Two-scale homogenization of abstract linear time-dependent PDEs

Abstract

Many time-dependent linear partial differential equations of mathematical physics and continuum mechanics can be phrased in the form of an abstract evolutionary system defined on a Hilbert space. In this paper we discuss a general framework for homogenization (periodic and stochastic) of such systems. The method combines a unified Hilbert space approach to evolutionary systems with an operator theoretic reformulation of the well-established periodic unfolding method in homogenization. Regarding the latter, we introduce a well-structured family of unitary operators on a Hilbert space that allows to describe and analyze differential operators with rapidly oscillating (possibly random) coefficients. We illustrate the approach by establishing periodic and stochastic homogenization results for elliptic partial differential equations, Maxwell’s equations, and the wave equation.

Keywords

Periodic and stochastic homogenization unfolding abstract evolutionary equations Maxwell’s equations

1. Introduction

Homogenization of partial differential equations with rapidly oscillating (periodic or random) coefficients is a classical field and manifold methods have been introduced since the 1970s, e.g. the oscillating test function method, which relies on the construction of correctors [23,27], weak-convergence methods such as periodic and stochastic two-scale convergence, e.g. [1,2,4,15,20,26,59], or unfolding methods, e.g. [8,9,16,22,25,46]. Homogenization requires that heterogeneities (on small scales) of the partial differential equation are spatially homogeneous on a larger scale. Thus, typically a structural condition such as periodicity (or stationarity and ergodicity in the case of random materials) is assumed. Classical methods, in particular, two-scale convergence and unfolding methods, rely on the assumed structural condition. Moreover, while they are especially well-suited for certain classes of equations (e.g. of elliptic or parabolic type), their application to non-standard models often requires a subtle ad hoc extension.

Motivated by this, in this paper we introduce a unified strategy for homogenization of abstract evolutionary equations. The approach is genuinly operator theoretic and one of the challenges in this endeavour is to recast the requirement of “spatial homogeneity” into the operator theoretic framework; while for partial differential equations this is naturally done in form of a condition on the coefficients, for operator equations it is a priori unclear how to implement the concept of “spatial homogeneity”. Our approach is based on a combination of three concepts: The abstract unified theory for evolutionary equations introduced in [28] (see also [21,32,37]), the homogenization theory developed in [56] with plenty of applications (see references below), and the stochastic unfolding procedure recently introduced in [16,25]. Based on this we recover a large variety of classical homogenization results within a unified argument, but more importantly, our method also applies to equations that have not been accessible before, e.g. equations with rapidly oscillating type (elliptic-parabolic or hyperbolic-parabolic), see Section 5.

According to [28] (see also [32,34,57] and the references therein) a large variety of linear evolutionary equations of mathematical physics can be recast in the following form: $\begin{matrix} (M (\partial_{t, ν}) + A) u = F, \end{matrix}$ where $z \mapsto M (z)$ denotes a family of linear bounded operators on a Hilbert space H and A is a skew-selfadjoint densely defined operator on H; $M (\partial_{t, ν})$ is defined in the sense of an explicit functional calculus for $\partial_{t, ν}$ , the time derivative, which is established as a boundedly-invertible operator. We refer to [28] (see also [32,38,42,43,57]) for existence and other essential results regarding the above equation. We briefly recall the most important ingredients of this framework in Section 4.1.

In many applications one is interested in describing physical properties of systems (e.g., composites, alloys, metamaterials) that feature material heterogeneities on a small length (or time) scale, say $ε ≪ 1$ . In the above described framework this may be described by equations of the form $\begin{matrix} (1) & (M_{ε} (\partial_{t, ν}) + A) u = F, \end{matrix}$ where $M_{ε}$ denotes a sequence of operators with “coefficients oscillating on scale ε”. The goal of homogenization is to derive (by analyzing the limit $ε \to 0$ ) a simplified, effective equation, say an equation of the form $\begin{matrix} (2) & (M_{0} (\partial_{t, ν}) + A) u = F, \end{matrix}$ that captures the large scale properties of the original system. A partial answer to this problem is given by H-compactness results, which establish the existence of a (possibly non-unique and abstract) limiting operator $M_{0}$ and the convergence of solutions to (1) to solutions to (2) (along subsequences). A complete homogenization result additionaly identifies $M_{0}$ uniquely and establishes a homogenization formula that allows one to compute $M_{0}$ as a function of “the oscillating coefficients” of $M_{ε}$ . The latter is particularly interesting for modelling of microstructured materials, where the operator $M_{ε}$ comes in form of a partial differential equation with rapidly oscillating coefficients that encode the specific microstructure of the material. In the context of abstract evolutionary equations, homogenization has been studied in [47–50,52,54–57]. Apart from [47,49,54] only H-compactness results have been obtained, while complete results have only been developed in [47,49,54] for particular equations and under restrictive assumptions on the microstructure.

In the present paper we establish a way to incorporate information on the microstructure into the general abstract operator theoretic description and thus yield complete homogenization results in a systematic approach for a very general class of abstract evolution equations. For this purpose we rephrase the stochastic unfolding method of [16,25] in an operator theoretic form: While the unfolding method of partial differential equations is based on a linear and bounded unfolding operator, in our approach, we replace the latter by an abstract family of operators, see Section 2. In contrast to earlier works on abstract evolutionary equations, in this way we obtain an explicit description of the effective model.

The abstract homogenization results we obtain cover a large variety of problems. To illustrate this, we specifically reconsider periodic homogenization of elliptic partial differential equations, and obtain (as simple corollaries of our abstract theorem) stochastic homogenization results for the Maxwell’s equations and the wave equation. Additionally, we derive corrector type results for the considered examples that are based on specific properties of the partial differential equation under consideration. We emphasise that both in the Maxwell model as well as in the wave equation, our approach permits the consideration of highly oscillatory mixed type equations. This is in contrast to earlier approaches for example based on semi-group theory. In particular, when Maxwell’s equations are concerned, this is the first time, where it is possible to treat stochastically oscillating material coefficients that lead to variations between the eddy current approximation (i.e. vanishing dielectricity, positive conductivity) and classical Maxwell (i.e. positive dielectricity). For a more detailed account on this, we refer to Section 5.4.

Structure of the paper. In order to motivate the theory, in Section 1.1 we recall the standard setting for stochastic homogenization and provide the definition of stochastic unfolding. Section 2 provides the definitions of the unfolding operator and two-scale convergence from an abstract point of view. In the rest of that section we present some important properties of the latter notions. Section 3 is devoted to the derivation of a homogenization result for an abstract elliptic type problem. In Section 4.1 we briefly recall the setting for abstract evolutionary equations. Section 4.2 provides a homogenization result for an abstract evolutionary equation. Section 5 treats some particular examples of the previously discussed abstract theory.

1.1. A brief reminder on two-scale homogenization

In this section we recall some classical results and concepts from periodic and stochastic homogenization via two-scale methods with the intention to motivate our abstract framework of Section 2. To fix ideas, let $Q \subset R^{n}$ denote an open, bounded domain, $f \in L^{2} (Q)$ , and let $u_{ε} \in H_{0}^{1} (Q)$ denote the unique weak solution to $\begin{matrix} (3) & - \nabla \cdot (a (\frac{\cdot}{ε}) \nabla u_{ε}) = f in Q, \end{matrix}$ where $a : R^{n} \to R^{n \times n}$ denotes a uniformly elliptic coefficient field, i.e., $a : R^{n} \to R_{λ, Λ}^{n \times n}$ is measurable and $R_{λ, Λ}^{n \times n}$ denotes (for some fixed constants of ellipticity $0 < λ ⩽ Λ$ ) the set of matrices $a_{0} \in R^{n \times n}$ satisfying $a_{0} ξ \cdot ξ ⩾ λ | ξ |^{2}$ and $| a_{0} ξ | ⩽ Λ | ξ |$ (for all $ξ \in R^{n}$ ).

The periodic case. Periodic homogenization is concerned with the case that the coefficient field a in (3) is periodic, say $a (\cdot + k) = a (\cdot)$ a.e. in $R^{n}$ for all $k \in Z^{n}$ . Under this condition a classical result (e.g., [3,10,23]) states that there exists a uniformly elliptic matrix $a_{hom} \in R^{n \times n}$ such that ${(u_{ε})}_{ε > 0}$ weakly converges in $H_{0}^{1} (Q)$ (as $ε \to 0$ ) to the unique weak solution $u \in H_{0}^{1} (Q)$ to the homogenized equation $\begin{matrix} - \nabla \cdot (a_{hom} \nabla u) = f . \end{matrix}$ The homogenized coefficient matrix $a_{hom}$ is characterized by the formula $\begin{matrix} a_{hom} e_{i} = \int_{□} a (\nabla φ_{i} + e_{i}) d x (i \in {1, \dots, n}), \end{matrix}$ where ${□ : = [0, 1)}^{n}$ represents the reference cell of periodicity, $e_{i} \in R^{n}$ denotes the i’th unit vector, and $φ_{i} \in H_{loc}^{1} (R^{n})$ denotes a periodic solution to the corrector equation $\begin{matrix} (4) & - \nabla \cdot (a (\nabla φ_{i} + e_{i})) = 0 in R^{n} . \end{matrix}$ Thanks to the periodicity of a, (4) can be solved by lifting the equation to the Sobolev space of periodic functions with zero mean (where Poincaré’s inequality and uniform ellipticity of a implies coercivity of the elliptic operator $- \nabla \cdot (a \nabla)$ ). The corrector equation (4) is a key object in homogenization theory for elliptic equations, since its solution – the corrector $φ_{i}$ – captures the spatial oscillations of $u_{ε}$ induced by the heterogeneity of the coefficient field a. This can be expressed in terms of the (formal) asymptotic expansions (using Einstein’s summation convention) $\begin{matrix} u_{ε} (x) \approx u (x) + ε φ_{i} (\frac{x}{ε}) \partial_{i} u (x), \nabla u_{ε} (x) \approx \nabla u (x) + \nabla φ_{i} (\frac{x}{ε}) \partial_{i} u (x) \end{matrix}$ or (more precisely) in form of the two-scale convergence statement $\begin{matrix} \nabla u_{ε} \overset{2}{\to} \nabla u + \nabla φ_{i} \partial_{i} u, \end{matrix}$ where $\overset{2}{\to}$ denotes the notion of (strong) two-scale convergence introduced in [26] and further investigated in [1] (see also [20]) and $\nabla u + \nabla φ_{i} \partial_{i} u$ denotes the function $(x, y) \mapsto \nabla u (x) + \nabla φ_{i} (y) \partial_{i} u (x)$ . Also, another method closely related to two-scale convergence is the periodic unfolding method [8,9,46], which is based on an unfolding operator that equivalently characterizes two-scale convergence (see also [22]).

The stochastic case. In this case the coefficient field a is assumed to be a random object, i.e., ${(a (x))}_{x \in R^{n}}$ is viewed as a family of $R_{λ, Λ}^{n \times n}$ -valued random variables. Minimial requirements for stochastic homogenization (towards a deterministic limit) are stationarity and ergodicity. The former means that for any $x_{1}, \dots, x_{N} \in R^{n}$ and $z \in R^{n}$ , the distribution of $(a (x_{1} + z), \dots, a (x_{N} + z))$ is independent of the shift $z \in R^{n}$ . Ergodicity means that any shift-invariant (measurable) subset of the coefficients has probability 1 or 0. In their seminal work [27], Papanicolaou and Varadhan rephrased these conditions in an analytic framework that by now became a standard in stochastic homogenization. In the following we recall their framework and some objects involved in their approach. We refer to [17,27] for details (see also the lecture notes [24] for a self-contained presentation). The idea is to equip $Ω : = {a : R^{n} \to R_{λ, Λ}^{n \times n} : a measurable}$ , the set of all uniformly elliptic coefficient fields, with a probability measure μ and to draw a randomly from Ω according to μ. The precise setup is as follows:

Assumption 1.
Let $(Ω, Σ, μ)$ denote a probability space with a countably generated σ-algebra (that implies separability of $L^{2} (Ω)$ ), and let $τ = {τ_{x} : x \in R^{n}}$ be a group of measurable bijections $τ_{x} : Ω \to Ω$ such that:
(Group property). $τ_{0} = {id}_{Ω}$ and $τ_{x + y} = τ_{x} \circ τ_{y}$ for all $x, y \in R^{n}$ .

(Measure preservation). $μ (τ_{x} A) = μ (A)$ for all $A \in Σ$ and $x \in R^{n}$ .

(Measurability). $(ω, x) \mapsto τ_{x} ω$ is $(Σ \otimes L (R^{n}), Σ)$ -measurable ( $L (R^{n})$ denotes the Lebesgue-σ-algebra on $R^{n}$ ).

Let $a_{0} : Ω \to R_{λ, Λ}^{n \times n}$ be measurable, then $a (x, ω) : = a_{0} (τ_{x} ω)$ defines a uniformly elliptic, random coefficient field that is stationary (thanks to property (b)). Moreover, in this framework, the assumption of ergodicity reads: any shift-invariant set $A \in Σ$ (i.e., $τ_{x} A \subset A$ for all $x \in R^{n}$ ) satisfies $μ (A) \in {0, 1}$ .

