} lim_{n \to \infty} ξ_{n} = ξ, P -a.s.; \end{matrix}$ the symbol $L (\cdot)$ stands for the law of ·.

Let us introduce the space $\begin{array}{l} Z & = {ψ : sup_{0 ⩽ t ⩽ T} {‖ ψ (t) ‖}_{L^{2} (Q)}^{2} ⩽ C_{1}, \int_{0}^{T} {‖ ψ (t) ‖}_{H_{0}^{1} (Q)}^{2} d t ⩽ C_{2} \\ and sup_{n} \frac{1}{ν_{n}} sup_{θ ⩽ μ_{n}} {(\int_{0}^{T} {‖ ψ (t + θ) - ψ (t) ‖}_{H^{- 1} (Q)}^{2})}^{\frac{1}{2}} < \infty}, \end{array}$ where $μ_{n}$ , $ν_{n}$ are sequences of positive real numbers such that $μ_{n}, ν_{n} \to 0$ as $n \to \infty$ . The following result is due to Bensoussan [3]

Lemma 9.
The above constructed space Z is a compact subset of $L^{2} (0, T; L^{2} (Q))$
Proof.
Lemma 6 together with a suitable argument due to Bensoussan [3] give the compactness of Z in $L^{2} (0, T; L^{2} (Q))$ . □

We now consider the space $X = C (0, T; R^{m}) \times L^{2} (0, T; L^{2} (Q))$ and $B (X)$ the σ-algebra of its Borel sets. Let $Ψ_{ϵ}$ be a $(X, B (X))$ -valued measurable map defined on $(Ω, F, P)$ by $\begin{matrix} Ψ_{ϵ} : ω \mapsto (W (ω), u^{ϵ} (ω)) . \end{matrix}$ Define on $(X, B (X))$ a family of probability measures $(Π_{ϵ})$ by $\begin{matrix} Π_{ϵ} (A) = P (Ψ_{ϵ}^{- 1} (A)) for all A \in B (X) . \end{matrix}$

The proof of the following Lemma was obtained in [[18], Lemma 4.5], in fixed domains.
Lemma 10.
The family $(Π_{ϵ})$ is tight on $X$ .

From Lemmas 10 and 7, we are able to extract a subsequence $Π_{ϵ_{j}}$ from $Π_{ϵ}$ that weakly converges to a probability measure Π on $X$ . Skorohod’s Lemma 8 facilitates finding a probability space $(\hat{Ω}, \hat{F}, \hat{P})$ and $X$ -valued random variables $(W_{ϵ_{j}}, \tilde{Q^{ϵ_{j}}} (u^{ϵ_{j}}))$ , $(\hat{W}, u)$ such that the probability law of $(W_{ϵ_{j}}, \tilde{Q^{ϵ_{j}}} (u^{ϵ_{j}}))$ is $Π_{ϵ_{j}}$ and that of $(\hat{W}, u)$ is Π. Furthermore, we have $\begin{array}{l} (29) & W_{ϵ_{j}} \to \hat{W} in C (0, T; R^{m}), \hat{P} -a.s. \\ (30) & \tilde{Q^{ϵ}} (u^{ϵ}) \to u in L^{2} (0, T; L^{2} (Q)), \hat{P} -a.s. \end{array}$ It is also easy to see that $(W_{ϵ_{j}}, \tilde{Q^{ϵ_{j}}} (u^{ϵ_{j}}))$ satisfies $\hat{P} - a . s$ . the problem $(P_{ϵ_{j}})$ . From the strong convergence (30) and Proposition 1, we have $\begin{array}{l} (31) & T^{ϵ} (\tilde{Q^{ϵ}} (u^{ϵ})) \to u, strongly in L^{2} (0, T; L^{2} (Q \times Y)) P -a.s., \end{array}$ where we consider the usual unfolding operator. Now (27) and (31), give $\begin{array}{l} (32) & T_{*}^{ϵ} (u^{ϵ}) \to u, strongly in L^{2} (0, T; L^{2} (Q \times Y)) P -a.s., \end{array}$
5. The homogenization result

In this section, we use the periodic unfolding presented in Section 2 to study the asymptotic behaviour of Robin problem for linear parabolic stochastic equations in perforated domains.