Under the conditions of Assumption 1 and ergodicity, Papanicolaou and Varadhan proved the following homogenization result: For $f \in L^{2} (Q)$ and $ε > 0$ , let $u_{ε} \in H_{0}^{1} (Q) \otimes L^{2} (Ω)$ denote the unique (Lax–Milgram) solution to $\begin{matrix} (5) & - \nabla \cdot (a_{0} (τ_{\frac{x}{ε}} ω) \nabla u_{ε} (x, ω)) = f (x) in Q \times Ω . \end{matrix}$ Then there exists a uniformly elliptic coefficient matrix $a_{hom} \in R^{n \times n}$ (only depending on μ and $a_{0}$ ) such that $u_{ε}$ weakly converges in $H_{0}^{1} (Q) \otimes L^{2} (Ω)$ to the unique solution $u \in H_{0}^{1} (Q)$ of the homogenized equation $- \nabla \cdot (a_{hom} \nabla u) = f$ in Q. The proof of Papanicolaou and Varadhan is based on Tartar’s method of oscillating test functions and the main difficulty is to give sense to the corrector equation (4) and the corrector $φ_{i}$ in the stochastic case.

Stochastic two-scale convergence and unfolding. An alternative proof of the above homogenization result was introduced in [4], based on a stochastic counterpart of two-scale convergence – the notion of stochastic two-scale convergence in the mean (see also [2]). In particular, a bounded sequence ${(u_{ε})}_{ε}$ in $L^{2} (Q) \otimes L^{2} (Ω)$ is said to two-scale converge in the mean to a function $u \in L^{2} (Q) \otimes L^{2} (Ω)$ , if for all test functions $η \in C_{c}^{\infty} (Q)$ and $φ \in L^{2} (Ω)$ , $\begin{matrix} (6) & \int_{Ω} \int_{Q} u_{ε} (x, ω) η (x) φ (τ_{\frac{x}{ε}} ω) d x d μ (ω) \to \int_{Ω} \int_{Q} u (x, ω) η (x) φ (ω) d x d μ (ω) . \end{matrix}$

Recently, in [16,25] Heida and the first two authors reconsidered the notion of two-scale convergence in the mean from the perspective of an unfolding operator (see also [45]). This approach is motivated by (and shares many similarities with) the well-established notion of periodic unfolding [8]. In the following we briefly recall its definition (for more detail, see [16]).

For $η \in L^{2} (Q)$ and $φ \in L^{2} (Ω)$ , we define $\begin{matrix} (7) & T_{ε} (η \otimes φ) (x, ω) = η (x) φ (τ_{- \frac{x}{ε}} ω) . \end{matrix}$ Using the measure preserving property from Assumption 1 (b), it follows that $\begin{matrix} {‖ T_{ε} (η \otimes φ) ‖}_{L^{2} (Q) \otimes L^{2} (Ω)} = ‖ η \otimes φ ‖_{L^{2} (Q) \otimes L^{2} (Ω)}, \end{matrix}$ and by the density of the linear span of simple tensor products $\begin{matrix} L^{2} (Q) \overset{a}{\otimes} L^{2} (Ω) : = lin {(x, ω) \mapsto φ (x) ψ (ω); φ \in L^{2} (Q), ψ \in L^{2} (Ω)} \subset L^{2} (Q) \otimes L^{2} (Ω) dense, \end{matrix}$ $T_{ε}$ extends to a linear isometry $T_{ε} : L^{2} (Q) \otimes L^{2} (Ω) \to L^{2} (Q) \otimes L^{2} (Ω)$ . Moreover, the equality $T_{ε} T_{- ε} = 1$ on simple tensors implies that $T_{ε}$ is in fact unitary. A simple consequence of this definition is that for a bounded sequence $u_{ε} \in L^{2} (Q) \otimes L^{2} (Ω)$ the weak convergence $T_{ε} u_{ε} ⇀ u$ is equivalent to stochastic two-scale convergence in the mean of $u_{ε}$ to u (in sense of (6)). Also, note that using the stochastic unfolding operator, we might rephrase equation (5) as $\begin{matrix} (8) & - \nabla \cdot (T_{- ε} a_{0} T_{ε} \nabla u) = f . \end{matrix}$ Let us anticipate that in Section 2 we go one step further and reconsider the idea of unfolding on an abstract operator theoretic level where
$L^{2} (Q)$ and $L^{2} (Ω)$ are replaced by general Hilbert spaces $H_{d}$ and $H_{s}$ ,

the stochastic unfolding operator is replaced by a family of well-structured unitary operators,

$- \nabla \cdot$ and ∇ are replaced by densely defined closed (unbounded) linear operators $C_{d}^{*}$ and $C_{d}$ .
Note that in explicit applications $C_{d}$ might also represent differential operator differently from the gradient ∇ see Remark 4.6(b).
1.2. Model example: Homogenization of Maxwell’s equation

Our motivation is to develop a unified, operator theoretic approach to homogenization (in the mean) of linear evolutionary problems with periodic, quasiperiodic or random (stationary) coefficients. The abstract unfolding strategy that we propose applies to a variety of linear PDEs and we present some examples in Section 5. Here we briefly explain one of the examples—stochastic homogenization of Maxwell’s equations, which we discuss in detail in Section 5.4. In particular, for $Q \subseteq R^{3}$ open, we consider the following system of equations: Find $(u_{ε}, q_{ε}) : R \times Q \to C^{3} \times C^{3}$ such that $\begin{array}{l} (9) & \begin{array}{l} \partial_{t} (η_{ε} u_{ε}) + σ_{ε} u_{ε} - curl q_{ε} = f, \\ \partial_{t} (μ_{ε} q_{ε}) + curl u_{ε} = g, \end{array} \end{array}$ where $(f, g) : R \to L^{2} {(Q)}^{3} \oplus L^{2} {(Q)}^{3}$ is a datum and the random, oscillating coefficients are given in the form $η_{ε} (x, ω) = η_{0} (x, τ_{\frac{x}{ε}} ω)$ , $σ_{ε} (x, ω) = σ_{0} (x, τ_{\frac{x}{ε}} ω)$ , $μ_{ε} (x, ω) = μ_{0} (x, τ_{\frac{x}{ε}} ω)$ with $η_{0}, σ_{0}, μ_{0} \in L^{\infty} {(Q \times Ω)}^{3 \times 3}$ that satisfy suitable assumptions. Note that the solution depends on $ω \in Ω$ , which we see as a random configuration of the medium, and therefore we view the solution also as a random field, i.e., we seek functions such that at (almost) each time instance $t \in R$ satisfy $(u_{ε} (t), q_{ε} (t)) \in {(L^{2} (Q) \otimes L^{2} (Ω))}^{3} \oplus {(L^{2} (Q) \otimes L^{2} (Ω))}^{3}$ . In fact, we phrase (9) in the form of the operator equation (1) given on the functional space $L_{ν}^{2} (R; {(L^{2} (Q) \otimes L^{2} (Ω))}^{3} \oplus {(L^{2} (Q) \otimes L^{2} (Ω))}^{3})$ , which is an exponentially weighted $L^{2}$ -space with a parameter $ν \in R$ (see Section 4.1). In the limit $ε \to 0$ , we derive a two-scale homogenized system, that in the case of ergodic coefficients reads: Find $(u_{0}, χ_{1}, q_{0}, χ_{2}) \in L_{ν}^{2} (R; L^{2} {(Q)}^{3} \oplus (L^{2} (Q) \otimes L_{pot}^{2} (Ω)) \oplus L^{2} {(Q)}^{3} \oplus (L^{2} (Q) \otimes L_{pot}^{2} (Ω)))$ such that $\begin{array}{l} \partial_{t} (E [η_{0} (u_{0} + χ_{1})]) + E [σ_{0} (u_{0} + χ_{1})] - curl q_{0} = f, \\ \partial_{t} (E [μ_{0} (q_{0} + χ_{2})]) + curl u_{0} = g, \\ - {div}_{ω} (\partial_{t} (η_{0} (u_{0} + χ_{1})) + σ_{0} (u_{0} + χ_{1})) = 0, \\ - {div}_{ω} (\partial_{t} (μ_{0} (q_{0} + χ_{2}))) = 0, \end{array}$ where $E [\cdot]$ denotes the mathematical expectation in $(Ω, Σ, μ)$ and ${div}_{ω}$ is the stochastic divergence, which is defined in Section 5.1. We may view the first two equations as the effective Maxwell system for the deterministic variables $(u_{0}, q_{0})$ and the correctors $(χ_{1}, χ_{2})$ , that account for the microstructure evolution in the material, are determined by the last two corrector equations, cf. Remark 5.16. In particular, we get, as $ε \to 0$ , $\begin{matrix} T_{ε} (u_{ε}, q_{ε}) ⇀ (u_{0} + χ_{1}, q_{0} + χ_{2}) weakly in L_{ν}^{2} (R; {(L^{2} (Q) \otimes L^{2} (Ω))}^{3} \oplus {(L^{2} (Q) \otimes L^{2} (Ω))}^{3}) . \end{matrix}$ A similar result in a periodic-stochastic setting has been obtained in [39]. With our operator theoretic approach we are able to dispose of the continuity condition on the coefficients in the slow variable. Next, we can treat highly oscillatory mixed type equations by only requiring the sum of $σ_{0}$ and $η_{0}$ to be positive. Moreover, for suitably regular right-hand sides $(f, g)$ , we obtain the following corrector result: $\begin{matrix} ‖ u_{ε} - T_{- ε} χ_{1} - u_{0} ‖_{L_{ν}^{2} (R; L^{2} (Q) \otimes L^{2} {(Ω)}^{3})}^{2} + ‖ q_{ε} - T_{- ε} χ_{2} - q_{0} ‖_{L_{ν}^{2} (R; L^{2} (Q) \otimes L^{2} {(Ω)}^{3})}^{2} \to 0 as ε \to 0 . \end{matrix}$ See Section 5.4 for details.

2. The operator theoretic setting for unfolding

In this section we introduce the setting for abstract two-scale convergence and provide some compactness results which will be useful in the following sections. Throughout this section, we let $H_{d}$ and $H_{s}$ be Hilbert spaces, $m, n \in N$ ; ‘d’ and ‘s’ are a reminder of ‘deterministic’ and ‘stochastic’.

Abstract “differential” operators. We consider densely defined closed linear operators $\begin{matrix} C_{d} : dom (C_{d}) \subseteq H_{d}^{m} \to H_{d}^{n}, C_{s} : dom (C_{s}) \subseteq H_{s}^{m} \to H_{s}^{n} . \end{matrix}$ For examples ranging from classical operators from vector analysis to differential operators on manifolds or more involved situations, we refer to Remark 4.6 below. Given a Hilbert space H the canonical extension of $C_{d}$ to operators from the Hilbert space tensor product $H_{d}^{m} \otimes H$ to $H_{d}^{n} \otimes H$ will be—as a rule—again denoted by $C_{d}$ ; and similarly for $C_{s}$ . In the applications, we have in mind, we extend $C_{d}$ to the space $H_{d}^{m} \otimes H_{s}$ attaining values in $H_{d}^{n} \otimes H_{s}$ and $C_{s}$ as closed, densely defined linear operator from $H_{d} \otimes H_{s}^{m}$ to $H_{d} \otimes H_{s}^{n}$ . Note that we will further identify $H_{d}^{m} \otimes H_{s} = {(H_{d} \otimes H_{s})}^{m} = H_{d} \otimes H_{s}^{m}$ . Thus, the extended operators $C_{d}$ and $C_{s}$ are both operators defined on (subsets of) ${(H_{d} \otimes H_{s})}^{m}$ with values in ${(H_{d} \otimes H_{s})}^{n}$ . With this in mind, we require the following compatibility conditions for $C_{s}$ and $C_{d}$ : $\begin{array}{l} (10) & \begin{array}{l} \exists D_{1} \subseteq H_{d} dense \forall x \in D_{1}, y \in H_{s}^{m} : x \otimes y \in dom (C_{d}), \\ \exists D_{2} \subseteq H_{d} dense \forall x \in D_{2}, y \in H_{s}^{n} : x \otimes y \in dom (C_{d}^{*}), \\ \exists E_{1} \subseteq H_{s} dense \forall y \in E_{1}, x \in H_{d}^{m} : x \otimes y \in dom (C_{s}), \\ \exists E_{2} \subseteq H_{s} dense \forall y \in E_{2}, x \in H_{d}^{n} : x \otimes y \in dom (C_{s}^{*}) . \end{array} \end{array}$ The first and third conditions are trivially satisfied, if $m = 1$ , the second and fourth if $n = 1$ . In applications discussed later on, $D_{1} = D_{2} = C_{c}^{\infty} (Q)$ for some open $Q \subseteq R^{n}$ (a similar choice can be made for $E_{1}$ , $E_{2}$ ).

Remark 2.1.
Note the following consequence of our notation convention. Given a bounded linear operator $T \in L (H, K)$ for H, K Hilbert spaces, we have $T C_{d} \subseteq C_{d} T$ , and $T C_{s} \subseteq C_{s} T$ , see also Lemma 2.5 below.

Unfolding family. We call a strongly continuous map $T : R ∖ {0} \to L (H_{d} \otimes H_{s})$ taking values in the set of unitary operators an unfolding family, if the following structural hypotheses are satisfied (where the action of $T_{ε}$ on powers of $H_{d} \otimes H_{s}$ is understood “component-wise”, re-using the notation): $\begin{array}{l} (11) & T_{ε} = T_{- ε}^{- 1} (ε \in R ∖ {0}), \\ (12) & ε C_{d} T_{- ε} φ = ε T_{- ε} C_{d} φ + C_{s} T_{- ε} φ (φ \in dom (C_{d}) \cap dom (C_{s}) \subseteq {(H_{d} \otimes H_{s})}^{m}, ε \in R ∖ {0}), \\ (13) & T_{ε} C_{s} \subseteq C_{s} T_{ε} (ε \in R ∖ {0}), \\ (14) & T_{ε} v = v (v \in ker (C_{s}), ε \in R ∖ {0}) . \end{array}$ We remark here that (12) (in conjunction with (10)) implies, $\begin{matrix} (15) & ε C_{d}^{} T_{- ε} φ = ε T_{- ε} C_{d}^{} φ + T_{- ε} C_{s}^{} φ (φ \in dom (C_{s}^{}) \cap dom (C_{d}^{}) \subseteq {(H_{d} \otimes H_{s})}^{n}, ε \in R ∖ {0}) . \end{matrix}$ The latter will be particularly important, when we discuss time-dependent problems.
Remark 2.2.
The stochastic unfolding operator introduced in Section 1.1 together with $C_{d} = \nabla$ , and $C_{s}$ denoting the stochastic gradient, satisfies the above assumptions (see Section 5).