Theorem 2.
Suppose that the assumptions (A1)–(A7) are satisfied and $\begin{array}{l} (33) & {\tilde{b}}^{ϵ_{j}} & ⇀ b, weakly in L^{2} (Q) \\ (34) & {\tilde{f}}^{ϵ_{j}} & ⇀ f, weakly in L^{2} (Q \times (0, T)), \\ (35) & {\tilde{g}}^{ϵ_{j}} & ⇀ g, weakly in L^{2} (Q \times (0, T)) . \end{array}$ Then there exists a probability space $(\hat{Ω}, \hat{F}, \hat{P}, {({\hat{F}}_{t})}_{0 ⩽ t ⩽ T})$ with random variables $(u^{ϵ_{j}}, W_{ϵ_{j}})$ and $(u, \hat{W})$ such that the convergences ( 29 ) and ( 30 ) hold. Furthermore there exists $\hat{u} \in L^{2} ((0, T) \times Q; H_{per} (Y))$ such that $\begin{array}{l} (36) & T_{}^{ϵ} (u^{ϵ}) ⇀ u {weakly}^{} in L^{\infty} (0, T; L_{loc}^{2} (Q; H^{1} (Y^{}))) \hat{P} -a.s., \\ (37) & T_{b}^{ϵ} (u^{ϵ}) ⇀ u weakly in L^{2} (0, T; L_{loc}^{2} (Q \times \partial S)) \hat{P} -a.s., \\ (38) & \frac{1}{ϵ} T_{}^{ϵ} (R^{ϵ} (u^{ϵ})) ⇀ \hat{u} {weakly}^{} in L^{\infty} (0, T; L_{loc}^{2} (Q; H^{1} (Y^{}))) \hat{P} -a.s., \\ (39) & T_{}^{ϵ} (\nabla_{x} u^{ϵ}) ⇀ \nabla_{x} u + \nabla_{y} \hat{u} weakly in L^{2} (0, T; L_{loc}^{2} (Q; L^{2} (Y^{}))) \hat{P} -a.s., \end{array}$ where $(u, \hat{W})$ satisfies the homogenized problem $(P)$ .
Proof.
By unfolding the weak formulation of problem $(P_{ϵ})$ , we have $\begin{array}{l} \int_{0}^{T} \int_{\tilde{Q^{ϵ}} \times Y^{}} T_{}^{ϵ} (u^{ϵ}) T_{}^{ϵ} (ϕ) ψ d y d x d t + \int_{0}^{T} \int_{\tilde{Q^{ϵ}} \times Y^{}} A (y) T_{}^{ϵ} (\nabla u^{ϵ}) T_{}^{ϵ} (\nabla ϕ) ψ d y d x d t \\ + α ϵ \int_{0}^{T} \int_{\tilde{Q^{ϵ}} \times \partial S^{ϵ}} T_{b}^{ϵ} (u^{ϵ}) T_{b}^{ϵ} (ϕ) ψ d σ_{y} d x d t \\ = \int_{0}^{T} \int_{\tilde{Q^{ϵ}} \times Y^{}} T_{}^{ϵ} (b^{ϵ}) T_{}^{ϵ} (ϕ) ψ d y d x d t + \int_{0}^{T} \int_{Q \times Y^{}} T_{}^{ϵ} (f^{ϵ}) T_{}^{ϵ} (ϕ) d y d x d t \\ (40) & + ϵ \int_{0}^{T} \int_{\tilde{Q^{ϵ}} \times \partial S^{ϵ}} T_{b}^{ϵ} (h^{ϵ}) T_{b}^{ϵ} (ϕ) ψ d σ_{y} d x d t + \int_{0}^{T} \int_{\tilde{Q^{ϵ}} \times Y^{}} T_{}^{ϵ} g^{ϵ} T_{}^{ϵ} (ϕ) ψ d y d x d W_{ϵ} (t) \end{array}$ where $ϕ \in D (Q)$ and $ψ \in D (0, T)$ . In view of Lemma 1, Remark 1 and Estimates (14), (15) and (16), we obtain the existence of $u \in L^{\infty} (0, T; H_{0}^{1} (Q))$ and $\hat{u} \in L^{\infty} (0, T; L^{2} (Q; H^{1} (Y^{})))$ , such that up to a subsequence convergences (36), (37), (38) and (39) hold. Now, from (36) and (6), we have $\begin{array}{l} lim_{ϵ \to 0} \int_{0}^{T} \int_{Q^{ϵ}} d u^{ϵ} ϕ ψ d x & = lim_{ϵ \to 0} \frac{1}{| Y |} \int_{0}^{T} \int_{\tilde{Q^{ϵ}} \times Y^{}} T_{}^{ϵ} (u^{ϵ}) T_{}^{ϵ} (ϕ) ψ_{t} d y d x d t \\ (41) & = θ \int_{0}^{T} \int_{Q} u ϕ ψ d x d t \hat{P} -a.s., \end{array}$ where $ϕ \in D (Q)$ and $ψ \in D (0, T)$ . Using convergences (6), and (36) we, get $\begin{array}{l} lim_{ϵ \to 0} \int_{0}^{T} \int_{Q^{ϵ}} A_{ϵ} \nabla u^{ϵ} \nabla ϕ ψ d x d t & = lim_{ϵ \to 0} \frac{1}{| Y |} \int_{0}^{T} \int_{\tilde{Q^{ϵ}} \times Y^{}} A (y) T_{}^{ϵ} (\nabla u^{ϵ}) T_{}^{ϵ} (\nabla ϕ) ψ d y d x d t \\ (42) & = \frac{1}{| Y |} \int_{0}^{T} \int_{Q \times Y^{}} A (\nabla u + \nabla \hat{u}) \nabla ϕ ψ d y d x d t \hat{P} -a.s., \end{array}$ for $ϕ \in D (Q)$ and $ψ \in D (0, T)$ . We also have by Proposition 2 (2) and (3), the following convergence $\begin{array}{l} lim_{ϵ \to 0} ϵ \int_{0}^{T} \int_{\partial S^{ϵ}} u^{ϵ} ϕ ψ d σ_{x} d t & = lim_{ϵ \to 0} \frac{α}{| Y |} \int_{0}^{T} \int_{\tilde{Q^{ϵ}} \times \partial S} T_{b}^{ϵ} (u^{ϵ}) T_{b}^{ϵ} (ϕ) ψ d σ_{y} d x d t \\ (43) & = α \frac{| \partial S |}{| Y |} \int_{0}^{T} \int_{Q} u ϕ ψ d x d t \hat{P} -a.s., \end{array}$ for $ϕ \in D (Q)$ and $ψ \in D (0, T)$ . Similarly we have the convergences $\begin{array}{l} lim_{ϵ \to 0} \int_{0}^{T} \int_{Q^{ϵ}} b^{ϵ} ϕ ψ d x d t & = lim_{ϵ \to 0} \frac{1}{| Y |} \int_{0}^{T} \int_{\tilde{Q^{ϵ}} \times Y^{}} T_{}^{ϵ} (b^{ϵ}) T_{}^{ϵ} (ϕ) ψ d y d x d t \\ (44) & = θ \int_{0}^{T} \int_{Q} b ϕ ψ d x d t \hat{P} -a.s., \end{array}$ and $\begin{array}{l} lim_{ϵ \to 0} \int_{0}^{T} \int_{Q^{ϵ}} f^{ϵ} ϕ ψ d x d t & = lim_{ϵ \to 0} \frac{1}{| Y |} \int_{0}^{T} \int_{\tilde{Q^{ϵ}} \times Y^{}} T_{}^{ϵ} (f^{ϵ}) T_{}^{ϵ} (ϕ) ψ d y d x d t \\ (45) & = θ \int_{0}^{T} \int_{Q} f ϕ ψ d x d t \hat{P} -a.s. . \end{array}$ By Proposition 3, we have $\begin{array}{l} (46) & lim_{ϵ \to 0} ϵ \int_{0}^{T} \int_{\partial S^{ϵ}} h^{ϵ} (x, t) ϕ ψ d σ_{x} d t = \frac{| \partial S |}{| Y |} T M_{(\partial S \times (0, T))} (h) \int_{0}^{T} \int_{Q} ϕ ψ d x d t, \hat{P} -a.s. \end{array}$