Within the above setting we define (stochastic) two-scale convergence as follows.
Definition 2.3.
Let ${(u_{ε})}_{ε > 0}$ in $H_{d} \otimes H_{s}$ . Then ${(u_{ε})}_{ε}$ is said to strongly (weakly) two-scale converge to* $u \in H_{d} \otimes H_{s}$ (we also use the notation $u_{ε} \overset{2}{\to} u$ ( $u_{ε} \overset{2}{⇀} u$ )), if $\begin{matrix} T_{ε} u_{ε} \to u (T_{ε} u_{ε} ⇀ u) \end{matrix}$ strongly (weakly) as $ε \to 0$ .
Remark 2.4.
Strictly speaking two-scale convergence is only defined for families ${(u_{ε})}_{ε > 0}$ . In the following, we will however also say that “a subsequence of ${(u_{ε})}_{ε > 0}$ (weakly/strongly) two-scale converges to some u”. By this, we mean that there exists a sequence ${(ε_{k})}_{k}$ in $(0, \infty)$ converging to 0 such that $T_{ε_{k}} u_{ε_{k}} \to u$ weakly as $k \to \infty$ . As the particular subsequence will not be important in the considerations we are aiming for, we shall however re-use ε and dispense with k.

In the following we establish various properties that we shall exploit in our abstract homogenization scheme, and highlight analogies to stochastic two-scale convergence in the mean and periodic unfolding, respectively. We begin with an abstract counter part of [4, Theorem 3.7]. In order to avoid cluttered notation as much as possible, we often write $H_{d} \otimes H_{s}$ regardless of the number of components of the objects under consideration. We begin with an auxiliary result.

Unless explicitly stated otherwise, we shall not assume condition (14) in this section. The conditions (10)–(13), however, are assumed to be in effect. Lemma 2.5.
Let P be the orthogonal projection onto $ker (C_{s}) \subseteq {(H_{d} \otimes H_{s})}^{m}$ . Then $P T_{ε} = T_{ε} P$ for all $ε \in R ∖ {0}$ .
Proof.
Let $ε \in R ∖ {0}$ . Then we have $T_{ε} C_{s} \subseteq C_{s} T_{ε}$ by (13). Since $T_{ε}^{} = T_{- ε}$ by (11), we have $C_{s}^{} T_{- ε} = {(T_{ε} C_{s})}^{} \supseteq {(C_{s} T_{ε})}^{} = T_{- ε} C_{s}^{}$ . Hence, we deduce for all $ε \in R ∖ {0}$ $\begin{matrix} T_{ε} C_{s}^{} C_{s} \subseteq C_{s}^{} T_{ε} C_{s} \subseteq C_{s}^{} C_{s} T_{ε} . \end{matrix}$ Since $C_{s}^{} C_{s}$ is self-adjoint and $T_{ε}$ continuous, we deduce that for all bounded, measurable functions $f : σ (C_{s}^{} C_{s}) \to R$ , $T_{ε} f (C_{s}^{} C_{s}) = f (C_{s}^{} C_{s}) T_{ε}$ . In particular, $f = χ_{{0}}$ is a possible choice. $χ_{{0}} (C_{s}^{} C_{s})$ is the orthogonal projection onto $ker (C_{s}^{} C_{s})$ . Note that $ker (C_{s}^{} C_{s}) = ker (C_{s})$ . Indeed, $ker (C_{s}) \subseteq ker (C_{s}^{} C_{s})$ is trivial; for $φ \in ker (C_{s}^{} C_{s})$ , we test the equation $C_{s}^{} C_{s} φ = 0$ with $φ \in ker (C_{s}^{} C_{s}) \subseteq dom (C_{s}^{} C_{s}) \cap dom (C_{s})$ to obtain $⟨ C_{s} φ, C_{s} φ ⟩ = 0$ , that is, $ker (C_{s}) \supseteq ker (C_{s}^{} C_{s})$ . We infer $P = χ_{{0}} (C_{s}^{} C_{s})$ , which yields the assertion. □
Lemma 2.6.
Let $K \subseteq H_{s}$ be a closed subspace and P the projection onto K. Then $P C_{d} \subseteq C_{d} P$ .
Proof.
We have identified $C_{d}$ with $C_{d} \otimes 1_{H_{s}}$ and P with $1_{H_{d}} \otimes P$ . Thus, the assertion follows from standard tensor-product theory of operators in Hilbert spaces, see, e.g., [56, Appendix]. □
Lemma 2.7.
Let ${(u_{ε})}_{ε}$ and ${(ε C_{d} u_{ε})}_{ε}$ be bounded in ${(H_{d} \otimes H_{s})}^{m}$ and ${(H_{d} \otimes H_{s})}^{n}$ , respectively. Then there are subsequences of ${(u_{ε})}_{ε}$ and ${(ε C_{d} u_{ε})}_{ε}$ as well as $u \in dom (C_{s})$ such that ${(u_{ε})}_{ε}$ weakly two-scale converges to u and ${(ε C_{d} u_{ε})}_{ε}$ weakly two-scale converges to $C_{s} u$ .
Proof.
There exist subsequences such that ${(T_{ε} u_{ε})}_{ε}$ weakly converges to $u \in {(H_{d} \otimes H_{s})}^{m}$ and ${(ε T_{ε} C_{d} u_{ε})}_{ε}$ weakly converges to $v \in {(H_{d} \otimes H_{s})}^{n}$ , respectively. We will show that $u \in dom (C_{s})$ and $C_{s} u = v$ . For this, we let $g \in dom (C_{s}^{}) \cap dom (C_{d}^{})$ and compute (using (15)) $\begin{array}{l} ⟨ ε T_{ε} C_{d} u_{ε}, g ⟩ & = ⟨ u_{ε}, ε C_{d}^{} T_{- ε} g ⟩ \\ = ⟨ u_{ε}, ε T_{- ε} C_{d}^{} g + T_{- ε} C_{s}^{} g ⟩ \\ = ⟨ ε T_{ε} u_{ε}, C_{d}^{} g ⟩ + ⟨ T_{ε} u_{ε}, C_{s}^{} g ⟩ \\ \to ⟨ u, C_{s}^{} g ⟩ . \end{array}$ Since $dom (C_{s}^{}) \cap dom (C_{d}^{})$ is an operator core for $C_{s}^{*}$ by (10), we obtain $u \in dom (C_{s})$ and $C_{s} u = (w-) {lim}_{ε \to 0} ε T_{ε} C_{d} u_{ε}$ . □
Theorem 2.8 (Compactness).

Assume ( 10 )–( 13 ).

Let ${(u_{ε})}_{ε}$ be uniformly bounded in $dom (C_{d})$ . Assume that ${(u_{ε})}_{ε}$ and ${(C_{d} u_{ε})}_{ε}$ weakly two-scale converge. Then there exist $u \in dom (C_{d}) \cap ker (C_{s})$ and $v \in \overline{ran} (C_{s})$ such that $u_{ε} \overset{2}{⇀} u$ and $C_{d} u_{ε} \overset{2}{⇀} C_{d} u + v$ .

Let ${(q_{ε})}_{ε}$ be uniformly bounded in $dom (C_{d}^{*})$ . Assume that ${(q_{ε})}_{ε}$ and ${(C_{d}^{*} q_{ε})}_{ε}$ weakly two-scale converge. Then there exist $q \in dom (C_{d}^{*}) \cap ker (C_{s}^{*})$ and $w \in \overline{ran} (C_{s}^{*})$ such that $q_{ε} \overset{2}{⇀} q$ and $C_{d}^{*} q_{ε} \overset{2}{⇀} C_{d}^{*} q + w$ .

Assume, in addition, $m = 1$ and that ( 14 ) holds. Let ${(u_{ε})}_{ε}$ be uniformly bounded in $dom (C_{d})$ . Then there exists $u \in dom (C_{d}) \cap ker (C_{s})$ and $v \in \overline{ran} (C_{s}) \cap \overline{ran} {(C_{s}^{*})}^{n} \subseteq {(H_{d} \otimes H_{s})}^{n}$ such that (a subsequence of) ${(u_{ε})}_{ε}$ weakly two-scale converges to u and ${(C_{d} u_{ε})}_{ε}$ weakly two-scale converges to $C_{d} u + v$ .

Proof.
(a) For suitable subsequences, denote by u and $\overline{v}$ the corresponding weak two-scale-limits of ${(u_{ε})}_{ε}$ and ${(C_{d} u_{ε})}_{ε}$ . By Lemma 2.7, we deduce that $u \in ker (C_{s})$ . By weak continuity of P, it is easy to see that ${(P T_{ε} u_{ε})}_{ε}$ weakly converges to $P u = u$ . By Lemma 2.5, we have $P T_{ε} = T_{ε} P$ . Thus, since $P C_{d} \subseteq C_{d} P$ , we get $P u_{ε} \in dom (C_{d}) \cap ker (C_{s}) \subseteq dom (C_{d}) \cap dom (C_{s})$ and so, by (12), $\begin{matrix} C_{d} P T_{ε} u_{ε} = T_{ε} C_{d} P u_{ε} = T_{ε} P C_{d} u_{ε} . \end{matrix}$ Thus, since the right-hand side is uniformly bounded in $ε > 0$ , so is the left-hand side. From the closedness of $C_{d} P$ , it follows that $u = P u \in dom (C_{d})$ .

We define $v : = \overline{v} - C_{d} u$ . Then we compute for suitable $ε > 0$ and all $w \in ker (C_{s}^{}) \cap dom (C_{d}^{})$ : $\begin{matrix} ⟨ T_{ε} C_{d} u_{ε}, w ⟩ = ⟨ T_{ε} u_{ε}, C_{d}^{} w ⟩ . \end{matrix}$ Letting $ε \to 0$ on both sides, we obtain $\begin{matrix} ⟨ C_{d} u + v, w ⟩ = ⟨ C_{d} u, w ⟩, \end{matrix}$ which leads to $\begin{matrix} ⟨ v, w ⟩ = 0 (w \in ker (C_{s}^{}) \cap dom (C_{d}^{})) . \end{matrix}$ Since $ker (C_{s}^{}) \cap dom (C_{d}^{})$ is dense in $ker (C_{s}^{})$ by (10), we infer $v \in ker {(C_{s}^{})}^{⊥} = \overline{ran} (C_{s})$ .

(b) By symmetry of the conditions (10)–(13) in $C_{d}$ and $C_{d}^{}$ , the proof follows analogously to (a).

(c) Choose subsequences of ${(T_{ε} u_{ε})}_{ε}$ and ${(T_{ε} C_{d} u_{ε})}_{ε}$ that weakly converge to some u and $\overline{v}$ . By Lemma 2.7, we deduce that $C_{s} u = (w-) {lim}_{ε \to 0} ε T_{ε} C_{d} u_{ε} = 0 \cdot \overline{v}$ , which yields $u \in ker (C_{s})$ . Moreover, we may choose a weakly convergent subsequence of ${(u_{ε})}_{ε}$ in $dom (C_{d})$ . Denote the limit by $\tilde{u}$ . Let P be the orthogonal projection onto $ker (C_{s})$ . As $m = 1$ , $ker (C_{s}) \subseteq H_{s}$ and Lemma 2.6 yields $P C_{d} \subseteq C_{d} P$ . In particular, we obtain that ${(P u_{ε})}_{ε}$ weakly converges to $P \tilde{u}$ in $dom (C_{d})$ . Furthermore, by $T_{ε} v = v$ on $ker (C_{s})$ (see (14)), we infer $T_{ε} P = P$ . Moreover, from Lemma 2.5, we have $T_{ε} P = P T_{ε}$ . Thus, we get that $\begin{matrix} u = P w- lim_{ε \to 0} T_{ε} u_{ε} = w- lim_{ε \to 0} P T_{ε} u_{ε} = w- lim_{ε \to 0} T_{ε} P u_{ε} = w- lim_{ε \to 0} P u_{ε} = P \tilde{u} \in dom (C_{d}) . \end{matrix}$ We define $v : = \overline{v} - C_{d} u$ . Then we compute for suitable $ε > 0$ and all $w \in ker (C_{s}^{}) \cap dom (C_{d}^{})$ : $\begin{matrix} ⟨ T_{ε} C_{d} u_{ε}, w ⟩ = ⟨ u_{ε}, T_{- ε} C_{d}^{} w + \frac{1}{ε} T_{- ε} C_{s}^{} w ⟩ = ⟨ T_{ε} u_{ε}, C_{d}^{} w ⟩ . \end{matrix}$ Letting $ε \to 0$ on both sides, we obtain $\begin{matrix} ⟨ C_{d} u + v, w ⟩ = ⟨ C_{d} u, w ⟩, \end{matrix}$ which leads to $\begin{matrix} ⟨ v, w ⟩ = 0 (w \in ker (C_{s}^{}) \cap dom (C_{d}^{})) . \end{matrix}$ Since $ker (C_{s}^{}) \cap dom (C_{d}^{})$ is dense in $ker (C_{s}^{})$ by (10), we infer $v \in ker {(C_{s}^{})}^{⊥} = \overline{ran} (C_{s})$ .