In what follows, we show that $\begin{array}{l} lim_{ϵ \to 0} \int_{0}^{T} \int_{Q^{ϵ}} g^{ϵ} ϕ ψ d x d W_{ϵ} & = lim_{ϵ \to 0} \frac{1}{| Y |} \int_{0}^{T} \int_{\tilde{Q^{ϵ}} \times Y^{}} T_{}^{ϵ} (g^{ϵ}) T_{}^{ϵ} (ϕ) ψ d y d x d W^{ϵ} \\ (47) & = θ \int_{0}^{T} \int_{Q} g ϕ ψ d x d \hat{W} \hat{P} -a.s., \end{array}$ for this, we write $\begin{array}{l} \hat{E} \int_{0}^{T} \int_{\tilde{Q^{ϵ}} \times Y^{}} T_{}^{ϵ} (g^{ϵ}) T_{}^{ϵ} (ϕ) ψ d x d y d W_{ϵ} \\ = \hat{E} \int_{0}^{T} \int_{\tilde{Q^{ϵ}} \times Y^{}} T_{}^{ϵ} (g^{ϵ}) T_{}^{ϵ} (ϕ) ψ d x d y d (W_{ϵ} - \hat{W}) \\ (48) & + \hat{E} \int_{0}^{T} \int_{\tilde{Q^{ϵ}} \times Y^{}} T_{}^{ϵ} (g^{ϵ}) T_{}^{ϵ} (ϕ) \hat{W} . \end{array}$

In view of the unbounded variation of $W_{ϵ} - \hat{W}$ , the convergence of the first term on the right-hand side of (48) requires appropriate care, in order to take advantage of the $\hat{P}$ -a.s. uniform convergence of $W_{ϵ}$ to $\hat{W}$ in $C ([0, T])$ . We adopt the idea of regularization of $T_{}^{ϵ} (g^{ϵ})$ with respect to the variable t, by means of the following sequence $\begin{matrix} (49) & T_{}^{ϵ, λ} (g^{ϵ}) = \frac{1}{λ} \int_{0}^{T} ρ (- \frac{t - s}{λ}) T_{}^{ϵ} (g (s, \frac{x}{ε})) d s for λ > 0, \end{matrix}$ where ρ is a standard mollifier. We have that $T_{}^{ϵ, λ} (g^{ϵ}) (t)$ is a differentiable function of t and satisfies the relations $\begin{matrix} (50) & \hat{E} \int_{0}^{T} {‖ T_{}^{ϵ, λ} (g^{ϵ}) (t) ‖}_{L_{2} (\tilde{Q^{ϵ}} \times Y^{})}^{2} d t ⩽ \hat{E} \int_{0}^{T} {‖ T_{}^{ϵ} (g (s, \frac{x}{ε})) ‖}_{L_{2} (\tilde{Q^{ϵ}} \times Y^{})}^{2} d t, for any λ > 0, \end{matrix}$ and for any $ε > 0$ $\begin{matrix} (51) & T_{}^{ϵ, λ} (g^{ϵ}) (t) \to T_{}^{ϵ} (g^{ϵ} (t, x)) strongly in L^{2} (L^{2} ((0, T); L^{2} (\tilde{Q^{ϵ}} \times Y^{}))) as λ \to 0 . \end{matrix}$ We split the first term in the right-hand side of (48) as $\begin{array}{l} \hat{E} \int_{0}^{T} \int_{\tilde{Q^{ϵ}} \times Y^{}} T_{}^{ϵ} (g^{ϵ}) T_{}^{ϵ} (ϕ) ψ d x d y d (W_{ϵ} - \hat{W}) \\ = \hat{E} \int_{0}^{T} \int_{\tilde{Q^{ϵ}} \times Y^{}} T_{}^{ϵ, λ} (g^{ϵ}) (t) T_{}^{ϵ} (ϕ) ψ d x d y d (W_{ϵ} - \hat{W}) \\ (52) & + \hat{E} \int_{0}^{T} \int_{\tilde{Q^{ϵ}} \times Y^{}} [T_{}^{ϵ} (g^{ϵ}) - T_{}^{ϵ, λ} (g^{ϵ}) (t)] T_{}^{ϵ} (ϕ) ψ d x d y d (W_{ϵ} - \hat{W}) . \end{array}$ Owing to (51), and Burkholder–Davis–Gundy’s inequality, it readily follows that the second term in right hand side of (52) is bounded by a function $σ_{1} (λ)$ which converges to zero as $λ \to 0$ . In the first term in the same relation, we take advantage of the differentiability of $T_{λ}^{ϵ} (g^{ϵ}) (t)$ with respect to t in order to integrate by parts. As a result, we get $\begin{array}{l} \hat{E} \int_{0}^{T} \int_{\tilde{Q^{ϵ}} \times Y^{}} T_{}^{ϵ, λ} (g^{ϵ}) (t) T_{}^{ϵ} (ϕ) ψ d x d y d (W_{ϵ} - \hat{W}) \\ = \hat{E} \int_{0}^{T} \int_{\tilde{Q^{ϵ}} \times Y^{}} (W_{ϵ} - \hat{W}) \frac{\partial}{\partial t} [T_{λ}^{ϵ, λ} (g^{ϵ}) (t) ψ (t)] T_{}^{ϵ} (ϕ) d x d y d t \\ (53) & + \hat{E} \int_{\tilde{Q^{ϵ}} \times Y^{}} T_{}^{ϵ, λ} (g^{ϵ}) (T) ψ (T) T_{}^{ϵ} (ϕ) (W_{ϵ} (T) - \hat{W} (T)) d x d y . \end{array}$ Thanks to the conditions on ϕ, ψ and g and the uniform convergence obtained from the application of Skorokhod’s compactness result, namely $\begin{matrix} (54) & W_{ϵ} \to \hat{W} uniformly in C ([0, T]), \hat{P} -a.s., \end{matrix}$ we get that both terms on the right-hand side of (53) are bounded by the product $σ_{2} (λ) η_{1} (ε)$ such that $σ_{2} (λ)$ is finite and $η_{1} (ε)$ vanishes as ε tends to zero. we now deduce from (52) that $\begin{matrix} (55) & | \hat{E} \int_{0}^{T} \int_{\tilde{Q^{ϵ}} \times Y^{}} T_{}^{ϵ} (g^{ϵ}) T_{}^{ϵ} (ϕ) ψ d x d y d (W_{ϵ} - \hat{W}) | ⩽ σ_{1} (λ) + σ_{2} (λ) η_{1} (ε) . \end{matrix}$

Thus, we infer from (48) that $\begin{array}{l} | \hat{E} \int_{0}^{T} \int_{\tilde{Q^{ϵ}} \times Y^{}} T_{}^{ϵ} (g^{ϵ}) T^{ϵ} (ϕ) ψ d x d y d W_{ϵ} \\ - \hat{E} \int_{0}^{T} \int_{\tilde{Q^{ϵ}} \times Y^{}} T_{}^{ϵ} (g^{ϵ}) T_{}^{ϵ} (ϕ) ψ d x d y d \hat{W} | \\ (56) & ⩽ σ_{1} (λ) + σ_{2} (λ) η_{1} (ε) \end{array}$ Taking the limit in (56) as $ε \to 0$ , we get $\begin{array}{l} lim_{ε \to 0} | \hat{E} \int_{0}^{T} \int_{\tilde{Q^{ϵ}} \times Y^{}} T_{}^{ϵ} (g^{ϵ}) T^{ϵ} (ϕ) ψ d x d y d W_{ϵ} \\ - \hat{E} \int_{0}^{T} \int_{\tilde{Q^{ϵ}} \times Y^{}} T_{}^{ϵ} (g^{ϵ}) T_{}^{ϵ} (ϕ) ψ d x d y d \hat{W} | ⩽ σ_{1} (λ); \end{array}$ but the left-hand side of this relation is independent of λ, thus we can pass to the limit on both sides as $λ \to 0$ to arrive at the crucial statement $\begin{array}{l} lim_{ε \to 0} \hat{E} \int_{0}^{T} \int_{\tilde{Q^{ϵ}} \times Y^{}} T_{}^{ϵ} (g^{ϵ}) T_{}^{ϵ} (ϕ) ψ d x d y d W_{ϵ} \\ (57) & = lim_{ε \to 0} \hat{E} \int_{0}^{T} \int_{\tilde{Q^{ϵ}} \times Y^{}} T_{}^{ϵ} (g^{ϵ}) T_{}^{ϵ} (ϕ) ψ d x d y d \hat{W} . \end{array}$ Owing to convergences (6) and (7), we can call upon the convergence theorem for stochastic integrals due to Rozovskii [30, Theorem 4, p. 63] to claim that $\begin{matrix} \hat{E} \int_{0}^{T} \int_{\tilde{Q^{ϵ}} \times Y^{}} T_{}^{ϵ} (g^{ϵ}) T_{}^{ϵ} (ϕ) ψ d x d y d \hat{W} \to \hat{E} \int_{0}^{T} \int_{Q \times Y^{}} g ϕ ψ d x d y d \hat{W} . \end{matrix}$ Hence, we deduce from (57) that, $\begin{matrix} (58) & \frac{1}{| Y |} \int_{0}^{T} \int_{\tilde{Q^{ϵ}} \times Y^{}} T_{}^{ϵ} (g^{ϵ}) T_{}^{ϵ} (ϕ) ψ d x d y d \hat{W} \to θ \int_{0}^{T} \int_{Q} g ϕ ψ d x d y d \hat{W}, \hat{P} -a.s. \end{matrix}$