Finally, we let $w \in ker (C_{s}) \cap dom (C_{d}^{})$ . Then, using $T_{ε} v = v$ for all $v \in ker (C_{s})$ , we have for suitable $ε > 0$ $\begin{matrix} ⟨ T_{ε} C_{d} u_{ε}, w ⟩ = ⟨ C_{d} u_{ε}, w ⟩ = ⟨ u_{ε}, C_{d}^{} w ⟩ = ⟨ u_{ε}, T_{- ε} C_{d}^{} w ⟩ = ⟨ T_{ε} u_{ε}, C_{d}^{} w ⟩ . \end{matrix}$ Letting $ε \to 0$ , we get $\begin{matrix} ⟨ C_{d} u + v, w ⟩ = ⟨ u, C_{d}^{} w ⟩ = ⟨ C_{d} u, w ⟩ . \end{matrix}$ Hence, since $dom (C_{d}^{}) \cap ker (C_{s})$ is dense in $ker (C_{s})$ as $m = 1$ , we deduce $\begin{matrix} ⟨ v, w ⟩ = 0 (w \in ker (C_{s})), \end{matrix}$ which leads to $v \in \overline{ran} (C_{s}^{})$ . □

3. Homogenization of (abstract) elliptic problems

In order to illustrate our so far findings, we shall treat an elliptic homogenization problem. Note that this is the abstract variant of the classical result [4, Theorem 4.1.1] (see Section 5.2 for the periodic case). We assume throughout $m = 1$ and (10)–(14).

Theorem 3.1.
Let $f \in ker (C_{s})$ and assume that $ran (C_{d})$ is closed and $C_{d}$ is injective. Let $A \in L ({(H_{d} \otimes H_{s})}^{n})$ satisfy $Re A = \frac{1}{2} (A + A^{}) ⩾ c$ for some* $c > 0$ . For $ε > 0$ consider $\begin{matrix} (16) & C_{d}^{} T_{- ε} A T_{ε} C_{d} u_{ε} = f . \end{matrix}$ Then* ${(u_{ε})}_{ε}$ is well-defined and uniformly bounded in $dom (C_{d})$ and strongly two-scale converges to some $u \in dom (C_{d}) \cap ker (C_{s})$ . Moreover, ${(C_{d} u_{ε})}_{ε}$ weakly two-scale converges to $C_{d} u + v$ for some $v \in \overline{ran} (C_{s}) \cap \overline{ran} {(C_{s}^{})}^{n}$ , where u, v are the unique solutions of the following system of equations* $\begin{array}{l} (17) & C_{d}^{} P A (C_{d} u + v) = f, \\ (18) & C_{s}^{} A (C_{d} u + v) = 0 . \end{array}$ Here, P is the orthogonal projection onto $ker (C_{s})$ .

Before we come to a proof of Theorem 3.1, we address well-posedness of (16). For this, we will use the direct approach outlined in [44, Theorem 3.1]; see also [54, Theorem 2.9].
Theorem 3.2.
Assume the conditions of Theorem 3.1 to be in effect. Then for all $ε > 0$ there exists a unique $u_{ε} \in dom (C_{d})$ satisfying ( 16 ). Moreover, we have that ${(u_{ε})}_{ε}$ is uniformly bounded in $dom (C_{d})$ .
Proof.
Let $ι : ran (C_{d}) ↪ {(H_{d} \otimes H_{s})}^{n}$ denote the canonical embedding. Then $ι^{}$ is the orthogonal projection onto $ran (C_{d})$ , see [33, Lemma 3.2]. Moreover, from $ker (C_{d}^{}) = ran {(C_{d})}^{⊥}$ , it is easy to see that $\begin{matrix} C_{d}^{} T_{- ε} A T_{ε} C_{d} = C_{d}^{} ι ι^{} T_{- ε} A T_{ε} ι ι^{} C_{d} . \end{matrix}$ By the closed graph theorem $ι^{} C_{d}$ is continuously invertible; by [44, Lemma 2.4 and Corollary 2.5] ${(ι^{} C_{d})}^{} = C_{d}^{} ι$ is also continuously invertible. Next, from $Re A ⩾ c$ it follows that $Re T_{- ε} A T_{ε} ⩾ c$ . In consequence, we obtain $Re ι^{} T_{- ε} A T_{ε} ι ⩾ c ι^{} ι$ . Thus, using that $ran (C_{d}^{})$ is closed and dense in $H_{d} \otimes H_{s}$ by the injectivity of $C_{d}$ , we obtain $\begin{matrix} (19) & u_{ε} = {(ι^{} C_{d})}^{- 1} {(ι^{} T_{- ε} A T_{ε} ι)}^{- 1} {(C_{d}^{} ι)}^{- 1} f, \end{matrix}$ which yields uniqueness of solutions of (16). Multiplying this equality by $ι^{} C_{d}$ , we infer $\begin{matrix} (20) & ι^{} C_{d} u_{ε} = {(ι^{} T_{- ε} A T_{ε} ι)}^{- 1} {(C_{d}^{} ι)}^{- 1} f . \end{matrix}$ The equalities (19) and (20) together with $\begin{matrix} ‖ {(ι^{} T_{- ε} A T_{ε} ι)}^{- 1} ‖ ⩽ \frac{1}{c} \end{matrix}$ yield that ${(u_{ε})}_{ε}$ is uniformly bounded in $dom (C_{d})$ . □

The next result settles uniqueness of the homogenized equations stated in Theorem 3.1. The rationale is similar to the one in [4, Theorem 4.1.1]; however we do not need to impose the curl-condition nor do we use any variant of ‘Kozlov’s identity’ (see [4, Lemma 2.4 and the subsequent remark]).
Lemma 3.3.
Let* $u \in dom (C_{d}) \cap ker (C_{s})$ and $v \in \overline{ran} (C_{s}) \cap \overline{ran} {(C_{s}^{})}^{n}$ satisfy (* 17 ) and ( 18 ) with $f = 0$ . Assume that $C_{d}$ is injective. Then $u = 0$ and $v = 0$ .
Proof.
We define $ζ : = C_{d} u + v$ . Since $ζ - C_{d} u = v \in \overline{ran} (C_{s})$ and $A (C_{d} u + v) \in ker (C_{s}^{})$ , by (18), $\begin{matrix} 0 = ⟨ A ζ, v ⟩ = ⟨ A ζ, ζ - C_{d} u ⟩ . \end{matrix}$ Thus, we deduce from (17) with $f = 0$ using $u \in dom (C_{d}) \cap ker (C_{s})$ $\begin{matrix} ⟨ A ζ, ζ ⟩ = ⟨ A ζ, C_{d} u ⟩ = ⟨ A ζ, C_{d} P u ⟩ = ⟨ P A ζ, C_{d} u ⟩ = ⟨ C_{d}^{} P A ζ, u ⟩ = 0 . \end{matrix}$ Thus, by $Re A ⩾ c$ , we infer $ζ = 0$ . Since $ker {(C_{s})}^{n} ∋ C_{d} u ⊥ v \in \overline{ran} {(C_{s}^{})}^{n}$ , we obtain the assertion. □

The following simple lemma will be useful in the proof of Theorem 3.1. It provides a recovery construction for the weak two-scale limit of the sequence $C_{d} u_{ε}$ .
Lemma 3.4.
Let* $u \in dom (C_{d}) \cap ker (C_{s})$ and $v \in \overline{ran} (C_{s})$ . For $δ > 0$ , there exists $φ_{δ} \in dom (C_{d}) \cap dom (C_{s})$ such that $\begin{matrix} (21) & ‖ C_{s} φ_{δ} - v ‖ ⩽ δ . \end{matrix}$ Moreover, for $φ_{δ, ε} : = T_{- ε} u + ε T_{- ε} φ_{δ}$ , we obtain $\begin{matrix} T_{ε} φ_{δ, ε} \to u, T_{ε} C_{d} φ_{δ, ε} \to C_{d} u + C_{s} φ_{δ} (as ε \to 0) . \end{matrix}$
Proof.
By the definition of $\overline{ran} (C_{s})$ and using (10), we obtain that there exists $φ_{δ} \in dom (C_{d}) \cap dom (C_{s})$ which satisfies (21). Also, since $T_{ε} φ_{δ, ε} = u + ε φ_{δ}$ , it follows that $T_{ε} φ_{δ, ε} \to u$ . Furthermore, using (12)–(13) we compute $\begin{matrix} C_{d} φ_{δ, ε} = T_{- ε} C_{d} u + ε T_{- ε} C_{d} φ_{δ} + T_{- ε} C_{s} φ_{δ} . \end{matrix}$ This implies that $T_{ε} C_{d} φ_{δ, ε} \to C_{d} u + C_{s} φ_{δ}$ . □
Proof of Theorem 3.1.
The family ${(u_{ε})}_{ε > 0}$ is well-defined and bounded in $dom (C_{d})$ by Theorem 3.2. By Theorem 2.8 (c), we find a subsequence of ${(u_{ε})}_{ε > 0}$ , $u \in dom (C_{d}) \cap ker (C_{s})$ , $v \in \overline{ran} {(C_{s}^{})}^{n} \cap \overline{ran} (C_{s})$ such that ${(u_{ε})}_{ε}$ and ${(C_{d} u_{ε})}_{ε}$ weakly two-scale converge to u and $C_{d} u + v$ , respectively. Let $φ \in dom (C_{d})$ and $ψ \in dom (C_{s}) \cap dom (C_{d})$ . Then we compute for suitable $ε > 0$ using (16) $\begin{array}{l} ⟨ f, φ ⟩ + ⟨ ε f, T_{- ε} ψ ⟩ & = ⟨ T_{- ε} A T_{ε} C_{d} u_{ε}, C_{d} P φ + ε C_{d} T_{- ε} ψ ⟩ \\ = ⟨ T_{- ε} A T_{ε} C_{d} u_{ε}, C_{d} P φ + ε T_{- ε} C_{d} ψ + C_{s} T_{- ε} ψ ⟩ \\ = ⟨ A T_{ε} C_{d} u_{ε}, T_{ε} C_{d} P φ ⟩ + ⟨ A T_{ε} C_{d} u_{ε}, ε C_{d} ψ ⟩ + ⟨ A T_{ε} C_{d} u_{ε}, C_{s} ψ ⟩, \end{array}$ where we used $ε C_{d} T_{- ε} ψ = ε T_{- ε} C_{d} ψ + C_{s} T_{- ε} ψ$ and $T_{- ε} C_{s} ψ = C_{s} T_{- ε} ψ$ . Since $C_{d} P φ \in ker (C_{s})$ , we have $T_{ε} C_{d} P φ = T_{ε} P C_{d} φ = P C_{d} φ$ . Thus, we obtain $\begin{matrix} ⟨ f, φ ⟩ + ⟨ ε f, T_{- ε} ψ ⟩ = ⟨ A T_{ε} C_{d} u_{ε}, P C_{d} φ ⟩ + ⟨ A T_{ε} C_{d} u_{ε}, ε C_{d} ψ ⟩ + ⟨ A T_{ε} C_{d} u_{ε}, C_{s} ψ ⟩ . \end{matrix}$ In this equality we may let $ε \to 0$ and obtain the asserted equalities (17) and (18). As the solutions to (17) and (18) are unique by Lemma 3.3, we deduce that any subsequence of ${(u_{ε})}_{ε}$ and ${(C_{d} u_{ε})}_{ε}$ weakly two-scale converges to u and $C_{d} u + v$ , which implies that both ${(u_{ε})}_{ε}$ and ${(C_{d} u_{ε})}_{ε}$ weakly two-scale converge without the choice of subsequences.

In order to show strong convergence of ${(T_{ε} u_{ε})}_{ε}$ , we consider $φ_{δ}$ and $φ_{δ, ε}$ defined as in Lemma 3.4 (for u, v the solution of (17)–(18)). Since $C_{d}$ is injective and it has closed range, it follows (see, e.g., [44, Remark 3.2(b)]) that there exists $c_{0} > 0$ such that $‖ φ ‖ ⩽ c_{0} ‖ C_{d} φ ‖$ for all $φ \in dom (C_{d})$ . As a result of this, we obtain $\begin{matrix} (22) & ‖ u_{ε} - φ_{δ, ε} ‖ ⩽ c_{0} ‖ C_{d} (u_{ε} - φ_{δ, ε}) ‖ = c_{0} ‖ T_{ε} C_{d} (u_{ε} - φ_{δ, ε}) ‖ . \end{matrix}$ Using the assumption $Re A ⩾ c$ (in the inequality) and that $u_{ε}$ solves (16) (in the equality), we obtain $\begin{array}{l} c {‖ T_{ε} C_{d} (u_{ε} - φ_{δ, ε}) ‖}^{2} & ⩽ Re ⟨ A T_{ε} C_{d} (u_{ε} - φ_{δ, ε}), T_{ε} C_{d} (u_{ε} - φ_{δ, ε}) ⟩ \\ = Re ⟨ f, u_{ε} - φ_{δ, ε} ⟩ - Re ⟨ A T_{ε} C_{d} φ_{δ, ε}, T_{ε} C_{d} (u_{ε} - φ_{δ, ε}) ⟩ . \end{array}$ Since $f \in ker (T_{ε} - 1)$ (for any $ε \neq 0$ ), we conclude that the first term on the right-hand side equals $Re ⟨ f, T_{ε} u_{ε} - T_{ε} φ_{δ, ε} ⟩$ . As a result of this and using the properties of $φ_{δ, ε}$ (Lemma 3.4), we obtain that for fixed $δ > 0$ , $\begin{matrix} \underset{ε \to 0}{lim sup} {‖ T_{ε} C_{d} (u_{ε} - φ_{δ, ε}) ‖}^{2} ⩽ - Re ⟨ A (C_{d} u + C_{s} φ_{δ}), v - C_{s} φ_{δ} ⟩ . \end{matrix}$ Moreover, using the above and (22) we obtain (using that the second term vanishes in the limit $ε \to 0$ by Lemma 3.4) for some $c_{1} ⩾ 0$ and all $δ > 0$ $\begin{matrix} \underset{ε \to 0}{lim sup} (‖ u_{ε} - φ_{δ, ε} ‖^{2} + ‖ T_{ε} φ_{δ, ε} - u ‖^{2} + {‖ T_{ε} C_{d} (u_{ε} - φ_{δ, ε}) ‖}^{2}) ⩽ c_{1} ‖ C_{d} u + C_{s} φ_{δ} ‖ ‖ v - C_{s} φ_{δ} ‖ . \end{matrix}$ By the choice of $φ_{δ}$ (Lemma 3.4), in the limit $δ \to 0$ the right-hand side vanishes. Consequently, we find a diagonal sequence $δ (ε) \to 0$ (as $ε \to 0$ ) such that $\begin{matrix} (23) & ‖ u_{ε} - φ_{δ (ε), ε} ‖ + ‖ T_{ε} φ_{δ (ε), ε} - u ‖ + ‖ T_{ε} C_{d} (u_{ε} - φ_{δ (ε), ε}) ‖ \to 0 as ε \to 0 . \end{matrix}$ Finally, this yields $\begin{matrix} ‖ T_{ε} u_{ε} - u ‖ ⩽ ‖ T_{ε} u_{ε} - T_{ε} φ_{δ (ε), ε} ‖ + ‖ T_{ε} φ_{δ (ε), ε} - u ‖ \to 0 as ε \to 0 . \end{matrix}$ □

The above proof implies the following abstract corrector type result (see (23)). Corollary 3.5.
Assume the conditions of Theorem* 3.1 to be in effect and let $u_{ε}$ , u, v be given as in Theorem 3.1 . There exists a sequence $φ_{ε} \in dom (C_{d}) \cap dom (C_{s})$ such that $\begin{matrix} ε φ_{ε} \to 0, T_{ε} C_{d} (u_{ε} - ε T_{- ε} φ_{ε}) \to C_{d} u strongly in H_{d} \otimes H_{s} . \end{matrix}$

4. Homogenization of abstract evolutionary equations

4.1. A Hilbert space framework for evolutionary equations

For the application to time-dependent problems in the subsequent sections, we shall specify the particular framework, we are working in. For this, we recall some results from [28], where the setting was introduced for the first time. We shall also refer to [18, Section 2] (or [37] for a more introductory type text) for more details, when properties of the time-derivative to be introduced in the following are concerned.

To begin with, we define for $ν \in R_{⩾ 0}$ the space $L_{ν}^{2} (R; H)$ of (equivalence classes of) Hilbert space H-valued functions f being Bochner measurable and satisfy $\begin{matrix} ‖ f ‖_{ν}^{2} : = \int_{R} {‖ f (t) ‖}_{H}^{2} exp (- 2 ν t) d t < \infty . \end{matrix}$ $L_{ν}^{2} (R; H)$ is a Hilbert space under the norm $‖ \cdot ‖_{ν}$ . Denoting by $H_{ν}^{1} (R; H)$ the (first) Sobolev space of weakly differentiable functions with distributional derivative representable as an element in $L_{ν}^{2} (R; H)$ , we define the time-derivative as the operator $\begin{matrix} \partial_{t, ν} : H_{ν}^{1} (R; H) \subseteq L_{ν}^{2} (R; H) \to L_{ν}^{2} (R; H), f \mapsto f^{'} . \end{matrix}$ It turns out that $\partial_{t, ν}$ is a normal operator. More particularly, $\partial_{t, ν}$ admits a spectral representation as multiplication operator in $L^{2} (R; H)$ . For this, we define the multiplication-by-the-argument operator $\begin{matrix} m : dom (m) \subseteq L^{2} (R; H) \to L^{2} (R; H), φ \mapsto (ξ \mapsto ξ φ (ξ)), \end{matrix}$ where $\begin{matrix} dom (m) : = {φ \in L^{2} (R; H); (ξ \mapsto ξ φ (ξ)) \in L^{2} (R; H)} . \end{matrix}$ Define the (H-valued) Fourier transformation $F : L^{2} (R; H) \to L^{2} (R; H)$ as the unitary extension of $\begin{matrix} F f : = (ξ \mapsto \frac{1}{\sqrt{2 π}} \int_{R} exp (- i t ξ) f (t) d t) (f \in L^{1} (R; H) \cap L^{2} (R; H)) . \end{matrix}$ Moreover, we define for $ν \in R_{⩾ 0}$ the (obviously) unitary operator $\begin{matrix} exp (- ν m) : L_{ν}^{2} (R; H) \to L^{2} (R; H), f \mapsto (t \mapsto e^{- ν t} f (t)), \end{matrix}$ and we set $\begin{matrix} L_{ν} : = F exp (- ν m), \end{matrix}$ the Fourier–Laplace transformation. With the latter transformation at hand the explicit spectral theorem for $\partial_{t, ν}$ (see also [18, Corollary 2.5(c)]) reads $\begin{matrix} \partial_{t, ν} = L_{ν}^{*} (i m + ν) L_{ν} . \end{matrix}$ The latter equation yields a functional calculus for $\partial_{t, ν}$ (or for $\partial_{t, ν}^{- 1}$ .). In fact, let $ν_{0} \in R_{⩾ 0}$ and $ν > ν_{0}$ and let $M \in H (C_{Re > ν_{0}}; L (H, K))$ , where K is another Hilbert space and $\begin{matrix} H (C_{Re > ν_{0}}; L (H, K)) : = {M : C_{Re > ν_{0}} \to L (H, K); M analytic} . \end{matrix}$ Then we define $\begin{matrix} M (\partial_{t, ν}) : = L_{ν}^{*} (M (i m + ν)) L_{ν}, \end{matrix}$ where $\begin{matrix} (M (i m + ν) φ) (ξ) : = M (i ξ + ν) φ (ξ) (ξ \in R) \end{matrix}$ for all compactly supported, continuous functions $φ : R \to H$ .

The reason we focus on analytic functions $M$ rather than continuous or even just measurable functions is that analyticity of $M$ and causality of $M (\partial_{t, ν})$ are strongly related, which is apparent from the Paley–Wiener theorem, see e.g., [28, Section 2] or [34, Theorem 2.4], [57, Section 1.2].

We are now in the position to formulate the well-posedness theorem, which can be viewed as underlying structure of many linear equations in mathematical physics and continuum mechanics.

Theorem 4.1 ([28, Solution Theory]).

Let $ν_{0} \in R_{⩾ 0}$ , $ν > ν_{0}$ , H Hilbert space, $A : dom (A) \subseteq H \to H$ a skew-self-adjoint operator, i.e., $A = - A^{*}$ , and $M \in H (C_{Re > ν_{0}}; L (H))$ . Assume there exists $c > 0$ such that $\begin{matrix} Re ⟨ M (z) φ, φ ⟩ ⩾ c ⟨ φ, φ ⟩ (z \in C_{Re > ν_{0}}, φ \in H) . \end{matrix}$ Then the operator $B : = M (\partial_{t, ν}) + A$ is densely defined on $L_{ν}^{2} (R; H)$ . Moreover, B is closable and $\overline{B}$ is continuously invertible in $L_{ν}^{2} (R; H)$ , $‖ {\overline{B}}^{- 1} ‖ ⩽ 1 / c$ and ${\overline{B}}^{- 1}$ is causal, that is, $\begin{matrix} 1_{R_{⩽ a}} {\overline{B}}^{- 1} 1_{R_{⩽ a}} = 1_{R_{⩽ a}} {\overline{B}}^{- 1} (a \in R) . \end{matrix}$

Proof.
For $z \in C_{Re > ν_{0}}$ it follows from [40, Lemma 2.12] that $M (z) + A$ is continuously invertible in H with inverse satisfying $‖ {(M (z) + A)}^{- 1} ‖ ⩽ 1 / c$ . Thus, [43, Remark 2.3(a)] applies and we obtain the assertion. □
Remark 4.2 (Initial Value Problems).

The usage of exponentially weighted spaces poses an implicit homogeneous boundary condition at $- \infty$ ; due to the integrability requirement with respect to the temporal variable. This does not imply any restrictions on the problem class considered, though. In fact, the seemingly missing case of initial value problems can be dealt with by considering suitable right-hand sides instead; see [32, Chapter 6]. A similar functional analytic setting with $[0, \infty)$ does work, as well; see [41]. In this case, however, the time-derivative operator ceases to be a normal operator. Thus, the problem class and in particular the material law $M = M (\partial_{t, ν})$ cannot be formulated by means of the well-established and explicit functional calculus for normal operators outlined here. Hence, the focus on $R$ as time-domain both makes it possible to treat inhomogeneous initial value problems and is the more convenient functional analytic setting.

We define $\begin{matrix} H^{\infty} (C_{Re > ν_{0}}; L (H, K)) : = {M \in H (C_{Re > ν_{0}}; L (H, K)); M bounded}, \end{matrix}$ which turns into a Banach space if endowed with the supremum norm.

Proposition 4.3.
Assume the hypotheses of Theorem 4.1 . Then $\begin{matrix} [C_{Re > ν_{0}} ∋ z \mapsto {(M (z) + A)}^{- 1} \in L (H)] \in H^{\infty} (C_{Re > ν}; L (H)) . \end{matrix}$
Proof.
The claim follows from [40, Lemma 2.12] and the fact that composition of analytic mappings are analytic again. □

It will be the next theorem, which forms the basic result for the convergence results to be followed in the next section.
Theorem 4.4.
Let $ν_{0} \in R_{⩾ 0}$ , H, K separable Hilbert spaces. Let ${(S_{ε})}_{ε > 0}$ be a bounded family in $H^{\infty} (C_{Re > ν_{0}}; L (H, K))$ and let $S_{0} \in H^{\infty} (C_{Re > ν_{0}}; L (H, K))$ . Assume that for all $z \in C_{Re > ν_{0}}$ we have $\begin{matrix} S_{ε} (z) \to S_{0} (z) (ε \to 0) \end{matrix}$ in the weak operator topology of $L (H, K)$ .

Then $S_{ε} (\partial_{t, ν}) \to S_{0} (\partial_{t, ν})$ in the weak operator topology of $L (L_{ν}^{2} (R; H), L_{ν}^{2} (R; K))$ .

For the proof of Theorem 4.4, we shall use the following result. Theorem 4.5 ([50, Theorem 4.3] and [47, Lemma 3.5]).

Let $Ω \subseteq C$ open, H, K separable Hilbert spaces. Let ${(S_{ε})}_{ε}$ be a bounded family in $H^{\infty} (C_{Re > ν_{0}}; L (H, K))$ . Then there exists $\begin{matrix} T \in H^{\infty} (C_{Re > ν_{0}}; L (H, K)) \end{matrix}$ and a nullsequence ${(ε_{k})}_{k \in N}$ in $(0, \infty)$ satisfying for all $K \subseteq Ω$ compact $\begin{matrix} S_{ε_{k}} (z) \to T (z) (k \to \infty, z \in K) \end{matrix}$ in the weak operator topology of $L (H, K)$ . Moreover, if $Ω = C_{Re > ν_{0}}$ , then $\begin{matrix} S_{ε_{k}} (\partial_{t, ν}) \to T (\partial_{t, ν}) (k \to \infty) \end{matrix}$ in the weak operator topology of $L (L_{ν}^{2} (R; H), L_{ν}^{2} (R; K))$ for all $ν > ν_{0}$ .

Proof of Theorem 4.4.
Choose ${(ε_{k})}_{k}$ and T according to Theorem 4.5. Then, for $z \in C_{Re > ν_{0}}$ , we obtain $\begin{matrix} S_{0} (z) = (τ_{w} -) lim_{k \to \infty} S_{ε_{k}} (z) = T (z) . \end{matrix}$ Moreover, we get $S_{ε_{k}} (\partial_{t, ν}) \to S_{0} (\partial_{t, ν})$ in the weak operator topology of $L (L_{ν}^{2} (R; H), L_{ν}^{2} (R; K))$ . The subsequence principle concludes the proof. □

4.2. Applications to time-dependent-type problems

We shall treat a subclass of evolutionary equations discussed in the previous section. The particular cases treated here cover the heat equation, the wave equation, the Maxwell’s equations and even systems of mixed type formulated on possibly rough domains, which do not need to satisfy any boundedness conditions, as we shall demonstrate in the subsequent sections by means of examples.

More specifically, in this section, we confine ourselves to the following class of problems. Define $H_{0} : = {(H_{d} \otimes H_{s})}^{m}$ , $H_{1} : = {(H_{d} \otimes H_{s})}^{n}$ and $H = H_{0} \oplus H_{1}$ . Let $ν_{0} ⩾ 0$ , $M_{k} : C_{Re > ν_{0}} \to L (H_{k})$ be analytic and assume that $Re M_{k} (z) ⩾ c$ for all $z \in C_{Re > ν_{0}}$ , $k \in {0, 1}$ and some $c > 0$ . We need to restrict ourselves to a certain class of right-hand sides. For this we set $\begin{array}{l} H_{0} : = ker (C_{s}) \cap ⋂_{ε \in R ∖ {0}} ker (T_{ε} - 1) \subseteq H_{0}, H_{1} : = ker (C_{s}^{*}) \cap ⋂_{ε \in R ∖ {0}} ker (T_{ε} - 1) \subseteq H_{1} . \end{array}$ We remark that in the applications, e.g., to stochastic homogenization (see Sections 5.4 and 5.5), the above choice allows the consideration of deterministic right-hand sides (that is a standard assumption in stochastic homogenization), i.e., $L^{2} (Q) \otimes C \subseteq H_{i}$ .