Using convergences (41)–(47) in the variational formulation (40), we obtain $\begin{array}{l} θ \int_{0}^{T} \int_{Q} u ϕ ψ d x d t + \frac{1}{| Y |} \int_{0}^{T} \int_{Q \times Y^{}} A (\nabla u + \nabla \hat{u}) \nabla ϕ ψ d y d x d t \\ + α \frac{| \partial S |}{| Y |} \int_{0}^{T} \int_{Q} u ϕ ψ d x d t + θ \int_{0}^{T} \int_{Q} b ϕ ψ d x d t \\ + θ \int_{0}^{T} \int_{Q} f ϕ ψ d x d t + \frac{| \partial S |}{| Y |} T M_{(\partial S \times (0, T))} (h) \int_{0}^{T} \int_{Q} ϕ ψ d x d t \\ (59) & + θ \int_{0}^{T} \int_{Q} g ϕ ψ d x d \hat{W} \end{array}$

In order to obtain the homogenized problem, we identify $\hat{u}$ as follows. Let us write $φ_{ϵ} = ϵ Φ (x) Ψ_{ϵ} (x)$ , where $Φ \in D (Q)$ and $Ψ_{ϵ} (\cdot) = Ψ (\frac{\cdot}{ϵ}) \in H_{per}^{1} (Y^{})$ . Then $\nabla φ_{ϵ} = Φ (x) \nabla Ψ (\frac{\cdot}{ϵ}) + ϵ Ψ_{ϵ} \nabla_{x} Φ$ and by (6), we have $\begin{array}{l} (60) & T^{ϵ} (φ_{ϵ}) \to 0 strongly in L^{2} (Q \times Y^{}), \\ (61) & T^{ϵ} (\nabla φ_{ϵ}) \to \nabla_{y} \hat{ϕ} strongly in L^{2} (Q \times Y^{}) \end{array}$ where $\hat{ϕ} (x, y) = Φ (x) Ψ (y)$ . Using $φ_{ϵ} (x) ψ (t)$ with $ψ \in D (0, T)$ as test function in (40) and unfolding the resulting equation to obtain $\begin{array}{l} \int_{0}^{T} \int_{\tilde{Q^{ϵ}} \times Y^{}} T_{}^{ϵ} (u^{ϵ}) T_{}^{ϵ} (φ_{ϵ}) d y d x d t + \int_{0}^{T} \int_{\tilde{Q^{ϵ}} \times Y^{}} T_{}^{ϵ} (A_{ϵ}) T_{}^{ϵ} (\nabla u^{ϵ}) T_{}^{ϵ} (\nabla φ_{ϵ}) d y d x d t \\ + α ϵ \int_{0}^{T} \int_{\tilde{Q^{ϵ}} \times \partial S^{ϵ}} T_{b}^{ϵ} (u^{ϵ}) T_{b}^{ϵ} (φ_{ϵ}) d σ_{y} d x d t \\ = \int_{0}^{T} \int_{\tilde{Q^{ϵ}} \times Y^{}} T_{}^{ϵ} (b^{ϵ}) T_{}^{ϵ} (φ_{ϵ}) d y d x d t + \int_{0}^{T} \int_{\tilde{Q^{ϵ}} \times Y^{}} T_{}^{ϵ} (f^{ϵ}) T_{}^{ϵ} (φ_{ϵ}) d y d x d t \\ (62) & + ϵ \int_{0}^{T} \int_{\tilde{Q^{ϵ}} \times \partial S^{ϵ}} T_{b}^{ϵ} (h^{ϵ}) T_{b}^{ϵ} (φ_{ϵ}) d σ_{y} d x d t + \int_{0}^{T} \int_{\tilde{Q^{ϵ}} \times Y^{}} T_{}^{ϵ} g^{ϵ} T_{}^{ϵ} (φ_{ϵ}) d y d x d W_{ϵ} (t) . \end{array}$