For $(f, g) \in H : = H_{0} \oplus H_{1}$ , we consider for $ε > 0$ $\begin{matrix} (24) & (T_{- ε} (\begin{matrix} M_{0} (z) & 0 \\ 0 & M_{1} (z) \end{matrix}) T_{ε} + (\begin{matrix} 0 & C_{d}^{*} \\ - C_{d} & 0 \end{matrix})) (\begin{matrix} u_{ε} \\ q_{ε} \end{matrix}) = (\begin{matrix} f \\ g \end{matrix}) . \end{matrix}$ Note that the operator $\begin{matrix} {\overline{B}}_{ε} : = \overline{(T_{- ε} (\begin{matrix} M_{0} (\partial_{t, ν}) & 0 \\ 0 & M_{1} (\partial_{t, ν}) \end{matrix}) T_{ε} + (\begin{matrix} 0 & C_{d}^{*} \\ - C_{d} & 0 \end{matrix}))} \end{matrix}$ is continuously invertible in $L_{ν}^{2} (R; H)$ by Theorem 4.1 applied to $M = T_{- ε} diag (M_{0}, M_{1}) T_{ε}$ and $A = (\begin{matrix} 0 & C_{d}^{*} \\ - C_{d} & 0 \end{matrix})$ . We shall also introduce $\begin{matrix} (25) & S_{ε} (z) : = {\overline{(T_{- ε} (\begin{matrix} M_{0} (z) & 0 \\ 0 & M_{1} (z) \end{matrix}) T_{ε} + (\begin{matrix} 0 & C_{d}^{*} \\ - C_{d} & 0 \end{matrix}))}}^{- 1} \end{matrix}$ for all $ε > 0$ and $z \in C_{Re > ν_{0}}$ . By Proposition 4.3, we have that ${(S_{ε})}_{ε > 0}$ is a bounded family in $H^{\infty} (C_{Re > ν_{0}}; L (H))$ .

Our aim in this section will be to construct an operator-valued function $S_{0} : z \mapsto S_{0} (z)$ such that $\begin{matrix} T_{ε} S_{ε} (z) \to S_{0} (z) (ε \to 0) \end{matrix}$ in the weak operator topology (in an appropriate space). It will turn out that $S_{0} (z)$ can be written as $S_{0} (z) = {(\tilde{M} (z) + (\begin{matrix} 0 & C_{d}^{*} \\ - C_{d} & 0 \end{matrix}))}^{- 1}$ for suitable $\tilde{M} (z)$ to be described explicitly below. Finally, we shall conclude with an application of Theorem 4.4 to obtain a homogenization result for the full time-dependent problem.

We will suppress the dependence of $u_{ε}$ and $q_{ε}$ on z for the time being; at the end of this section we come back to this.

Remark 4.6.

We refer to [32,34,57] and the references therein for an instance of the many examples that are covered by this equation. We will consider some special cases in the next section. The rather involved homogenization result for a suitable class of non-diagonal M is treated in [52]. In that paper, however, a compactness assumption had to be introduced, which we do not assume here. Moreover, in [52] the local problem is given implicitly and there is no criterion ensuring convergence without the extraction of subsequences. However, in the framework of so-called ‘nonlocal H-convergence’ a convergence result for Maxwell’s equations was shown in [54]. For a setting strictly confined to periodic problems defined on the whole Euclidean space as underlying domain, quantitative results can be found in [11].

There is a large variety of possible choices for $C_{d}$ . Most notably these choices do not only include one of the possibilities div, curl, or grad. If one considers differential equations on manifolds, $C_{d}$ can be the exterior derivative on q-forms or acting on a list of q-forms, see [29]. Furthermore, the type of derivative may change on different areas of the domain. For instance, considering an elasto-acoustic interaction problem, the considered operators can be rather involved, see [30]. Other permitted cases are electro-magneto-elasto-dynamic transmission problems, where different regions of the domain follow Maxwell’s equations and others are of elastic waves type, see [31].

First of all we establish existence and boundedness of ${(u_{ε}, q_{ε})}_{ε}$ in $dom (C_{d}) \oplus dom (C_{d}^{})$ . The result is as plain as it is to establish the uniform bound in H. We provide some more details as follows.
Proposition 4.7.
For all* $ε > 0$ , $(u_{ε}, q_{ε})$ is well-defined as a solution of ( 24 ). Moreover, the family ${(u_{ε}, q_{ε})}_{ε}$ is uniformly bounded in $dom (C_{d}) \oplus dom (C_{d}^{})$ .
Proof.
Note that $(\begin{matrix} 0 & C_{d}^{} \\ - C_{d} & 0 \end{matrix})$ is skew-self-adjoint and $Re T_{- ε} M (z) T_{ε} ⩾ c$ for all $z \in C_{Re > ν}$ , $ε > 0$ and some $c > 0$ . Hence, the assertion follows from [40, Lemma 2.12] applied to $M = T_{- ε} M (z) T_{ε}$ and $A = (\begin{matrix} 0 & C_{d}^{} \\ - C_{d} & 0 \end{matrix})$ . □

The main result of this section is presented next, that is, we will now present the main step to establish convergence of ${(T_{ε} S_{ε} (z))}_{ε}$ . We shall show convergence of a subsequence first. Then, we will prove uniqueness of the limit, so that the following theorem actually also holds without the choice of subsequences. Below, $P_{ker (C_{s})}$ and $P_{ker (C_{s}^{})}$ denote the orthogonal projections to $ker (C_{s})$ and $ker (C_{s}^{})$ .
Theorem 4.8.
Assume (* 10 )–( 13 ). For $ε > 0$ let ${(u_{ε}, q_{ε})}_{ε}$ be given by ( 24 ). Then (a subsequence of) ${(u_{ε})}_{ε}$ , ${(q_{ε})}_{ε}$ weakly two-scale converge to some $u \in dom (C_{d}) \cap ker (C_{s})$ , $q \in dom (C_{d}^{}) \cap ker (C_{s}^{})$ . $(u, q)$ satisfies the following system of equations $\begin{array}{l} (26) & P_{ker (C_{s})} M_{0} (z) u + P_{ker (C_{s})} C_{d}^{} q = f, \\ (27) & P_{ker (C_{s}^{})} M_{1} (z) q - P_{ker (C_{s}^{})} C_{d} u = g . \end{array}$
Proof.
By Proposition 4.7, ${(u_{ε})}_{ε}$ and ${(q_{ε})}_{ε}$ are bounded in $dom (C_{d})$ and $dom (C_{d}^{})$ , respectively. Hence, by Theorem 2.8 (a) and (b), we find subsequences of ${(u_{ε})}_{ε}$ and ${(q_{ε})}_{ε}$ as well as $u \in dom (C_{d}) \cap ker (C_{s})$ , $v \in \overline{ran} (C_{s})$ , $q \in dom (C_{d}^{}) \cap ker (C_{s}^{})$ and $w \in \overline{ran} (C_{s}^{})$ such that ${(u_{ε})}_{ε}$ , ${(q_{ε})}_{ε}$ , ${(C_{d} u_{ε})}_{ε}$ , and ${(C_{d}^{} q_{ε})}_{ε}$ weakly two-scale converge to u, q, $C_{d} u + v$ , and $C_{d}^{} q + w$ , respectively. Next, using (24), we have for all $φ \in {(H_{d} \otimes H_{s})}^{m}$ and suitable $ε > 0$ $\begin{array}{l} ⟨ f, φ ⟩ & = ⟨ T_{ε} f, φ ⟩ \\ = ⟨ P_{ker (C_{s})} f, T_{- ε} φ ⟩ \\ = ⟨ T_{- ε} M_{0} (z) T_{ε} u_{ε} + C_{d}^{} q_{ε}, T_{- ε} P_{ker (C_{s})} φ ⟩ \\ = ⟨ M_{0} (z) T_{ε} u_{ε}, P_{ker (C_{s})} φ ⟩ + ⟨ T_{ε} C_{d}^{} q_{ε}, P_{ker (C_{s})} φ ⟩ \\ \to ⟨ M_{0} (z) u, P_{ker (C_{s})} φ ⟩ + ⟨ C_{d}^{} q + w, P_{ker (C_{s})} φ ⟩ \\ = ⟨ P_{ker (C_{s})} M_{0} (z) u, φ ⟩ + ⟨ P_{ker (C_{s})} C_{d}^{} q, φ ⟩ (ε \to 0) . \end{array}$ We obtain $\begin{matrix} P_{ker (C_{s})} M_{0} (z) u + P_{ker (C_{s})} C_{d}^{} q = f, \end{matrix}$ which yields (26). (27) follows analogously. □

The next result is a reformulation of the system (26)–(27). For this, we introduce the canonical embeddings $\begin{array}{l} ι_{s} : ker (C_{s}) ↪ {(H_{d} \otimes H_{s})}^{m}, \\ ι_{s^{}} : ker (C_{s}^{}) ↪ {(H_{d} \otimes H_{s})}^{n} . \end{array}$ Since $u = ι_{s} ι_{s}^{} u$ and $q = ι_{s^{}} ι_{s^{}}^{} q$ , we obtain the following.
Corollary 4.9.
Let $(u, q)$ satisfy the system ( 26 )–( 27 ). Then $\begin{array}{l} (28) & ι_{s}^{} M_{0} (z) ι_{s} ι_{s}^{} u + ι_{s}^{} C_{d}^{} ι_{s^{}} ι_{s^{}}^{} q = ι_{s}^{} f, \\ (29) & ι_{s^{}}^{} M_{1} (z) ι_{s^{}} ι_{s^{}}^{} q - ι_{s^{}}^{} C_{d} ι_{s} ι_{s}^{} u = ι_{s^{}}^{} g . \end{array}$
Remark 4.10.
The equations for u and q from Corollary 4.9 can be written in the following block operator matrix form $\begin{matrix} ((\begin{matrix} {\tilde{M}}_{0} (z) & 0 \\ 0 & {\tilde{M}}_{1} (z) \end{matrix}) + (\begin{matrix} 0 & ι_{s}^{} C_{d}^{} ι_{s^{}} \\ - ι_{s^{}}^{} C_{d} ι_{s} & 0 \end{matrix})) (\begin{matrix} \tilde{u} \\ \tilde{q} \end{matrix}) = (\begin{matrix} \tilde{f} \\ \tilde{g} \end{matrix}), \end{matrix}$ where ${\tilde{M}}_{0} (z) = ι_{s}^{} M_{0} (z) ι_{s}$ , ${\tilde{M}}_{1} (z) = ι_{s^{}}^{} M_{1} (z) ι_{s^{}}$ , $\tilde{u} = ι_{s}^{} u$ , $\tilde{q} = ι_{s^{}}^{} q$ , $\tilde{f} = ι_{s}^{} f$ and $\tilde{g} = ι_{s^{}}^{} g$ .

From this remark we obtain the homogenized* problem as follows. Namely, $\begin{matrix} ((\begin{matrix} {\tilde{M}}_{0} (z) & 0 \\ 0 & {\tilde{M}}_{1} (z) \end{matrix}) + (\begin{matrix} 0 & {\tilde{C_{d}}}^{} \\ - \tilde{C_{d}} & 0 \end{matrix})) (\begin{matrix} w \\ r \end{matrix}) = (\begin{matrix} h \\ j \end{matrix}), \end{matrix}$ where $\tilde{C_{d}} = \overline{ι_{s^{}}^{} C_{d} ι_{s}}$ and ${\tilde{C_{d}}}^{} = \overline{ι_{s}^{} C_{d}^{} ι_{s^{}}}$ . In particular, we deduce that $(\tilde{u}, \tilde{q})$ is uniquely determined, see [40, Lemma 2.12].

This observation combined with Corollary 4.9 yields that the claim of Theorem 4.8 is true even without* choosing subsequences. In any case, we can formulate the following homogenization result for time-dependent homogenization problems, which is one of the main results of this article. We define $S_{ε} (z) \in L (H, H)$ by $\begin{matrix} S_{ε} (z) (f, g) : = S_{ε} (z) (\begin{matrix} f \\ g \end{matrix}), \end{matrix}$ where $S_{ε} (z)$ is given by (25). We shall also define $S_{0} (z) \in L (H, H)$ via $\begin{matrix} S_{0} (z) (f, g) : = (\begin{matrix} ι_{s} & 0 \\ 0 & ι_{s^{}} \end{matrix}) {((\begin{matrix} {\tilde{M}}_{0} (z) & 0 \\ 0 & {\tilde{M}}_{1} (z) \end{matrix}) + (\begin{matrix} 0 & {\tilde{C_{d}}}^{} \\ - \tilde{C_{d}} & 0 \end{matrix}))}^{- 1} (\begin{matrix} ι_{s}^{} & 0 \\ 0 & ι_{s^{}}^{} \end{matrix}) (\begin{matrix} f \\ g \end{matrix}) . \end{matrix}$
Theorem 4.11.
Assume (* 10 )–( 13 ) and that H is separable. Then we have $\begin{matrix} T_{ε} S_{ε} (\partial_{t, ν}) \to S_{0} (\partial_{t, ν}) (ε \to 0) \end{matrix}$ in the weak operator topology of $L (L_{ν}^{2} (R; H), L_{ν}^{2} (R; H))$ .
Proof.
For $z \in C_{Re > ν_{0}}$ we obtain with Theorem 4.8 (in the version without choosing subsequences, which is justified by the subsequence principle in combination with Corollary 4.9 and the representation in Remark 4.10) that $T_{ε} S_{ε} (z) \to S_{0} (z)$ in the weak operator topology of $L (H, H)$ . Thus, the assertion follows from Theorem 4.4. □

Strong convergence. In the following we show that the above weak convergence may be improved to strong (two-scale) convergence upon assuming that the right-hand side is differentiable in time. Otherwise, we also obtain convergence of the appropriately mollified sequence of solutions. Interpreting this strong convergence suitably in the specific examples in Section 5 we obtain certain corrector type results.

In the following we denote by $(u_{ε}, q_{ε}) \in L_{ν}^{2} (R, H)$ the unique solution to $\begin{matrix} (30) & {\overline{B}}_{ε} (\binom{u_{ε}}{q_{ε}}) = (\binom{f}{g}), where B_{ε} = (T_{- ε} (\begin{matrix} M_{0} (\partial_{t, ν}) & 0 \\ 0 & M_{1} (\partial_{t, ν}) \end{matrix}) T_{ε} + (\begin{matrix} 0 & C_{d}^{} \\ - C_{d} & 0 \end{matrix})) . \end{matrix}$ Also, we write $(u, q) \in L_{ν}^{2} (R, ker (C_{s}) \oplus ker (C_{s}^{}))$ for the unique solution to $\begin{matrix} (31) & {\overline{B}}_{hom} (\binom{u}{q}) = (\binom{ι_{s}^{} f}{ι_{s^{}}^{} g}), where B_{hom} = (T_{- ε} (\begin{matrix} {\tilde{M}}_{0} (\partial_{t, ν}) & 0 \\ 0 & {\tilde{M}}_{1} (\partial_{t, ν}) \end{matrix}) T_{ε} + (\begin{matrix} 0 & {\tilde{C_{d}}}^{} \\ - \tilde{C_{d}} & 0 \end{matrix})) . \end{matrix}$

We recall a (standard) mollification procedure from [35] (see also [36,51]). For $δ > 0$ , we consider the bounded operator $F_{δ} : = {(1 + δ \partial_{t, ν})}^{- 1}$ . We remark that $F_{δ} \to 1$ as $δ \to 0$ in the strong operator topology and that for any $φ \in dom ({\overline{B}}_{ε})$ , we have $F_{δ} φ \in dom (B_{ε})$ and $F_{δ} {\overline{B}}_{ε} φ = B_{ε} F_{δ} φ$ and the same statements hold if we replace $B_{ε}$ by $B_{hom}$ .
Proposition 4.12.
We assume the assumptions of Theorem 4.11 . Besides the above assumptions we additionally assume that $M_{0}$ and $M_{1}$ admit the following form $\begin{matrix} (32) & M_{0} (z) = M_{0, 0} (z) z + M_{0, 1} (z), M_{1} (z) = M_{1, 0} (z) z + M_{1, 1} (z), \end{matrix}$ where, for $k, i \in {0, 1}$ , $M_{k, i} : C_{Re > ν_{0}} \to L (H_{k})$ are bounded functions. Let $δ > 0$ , $(u_{ε}, q_{ε})$ be the solution of ( 30 ) and $(u, q)$ be the solution of ( 31 ). Then $\begin{matrix} {‖ F_{δ} (u_{ε} - T_{- ε} ι_{s} u) ‖}_{L_{ν}^{2} (R; H_{0})} + {‖ F_{δ} (q_{ε} - T_{- ε} ι_{s^{}} q) ‖}_{L_{ν}^{2} (R; H_{1})} \to 0 as ε \to 0 . \end{matrix}$
Proof.
First of all, we note that since $F_{δ} (\binom{u}{q}) \in dom (B_{hom})$ , we have $F_{δ} u \in dom (\overline{ι_{s^{}}^{} C_{d} ι_{s}}) \subseteq ker (C_{s})$ and $F_{δ} q \in dom (\overline{ι_{s}^{} C_{d}^{} ι_{s^{}}}) \subseteq ker (C_{s}^{})$ . As a result of this, there exist sequences $u_{δ, γ} \in ker (C_{s})$ and $q_{δ, γ} \in ker (C_{s}^{})$ such that $ι_{s} u_{δ, γ} \in dom (C_{d})$ , $ι_{s^{}} q_{δ, γ} \in dom (C_{d}^{})$ and as $γ \to 0$ , $\begin{matrix} (33) & (u_{δ, γ}, q_{δ, γ}) \to (F_{δ} u, F_{δ} q), (ι_{s^{}}^{} C_{d} ι_{s} u_{δ, γ}, ι_{s}^{} C_{d}^{} ι_{s^{}} q_{δ, γ}) \to (\overline{ι_{s^{}}^{} C_{d} ι_{s}} F_{δ} u, \overline{ι_{s}^{} C_{d}^{} ι_{s^{}}} F_{δ} q) . \end{matrix}$ For $α > 0$ , in the following we consider $F_{α} u_{δ, γ} \in ker (C_{s})$ and $F_{α} q_{δ, γ} \in ker (C_{s}^{})$ , which satisfy $\begin{array}{l} (34) & ι_{s} F_{α} u_{δ, γ} \in dom (C_{d}) \cap dom (M_{0} (\partial_{t, ν})), ι_{s^{}} F_{α} q_{δ, γ} \in dom (C_{d}^{}) \cap dom (M_{1} (\partial_{t, ν})) . \end{array}$ By the triangle inequality we have $\begin{array}{l} {‖ F_{δ} (u_{ε} - T_{- ε} ι_{s} u) ‖}_{L_{ν}^{2} (R; H_{0})} + {‖ F_{δ} (q_{ε} - T_{- ε} ι_{s^{}} q) ‖}_{L_{ν}^{2} (R; H_{1})} \\ ⩽ ‖ F_{δ} u_{ε} - T_{- ε} ι_{s} F_{α} u_{δ, γ} ‖_{L_{ν}^{2} (R; H_{0})} + ‖ T_{- ε} ι_{s} F_{α} u_{δ, γ} - T_{- ε} ι_{s} F_{α} F_{δ} u ‖_{L_{ν}^{2} (R; H_{0})} \\ + ‖ T_{- ε} ι_{s} F_{α} F_{δ} u - T_{- ε} ι_{s} F_{δ} u ‖_{L_{ν}^{2} (R; H_{0})} + ‖ F_{δ} q_{ε} - T_{- ε} ι_{s^{}} F_{α} q_{δ, γ} ‖_{L_{ν}^{2} (R; H_{1})} \\ + ‖ T_{- ε} ι_{s^{}} F_{α} q_{δ, γ} - T_{- ε} ι_{s^{}} F_{α} F_{δ} q ‖_{L_{ν}^{2} (R; H_{1})} + ‖ T_{- ε} ι_{s^{}} F_{α} F_{δ} q - T_{- ε} ι_{s^{}} q ‖_{L_{ν}^{2} (R; H_{1})} \\ ⩽ ‖ F_{δ} u_{ε} - T_{- ε} ι_{s} F_{α} u_{δ, γ} ‖_{L_{ν}^{2} (R; H_{0})} + c (α) ‖ ι_{s} u_{δ, γ} - ι_{s} F_{δ} u ‖_{L_{ν}^{2} (R; H_{0})} + ‖ ι_{s} F_{α} F_{δ} u - ι_{s} F_{δ} u ‖_{L_{ν}^{2} (R; H_{0})} \\ + ‖ F_{δ} q_{ε} - T_{- ε} ι_{s^{}} F_{α} q_{δ, γ} ‖_{L_{ν}^{2} (R; H_{1})} + c (α) ‖ ι_{s^{}} q_{δ, γ} - ι_{s^{}} F_{δ} q ‖_{L_{ν}^{2} (R; H_{1})} \\ + ‖ ι_{s^{}} F_{α} F_{δ} q - ι_{s^{}} F_{δ} q ‖_{L_{ν}^{2} (R; H_{1})}, \end{array}$ where in the second inequality we use the boundedness of the operators $T_{- ε}$ and $F_{α}$ . First letting $γ \to 0$ and then $α \to 0$ , the second, third, fifth and sixth terms on the right-hand side vanish. In the following we treat the first and fourth terms, in particular, we show that they also vanish if we first let $ε \to 0$ , which completes the proof.

By $F_{δ} (\binom{u_{ε}}{q_{ε}}) \in dom (B_{ε})$ , (34) and using the properties of $M_{0}$ and $M_{1}$ it follows $\begin{array}{l} ‖ F_{δ} u_{ε} - T_{- ε} ι_{s} F_{α} u_{δ, γ} ‖_{L_{ν}^{2} (R; H_{0})}^{2} + ‖ F_{δ} q_{ε} - T_{- ε} ι_{s^{}} F_{α} q_{δ, γ} ‖_{L_{ν}^{2} (R; H_{1})}^{2} \\ ⩽ \frac{1}{c} Re ⟨T_{- ε} (\begin{matrix} M_{0} (\partial_{t, ν}) & 0 \\ 0 & M_{1} (\partial_{t, ν}) \end{matrix}) T_{ε} (\begin{matrix} F_{δ} u_{ε} - T_{- ε} ι_{s} F_{α} u_{δ, γ} \\ F_{δ} q_{ε} - T_{- ε} ι_{s^{}} F_{α} q_{δ, γ} \end{matrix}), (\begin{matrix} F_{δ} u_{ε} - T_{- ε} ι_{s} F_{α} u_{δ, γ} \\ F_{δ} q_{ε} - T_{- ε} ι_{s^{}} F_{α} q_{δ, γ} \end{matrix})⟩ . \end{array}$ The above combined with the fact that $(\begin{matrix} 0 & C_{d}^{} \\ - C_{d} & 0 \end{matrix})$ is skew-self-adjoint yields $\begin{array}{l} ‖ F_{δ} u_{ε} - T_{- ε} ι_{s} F_{α} u_{δ, γ} ‖_{L_{ν}^{2} (R; H_{0})}^{2} + ‖ F_{δ} q_{ε} - T_{- ε} ι_{s^{}} F_{α} q_{δ, γ} ‖_{L_{ν}^{2} (R; H_{1})}^{2} \\ ⩽ \frac{1}{c} Re ⟨B_{ε} (\begin{matrix} F_{δ} u_{ε} - T_{- ε} ι_{s} F_{α} u_{δ, γ} \\ F_{δ} q_{ε} - T_{- ε} ι_{s^{}} F_{α} q_{δ, γ} \end{matrix}), (\begin{matrix} F_{δ} u_{ε} - T_{- ε} ι_{s} F_{α} u_{δ, γ} \\ F_{δ} q_{ε} - T_{- ε} ι_{s^{}} F_{α} q_{δ, γ} \end{matrix})⟩ \\ = \frac{1}{c} Re ⟨F_{δ} {\overline{B}}_{ε} (\begin{matrix} u_{ε} \\ q_{ε} \end{matrix}), (\begin{matrix} F_{δ} u_{ε} - T_{- ε} ι_{s} F_{α} u_{δ, γ} \\ F_{δ} q_{ε} - T_{- ε} ι_{s^{}} F_{α} q_{δ, γ} \end{matrix})⟩ \\ (35) & - \frac{1}{c} Re ⟨B_{ε} (\begin{matrix} T_{- ε} ι_{s} F_{α} u_{δ, γ} \\ T_{- ε} ι_{s^{}} F_{α} q_{δ, γ} \end{matrix}), (\begin{matrix} F_{δ} u_{ε} - T_{- ε} ι_{s} F_{α} u_{δ, γ} \\ F_{δ} q_{ε} - T_{- ε} ι_{s^{}} F_{α} q_{δ, γ} \end{matrix})⟩ . \end{array}$