We pass to the limit in equation (62), using convergences (41)–(46), (60) and (61), we get $\begin{array}{l} \frac{1}{| Y |} \int_{0}^{T} \int_{Q \times Y} A (y) (\nabla u + \nabla_{y} \hat{u}) \nabla_{y} \hat{ϕ} ψ d x d y d t \\ (63) & = lim_{ϵ \to 0} \int_{0}^{T} \int_{\tilde{Q^{ϵ}} \times Y^{}} T_{}^{ϵ} g^{ϵ} T_{}^{ϵ} (φ_{ϵ}) d y d x d W_{ϵ} (t) . \end{array}$ Thanks to Burkhölder–Davis–Gundy’s inequality, the assumptions on $g^{ϵ}$ and (60), we have $\begin{array}{l} lim_{ϵ \to 0} \hat{E} sup_{t \in [0, T]} | \int_{0}^{t} \int_{\tilde{Q^{ϵ}} \times Y^{}} T_{}^{ϵ} (g^{ϵ}) T_{}^{ϵ} (φ_{ϵ}) ψ d x d y d W_{ϵ} | \\ ⩽ C lim_{ϵ \to 0} \hat{E} {(\int_{0}^{T} ψ {(\int_{\tilde{Q^{ϵ}} \times Y^{}} T_{}^{ϵ} (g^{ϵ}) T_{}^{ϵ} (φ_{ϵ}) d x d y)}^{2} d t)}^{\frac{1}{2}} \\ ⩽ C_{1} lim_{ϵ \to 0} \hat{E} {(\int_{0}^{T} ψ {‖ T_{}^{ϵ} (g^{ϵ}) ‖}_{L^{2} (\tilde{Q^{ϵ}} \times Y)}^{2} {‖ T_{}^{ϵ} (φ_{ϵ}) ‖}_{L^{2} (\tilde{Q^{ϵ}} \times Y^{})}^{2} d t)}^{\frac{1}{2}} \\ ⩽ C_{1} lim_{ϵ \to 0} {(\int_{0}^{T} {‖ T_{}^{ϵ} (g^{ϵ}) ‖}_{L^{2} (\tilde{Q^{ϵ}} \times Y^{})}^{2} {‖ T_{}^{ϵ} (φ_{ϵ}) ‖}_{L^{2} (\tilde{Q^{ϵ}} \times Y^{})}^{2} d t)}^{\frac{1}{2}} \\ \to 0 \hat{P} -a.s. \end{array}$ Therefore, we have $\begin{array}{l} (64) & \frac{1}{| Y |} \int_{0}^{T} \int_{Q \times Y} A (y) (\nabla u + \nabla_{y} \hat{u}) \nabla_{y} \hat{ϕ} ψ d x d y d t = 0 . \end{array}$ Equation (64) has a unique solution (see e.g [19]) given by $\begin{array}{l} (65) & \hat{u} (t, x, y) = - χ (y) \cdot \nabla_{x} u (t, x) + \tilde{u_{1}} (t, x), \end{array}$ where χ is defined by (1). Substituting (65) into (59), it is easy to see that (59) is the weak formulation of the equation $\begin{array}{l} θ d u_{t} - A_{0} Δ u d t + α \frac{| \partial S |}{| Y |} u d t \\ (66) & = \frac{| \partial S |}{| Y |} T M_{(\partial S \times (0, T))} (h) d t + θ f (t, x) d t + θ g (t, x) \hat{W} \end{array}$ where $\begin{array}{l} (67) & A_{0} = \int_{Y} (A (y) - A (y) \nabla_{y} χ (y)) d y . \end{array}$ The initial boundary value problem corresponding to (66) has a unique solution by [28]. Simple calculations can show that the initial condition satisfied.

We note that $(\hat{W}, u)$ is a probabilistic weak solution of $(P)$ which is unique. Thus by the infinite dimensional version of Yamada–Watanabe’s theorem (see [27]), we get that $(W, u)$ is a unique strong solution of (P). Thus, up to distribution (probability law) the whole sequence of the solutions of $(P_{ϵ})$ converges to the solution of problem (P). Thus the proof is complete. □

6. Corrector results

In this section, we obtain corrector results without any supplementary assumption of regularity on the solutions of the problem $(P_{ϵ})$ . We have the following remark

Remark 2.

In [19, Theorem 5] the assumption that $\nabla_{y} χ (y) \in [L^{r} {(Y)]}^{n}$ and $\nabla u \in L^{2} (0, T; {[L^{s} (Y)]}^{n})$ with $1 ⩽ r$ , $s < \infty$ such that $\begin{matrix} \frac{1}{r} + \frac{1}{s} = \frac{1}{2}, \end{matrix}$ was required for the proof of the corrector results. Here we are able to avoid this regularity assumption and still prove the corrector results.

Theorem 3.

Let the assumptions of theorem 2 be fulfilled. Then $\begin{array}{l} (68) & T^{ϵ} (\nabla u^{ϵ}) \to \nabla_{x} u - \nabla_{y} \hat{u} strongly in L^{2} (0, T; L_{loc}^{2} (Q \times Y^{*})) \hat{P} -a.s. \\ (69) & \nabla u^{ϵ} - \nabla_{x} u - U^{ϵ} (\nabla_{y} \hat{u}) \to 0 strongly in L^{2} (0, T; L_{loc}^{2} (Q \times Y^{*})) \hat{P} -a.s. \end{array}$

Proof.

From (59) and (64), we have $\begin{array}{l} θ \int_{0}^{T} \int_{Q} u ϕ ψ d x d t + \frac{1}{| Y |} \int_{0}^{T} \int_{Q \times Y^{*}} A (\nabla u + \nabla_{y} \hat{u}) (\nabla ϕ + \nabla_{y} \hat{ϕ}) ψ d y d x d t \\ + α \frac{| \partial S |}{| Y |} \int_{0}^{T} \int_{Q} u ϕ ψ d x d t \\ = θ \int_{0}^{T} \int_{Q} b ϕ ψ d x d t + θ \int_{0}^{T} \int_{Q} f ϕ ψ d x d t \\ (70) & + \frac{| \partial S |}{| Y |} T M_{(\partial S \times (0, T))} (h) \int_{0}^{T} \int_{Q} ϕ ψ d x d t + θ \int_{0}^{T} \int_{Q} g ϕ ψ d x d \hat{W} \end{array}$

We start by proving the following $\begin{array}{l} lim_{ϵ \to 0} \hat{E} \int_{0}^{T} \int_{Q^{ϵ}} A_{ϵ} \nabla u^{ϵ} \nabla u^{ϵ} d x d s \\ (71) & = \frac{1}{| Y |} \hat{E} \int_{0}^{T} \int_{Q \times Y^{*}} A (y) [\nabla_{x} u (t, x) + \nabla_{y} \hat{u} (t, x, y)] [\nabla_{x} u (t, x) + \nabla_{y} \hat{u} (t, x, y)] d y d x d s . \end{array}$ To do that, we define the set $U \subset \subset Q$ with $\begin{matrix} v ⩾ 0, v = 1 on U . \end{matrix}$ Now let $ϕ = u v$ and $\hat{ϕ} = \hat{u} v$ in (70) and take the expectation on both sides of the resulting equation, we have $\begin{array}{l} θ \hat{E} \int_{0}^{T} \int_{Q} u^{2} v ψ d x d t + \frac{1}{| Y |} \hat{E} \int_{0}^{T} \int_{Q \times Y^{*}} A (\nabla u + \nabla_{y} \hat{u}) (\nabla (u v) + \nabla_{y} (\hat{u} v)) ψ d y d x d t \\ + α \frac{| \partial S |}{| Y |} \hat{E} \int_{0}^{T} \int_{Q} u^{2} v ψ d x d t \\ = θ \hat{E} \int_{0}^{T} \int_{Q} b u v ψ d x d t + θ \hat{E} \int_{0}^{T} \int_{Q} f u v ψ d x d t \\ (72) & + \frac{| \partial S |}{| Y |} T M_{(\partial S \times (0, T))} (h) \hat{E} \int_{0}^{T} \int_{Q} u v ψ d x d t . \end{array}$ The vanishing of the expectation of the stochastic integrals is due to the fact that $(g, u)$ is square integrable in time (see assumption (A5) and estimate (14)). From the second term on the right-hand side of (72), we have $\begin{array}{l} θ \hat{E} \int_{0}^{T} \int_{Q} u^{2} v ψ d x d t + \frac{1}{| Y |} \hat{E} \int_{0}^{T} \int_{Q \times Y^{*}} A (\nabla u + \nabla_{y} \hat{u}) (u \nabla v + \nabla_{y} (\hat{u} v)) ψ d y d x d t \\ + \frac{1}{| Y |} \hat{E} \int_{0}^{T} \int_{Q \times Y^{*}} A (\nabla u + \nabla_{y} \hat{u}) u \nabla v ψ d y d x d t + α \frac{| \partial S |}{| Y |} \hat{E} \int_{0}^{T} \int_{Q} u^{2} v ψ d x d t \\ = θ \hat{E} \int_{0}^{T} \int_{Q} b u v ψ d x d t + θ \hat{E} \int_{0}^{T} \int_{Q} f u v ψ d x d t \\ (73) & + \frac{| \partial S |}{| Y |} T M_{(\partial S \times (0, T))} (h) \hat{E} \int_{0}^{T} \int_{Q} u v ψ d x d t . \end{array}$ We also use $ϕ = u^{ϵ} v$ as test function in the weak formulation of problem $(P_{ϵ})$ and take the expectation, to have $\begin{array}{l} \hat{E} \int_{0}^{T} \int_{Q^{ϵ}} {(u^{ϵ})}^{2} v ψ d x d t + \hat{E} \int_{0}^{T} \int_{Q^{ϵ}} A_{ϵ} \nabla u^{ϵ} u^{ϵ} (\nabla v) ψ d x d t \\ + \hat{E} \int_{0}^{T} \int_{Q} A_{ϵ} \nabla u^{ϵ} \nabla u^{ϵ} v ψ d x d t + α \hat{E} \int_{0}^{T} \int_{\partial S^{ϵ}} {(u^{ϵ})}^{2} v ψ d σ_{x} d t \\ = \hat{E} \int_{0}^{T} \int_{Q^{ϵ}} b^{ϵ} u^{ϵ} v ψ d x d t + \hat{E} \int_{0}^{T} \int_{Q^{ϵ}} f^{ϵ} u^{ϵ} v ψ d x d t \\ (74) & + ϵ \hat{E} \int_{0}^{T} \int_{\partial S^{ϵ}} h^{ϵ} u^{ϵ} v ψ d σ_{x} d t . \end{array}$

By convergences (32) and (33), we have $\begin{array}{l} lim_{ϵ \to 0} \hat{E} \int_{0}^{T} \int_{Q^{ϵ}} b^{ϵ} u^{ϵ} v ψ d x d t & = lim_{ϵ \to 0} \frac{\hat{E}}{| Y |} \int_{0}^{T} \int_{\tilde{Q^{ϵ}} \times Y^{*}} T_{*}^{ϵ} (b^{ϵ}) T_{*}^{ϵ} (u^{ϵ}) v ψ d y d x d t \\ (75) & = θ \hat{E} \int_{0}^{T} \int_{Q} b u v ψ d x d t \end{array}$ By convergences (32) and (34), we have $\begin{array}{l} \hat{E} \int_{0}^{T} \int_{Q^{ϵ}} f^{ϵ} u^{ϵ} v ψ d x d t & = lim_{ϵ \to 0} \frac{\hat{E}}{| Y |} \int_{0}^{T} \int_{\tilde{Q^{ϵ}} \times Y^{*}} T_{*}^{ϵ} (f^{ϵ}) T_{*}^{ϵ} (u^{ϵ}) v ψ d y d x d t \\ (76) & = θ \hat{E} \int_{0}^{T} \int_{Q} f u v ψ d x d t \end{array}$ As for the last term in right-hand side of (74), we have $\begin{array}{l} ϵ \int_{0}^{T} \int_{\partial S^{ϵ}} h (\frac{x}{ϵ}, t) u^{ϵ} v ψ d σ_{x} d t & = \frac{1}{| Y |} \int_{0}^{T} \int_{R^{n} \times \partial S} T_{b}^{ϵ} (h) (y, t) T_{b}^{ϵ} (u^{ϵ}) v ψ d x d σ_{y} d t \\ = \frac{1}{| Y |} \int_{0}^{T} \int_{\tilde{Q^{ϵ}} \times \partial S} T_{b}^{ϵ} (h) (y, t) T_{b}^{ϵ} (u^{ϵ}) v ψ d x d σ_{y} d t \\ = \frac{1}{| Y |} \int_{0}^{T} \int_{\tilde{Q^{ϵ}} \times \partial S} h (y, t) T_{b}^{ϵ} (u^{ϵ}) v ψ d x d σ_{y} d t \end{array}$ Now, 3. of Propostion 2 gives $\begin{array}{l} lim_{ϵ \to 0} \frac{1}{| Y |} \int_{0}^{T} \int_{\tilde{Q^{ϵ}} \times \partial S} h (y, t) T_{b}^{ϵ} (u^{ϵ}) v ψ d x d σ_{y} d t \\ (77) & = \frac{| \partial S |}{| Y |} T M_{(\partial S \times (0, T))} (h) \int_{0}^{T} \int_{Q} u ψ d x d t, \hat{P} -a.s. \end{array}$ Using convergence (75)–(77) with (73) and (74) one can easily obtain $\begin{array}{l} lim_{ϵ \to 0} \hat{E} \int_{0}^{T} \int_{Q^{ϵ}} A_{ϵ} \nabla u^{ϵ} (\nabla u^{ϵ}) v ψ d d x d t + lim_{ϵ \to 0} \hat{E} \int_{0}^{T} \int_{Q^{ϵ}} A_{ϵ} \nabla u^{ϵ} u^{ϵ} (\nabla v) ψ d x d t \\ = \frac{1}{| Y |} \hat{E} \int_{0}^{T} \int_{Q \times Y^{*}} A (\nabla u + \nabla_{y} \hat{u}) u \nabla v ψ d y d x d t \\ (78) & + \frac{1}{| Y |} \hat{E} \int_{0}^{T} \int_{Q \times Y^{*}} A (\nabla u + \nabla_{y} \hat{u}) (\nabla u + \nabla_{y} \hat{u}) v ψ d y d x d t \end{array}$ To identify the limits in the above equation, we use convergences (32) and (39) and Proposition 1, to get $\begin{array}{l} lim_{ϵ \to 0} \hat{E} \int_{0}^{T} \int_{Q^{ϵ}} A_{ϵ} \nabla u^{ϵ} u^{ϵ} (\nabla v) ψ d x d t \\ = lim_{ϵ \to 0} \hat{E} \frac{1}{| Y |} \int_{0}^{T} \int_{Q \times Y^{*}} A (y) T_{*}^{ϵ} (\nabla u^{ϵ}) T_{*}^{ϵ} (u^{ϵ}) T_{*}^{ϵ} (\nabla v) ψ d x d y d t \\ (79) & = \frac{1}{| Y |} \hat{E} \int_{0}^{T} \int_{Q \times Y^{*}} A (\nabla u + \nabla_{y} \hat{u}) u \nabla v ψ d y d x d t \end{array}$

From (78) and (79), we have $\begin{array}{l} lim_{ϵ \to 0} \hat{E} \int_{0}^{T} \int_{Q^{ϵ}} A_{ϵ} \nabla u^{ϵ} (\nabla u^{ϵ}) v ψ d x d t \\ (80) & = \frac{1}{| Y |} \hat{E} \int_{0}^{T} \int_{Q \times Y^{*}} A (\nabla u + \nabla_{y} \hat{u}) (\nabla u + \nabla_{y} \hat{u}) v ψ d y d x d t \end{array}$

Now using the ellipticity assumption on the matrix A, we get $\begin{array}{l} α \hat{E} \int_{0}^{T} {‖ T^{ϵ} (\nabla u^{ϵ}) - \tilde{\nabla u} - \nabla_{y} \hat{u} ‖}_{L^{2} (U \times Y^{*})}^{2} d t \\ ⩽ α \hat{E} \int_{0}^{T} v {‖ T^{ϵ} (\nabla u^{ϵ}) - \tilde{\nabla u} - \nabla_{y} \hat{u} ‖}_{L^{2} (R^{n} \times Y^{*})}^{2} d t \\ ⩽ \hat{E} \int_{0}^{T} \int_{R^{n} \times Y^{*}} A v (T^{ϵ} (\nabla u^{ϵ}) - \nabla u - \nabla_{y} \hat{u}) \cdot (T^{ϵ} (\nabla u^{ϵ}) - \nabla u - \nabla_{y} \hat{u}) d x d y d t \\ = \hat{E} \int_{0}^{T} \int_{R^{n} \times Y^{*}} A v T^{ϵ} (\nabla u^{ϵ}) T^{ϵ} (\nabla u^{ϵ}) d x d y d t \\ - \hat{E} \int_{0}^{T} \int_{R^{n} \times Y^{*}} A v T^{ϵ} (\nabla u^{ϵ}) A (\nabla u + \nabla_{y} \hat{u}) d x d y d t \\ - \hat{E} \int_{0}^{T} \int_{R^{n} \times Y^{*}} A v (\nabla u + \nabla_{y} \hat{u}) T^{ϵ} (\nabla u^{ϵ}) d x d y d t \\ (81) & + \hat{E} \int_{0}^{T} \int_{R^{n} \times Y^{*}} A v (\nabla u + \nabla_{y} \hat{u}) \cdot (\nabla u + \nabla_{y} \hat{u}) d x d t . \end{array}$ The right-hand side converges to zero using convergences (39) and (80). Thus (68) holds. As an immediate application to Lemma 2 (3ii) (69) also holds. Thus the proof is complete. □

Footnotes

Acknowledgements

The authors thank the anonymous reviewers for their useful comments that improved the paper. The first author would like to thank Deapship of scientific research at Majmaah University for supporting this work under the project number R-1441-4.

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