Since $(u_{ε}, q_{ε})$ solves (30) and $(f, g) \in ker (T_{ε} - 1)$ , it follows that the first term on the right-hand side of (35) satisfies $\begin{array}{l} \frac{1}{c} Re ⟨F_{δ} {\overline{B}}_{ε} (\begin{matrix} u_{ε} \\ q_{ε} \end{matrix}), (\begin{matrix} F_{δ} u_{ε} - T_{- ε} ι_{s} F_{α} u_{δ, γ} \\ F_{δ} q_{ε} - T_{- ε} ι_{s^{}} F_{α} q_{δ, γ} \end{matrix})⟩ & = \frac{1}{c} Re ⟨F_{δ} (\begin{matrix} f \\ g \end{matrix}), (\begin{matrix} F_{δ} u_{ε} - T_{- ε} ι_{s} F_{α} u_{δ, γ} \\ F_{δ} q_{ε} - T_{- ε} ι_{s^{}} F_{α} q_{δ, γ} \end{matrix})⟩ \\ = \frac{1}{c} Re ⟨F_{δ} (\begin{matrix} f \\ g \end{matrix}), (\begin{matrix} F_{δ} T_{ε} u_{ε} - ι_{s} F_{α} u_{δ, γ} \\ F_{δ} T_{ε} q_{ε} - ι_{s^{}} F_{α} q_{δ, γ} \end{matrix})⟩ . \end{array}$ By Theorem 4.11 as $ε \to 0$ the last expression converges to $\begin{matrix} \frac{1}{c} Re ⟨ F_{δ} (\binom{f}{g}), (\binom{F_{δ} ι_{s} u - ι_{s} F_{α} u_{δ, γ}}{F_{δ} ι_{s^{}} q - ι_{s^{}} F_{α} q_{δ, γ}}) ⟩ . \end{matrix}$ If we first let $γ \to 0$ and then $α \to 0$ , the latter expression vanishes. In the end, we treat the second term on the right-hand side of (35) as follows: $\begin{array}{l} ⟨B_{ε} (\begin{matrix} T_{- ε} ι_{s} F_{α} u_{δ, γ} \\ T_{- ε} ι_{s^{}} F_{α} q_{δ, γ} \end{matrix}), (\begin{matrix} F_{δ} u_{ε} - T_{- ε} ι_{s} F_{α} u_{δ, γ} \\ F_{δ} q_{ε} - T_{- ε} ι_{s^{}} F_{α} q_{δ, γ} \end{matrix})⟩ \\ = ⟨((\begin{matrix} M_{0} (\partial_{t, ν}) & 0 \\ 0 & M_{1} (\partial_{t, ν}) \end{matrix}) + T_{ε} (\begin{matrix} 0 & C_{d}^{} \\ - C_{d} & 0 \end{matrix}) T_{- ε}) (\begin{matrix} ι_{s} F_{α} u_{δ, γ} \\ ι_{s^{}} F_{α} q_{δ, γ} \end{matrix}), (\begin{matrix} F_{δ} T_{ε} u_{ε} - ι_{s} F_{α} u_{δ, γ} \\ F_{δ} T_{ε} q_{ε} - ι_{s^{}} F_{α} q_{δ, γ} \end{matrix})⟩ \\ (36) & = ⟨((\begin{matrix} M_{0} (\partial_{t, ν}) & 0 \\ 0 & M_{1} (\partial_{t, ν}) \end{matrix}) + (\begin{matrix} 0 & C_{d}^{} \\ - C_{d} & 0 \end{matrix})) (\begin{matrix} ι_{s} F_{α} u_{δ, γ} \\ ι_{s^{}} F_{α} q_{δ, γ} \end{matrix}), (\begin{matrix} F_{δ} T_{ε} u_{ε} - ι_{s} F_{α} u_{δ, γ} \\ F_{δ} T_{ε} q_{ε} - ι_{s^{}} F_{α} q_{δ, γ} \end{matrix})⟩, \end{array}$ where in the last equality we use the following computation $\begin{matrix} (37) & (\begin{matrix} 0 & C_{d}^{} \\ - C_{d} & 0 \end{matrix}) T_{- ε} (\begin{matrix} ι_{s} F_{α} u_{δ, γ} \\ ι_{s^{}} F_{α} q_{δ, γ} \end{matrix}) = (\begin{matrix} C_{d}^{} T_{- ε} ι_{s^{}} F_{α} q_{δ, γ} \\ - C_{d} T_{- ε} ι_{s} F_{α} u_{δ, γ} \end{matrix}) = (\begin{matrix} T_{- ε} C_{d}^{} ι_{s^{}} F_{α} q_{δ, γ} \\ - T_{- ε} C_{d} ι_{s} F_{α} u_{δ, γ} \end{matrix}), \end{matrix}$ where we used the commutation relations (12) and (15), and the fact that $F_{α} u_{δ, γ} \in ker (C_{s})$ and $F_{α} q_{δ, γ} \in ker (C_{s}^{})$ . Letting $ε \to 0$ , the right-hand side of (36) boils down to $\begin{array}{l} ⟨((\begin{matrix} M_{0} (\partial_{t, ν}) & 0 \\ 0 & M_{1} (\partial_{t, ν}) \end{matrix}) + (\begin{matrix} 0 & C_{d}^{} \\ - C_{d} & 0 \end{matrix})) (\begin{matrix} ι_{s} F_{α} u_{δ, γ} \\ ι_{s^{}} F_{α} q_{δ, γ} \end{matrix}), (\begin{matrix} F_{δ} ι_{s} u - ι_{s} F_{α} u_{δ, γ} \\ F_{δ} ι_{s^{}} q - ι_{s^{}} F_{α} q_{δ, γ} \end{matrix})⟩ \\ = ⟨(\begin{matrix} M_{0} (\partial_{t, ν}) & 0 \\ 0 & M_{1} (\partial_{t, ν}) \end{matrix}) (\begin{matrix} ι_{s} F_{α} u_{δ, γ} \\ ι_{s^{}} F_{α} q_{δ, γ} \end{matrix}) + (\begin{matrix} C_{d}^{} ι_{s^{}} F_{α} q_{δ, γ} \\ - C_{d} ι_{s} F_{α} u_{δ, γ} \end{matrix}), (\begin{matrix} F_{δ} ι_{s} u - ι_{s} F_{α} u_{δ, γ} \\ F_{δ} ι_{s^{}} q - ι_{s^{}} F_{α} q_{δ, γ} \end{matrix})⟩ \\ = ⟨(\begin{matrix} ι_{s}^{} M_{0} (\partial_{t, ν}) ι_{s} F_{α} u_{δ, γ} \\ ι_{s^{}}^{} M_{1} (\partial_{t, ν}) ι_{s^{}} F_{α} q_{δ, γ} \end{matrix}) + (\begin{matrix} ι_{s}^{} C_{d}^{} ι_{s^{}} F_{α} q_{δ, γ} \\ - ι_{s^{}}^{} C_{d} ι_{s} F_{α} u_{δ, γ} \end{matrix}), (\begin{matrix} F_{δ} u - F_{α} u_{δ, γ} \\ F_{δ} q - F_{α} q_{δ, γ} \end{matrix})⟩ . \end{array}$ Using the additional assumption (32), as $γ \to 0$ , the expression $(\begin{matrix} ι_{s}^{} M_{0} (\partial_{t, ν}) ι_{s} F_{α} u_{δ, γ} \\ ι_{s^{}}^{} M_{1} (\partial_{t, ν}) ι_{s^{}} F_{α} q_{δ, γ} \end{matrix})$ converges to $(\begin{matrix} ι_{s}^{} M_{0} (\partial_{t, ν}) ι_{s} F_{α} F_{δ} u \\ ι_{s^{}}^{} M_{1} (\partial_{t, ν}) ι_{s^{}} F_{α} F_{δ} q \end{matrix})$ . As a result of this and using (33), as $γ \to 0$ , the last expression converges to $\begin{array}{l} ⟨(\begin{matrix} ι_{s}^{} M_{0} (\partial_{t, ν}) ι_{s} F_{α} F_{δ} u \\ ι_{s^{}}^{} M_{1} (\partial_{t, ν}) ι_{s^{}} F_{α} F_{δ} q \end{matrix}) + (\begin{matrix} F_{α} \overline{ι_{s}^{} C_{d}^{} ι_{s^{}}} F_{δ} q \\ - F_{α} \overline{ι_{s^{}}^{} C_{d} ι_{s}} F_{δ} u \end{matrix}), (\begin{matrix} F_{δ} u - F_{α} F_{δ} u \\ F_{δ} q - F_{α} F_{δ} q \end{matrix})⟩ \\ = ⟨F_{α} ((\begin{matrix} ι_{s}^{} M_{0} (\partial_{t, ν}) ι_{s} F_{δ} u \\ ι_{s^{}}^{} M_{1} (\partial_{t, ν}) ι_{s^{}} F_{δ} q \end{matrix}) + (\begin{matrix} \overline{ι_{s}^{} C_{d}^{} ι_{s^{}}} F_{δ} q \\ - \overline{ι_{s^{}}^{} C_{d} ι_{s}} F_{δ} u \end{matrix})), (\begin{matrix} F_{δ} u - F_{α} F_{δ} u \\ F_{δ} q - F_{α} F_{δ} q \end{matrix})⟩ . \end{array}$ Finally, letting $α \to 0$ , the right-hand side vanishes, which completes the proof. □

If we, additionally, assume that $(f, g) \in dom (\partial_{t, ν})$ , it follows that $(u_{ε}, q_{ε}) \in dom (\partial_{t, ν})$ and $(u_{ε}, q_{ε}) \in dom (B_{ε})$ (the analogous claims hold if we replace $B_{ε}$ by $B_{hom}$ and $(u_{ε}, q_{ε})$ by $(u, q)$ ). As a result of this, we obtain the following: Corollary 4.13.
Assume the same assumptions as in Proposition 4.12 . Additionally, we assume that $(f, g) \in H_{ν}^{1} (R; H) (= dom (\partial_{t, ν}))$ . Then $\begin{matrix} ‖ u_{ε} - T_{- ε} ι_{s} u ‖_{L_{ν}^{2} (R; H_{0})}^{2} + ‖ q_{ε} - T_{- ε} ι_{s^{}} q ‖_{L_{ν}^{2} (R; H_{1})}^{2} \to 0 as ε \to 0 . \end{matrix}$
Proof.
Since $(f, g) \in H_{ν}^{1} (R; H)$ , we may apply the operator $(1 + \partial_{t, ν})$ to both sides of the equations (30) and (31) to obtain $\begin{array}{l} {\overline{B}}_{ε} (1 + \partial_{t, ν}) (\begin{matrix} u_{ε} \\ q_{ε} \end{matrix}) = (1 + \partial_{t, ν}) (\begin{matrix} f \\ g \end{matrix}), \\ {\overline{B}}_{hom} (1 + \partial_{t, ν}) (\begin{matrix} u \\ q \end{matrix}) = (1 + \partial_{t, ν}) (\begin{matrix} f \\ g \end{matrix}) . \end{array}$ We have that $w_{ε} : = (1 + \partial_{t, ν}) (\begin{matrix} u_{ε} \\ q_{ε} \end{matrix})$ and $w_{0} : = (1 + \partial_{t, ν}) (\begin{matrix} u \\ q \end{matrix})$ solve equations (30)and (31) with right-hand side $(1 + \partial_{t, ν}) (\begin{matrix} f \\ g \end{matrix})$ (instead of $(\begin{matrix} f \\ g \end{matrix})$ ). As a result of this, Proposition 4.12 implies that for any $δ > 0$ , with $w : = (\begin{matrix} ι_{s} 0 \\ 0 ι_{s^{}} \end{matrix}) w_{0}$ we have $\begin{matrix} {‖ F_{δ} (w_{ε} - T_{- ε} w) ‖}_{L_{ν}^{2} (R; H)}^{2} \to 0 as ε \to 0 . \end{matrix}$ Setting $δ = 1$ , the claim follows. □

5. Examples

In this section we present three specific examples of the unfolding operator: the stochastic, quasi-periodic and (a variant of the) periodic unfolding operator. Moreover, we provide specific examples in which Theorem 3.1 and Theorem 4.11 yield homogenization results. In particular, we consider homogenization problems for elliptic, Maxwell’s and wave equations. In the latter two cases we consider also equations of mixed type. Also, besides the essential homogenization results obtained by Theorem 4.11 for the evolutionary equations, we derive some corrector type results based on Corollary 4.13 and on the specific structure of the equations.

5.1. Examples of unfolding operators

Deterministic differential operators. First, we introduce the deterministic differential operators which we use in the description of the considered problems. In the following, we denote by $\partial_{i}$ , $\nabla = (\partial_{1}, \dots, \partial_{n})$ , and $\nabla \cdot$ the partial derivative, the gradient and the divergence, respectively, and use the notation $\begin{matrix} \nabla \times u : = (\partial_{2} u_{3} - \partial_{3} u_{2}, \partial_{3} u_{1} - \partial_{1} u_{3}, \partial_{1} u_{2} - \partial_{2} u_{1}) \end{matrix}$ for the curl of a smooth three-dimensional vector-field. Let $Q \subseteq R^{n}$ be open.

Gradient and divergence operators. The operator closure of $C_{c}^{\infty} (Q) \subseteq L^{2} (Q) \to L^{2} {(Q)}^{n}$ , $φ \mapsto \nabla φ$ and its adjoint are denoted by $\begin{array}{l} {\overset{o}{grad}}_{x} : dom ({\overset{o}{grad}}_{x}) = H_{0}^{1} (Q) \subseteq L^{2} (Q) \to L^{2} {(Q)}^{n} and - {div}_{x} = {({\overset{o}{grad}}_{x})}^{*} . \end{array}$

Curl operator. Let $n = 3$ . The operator closure of $C_{c}^{\infty} {(Q)}^{3} \subseteq L^{2} {(Q)}^{3} \to L^{2} {(Q)}^{3}$ , $u \mapsto \nabla \times u$ and its adjoint are denoted by $\begin{array}{l} {\overset{o}{curl}}_{x} : dom ({\overset{o}{curl}}_{x}) \subseteq L^{2} {(Q)}^{3} \to L^{2} {(Q)}^{3} and {curl}_{x} : = {({\overset{o}{curl}}_{x})}^{*} . \end{array}$

Periodic unfolding (in the mean). We present a periodic unfolding operator suited for homogenization problems involving periodic coefficients. We consider the unit torus $□ : = R^{n} / Z^{n}$ equipped with the push-forward of the Lebesgue measure $L {([0, 1)}^{n})$ and let $Q \subseteq R^{n}$ be open. We consider the following particular choice of the abstract Hilbert spaces from Section 2: $H_{d} = L^{2} (Q)$ , $H_{s} = L^{2} (□)$ . For $ε \in R ∖ {0}$ , we let $T_{ε} : L^{2} (Q) \overset{a}{\otimes} L^{2} (□) \to L^{2} (Q) \otimes L^{2} (□)$ be given by $\begin{matrix} (38) & T_{ε} u (x, y) = u (x, y - {\frac{x}{ε}}_{□}), \end{matrix}$ where ${\cdot}_{□} : R^{n} \to □$ is the canonical quotient map. $T_{ε}$ is a linear isometry and we call its continuous extension $T_{ε} : L^{2} (Q) \otimes L^{2} (□) \to L^{2} (Q) \otimes L^{2} (□)$ periodic unfolding operator. Thus, $T_{ε}$ is unitary and we have $T_{ε}^{- 1} = T_{- ε}$ .

Remark 5.1.
The above notion of periodic unfolding differs from the classical notion of unfolding introduced in [8,9]. In particular, in [8,9] the unfolding operator is defined as $\begin{matrix} \tilde{T_{ε}} : L^{2} (Q) \to L^{2} (Q) \otimes L^{2} (□), given by \tilde{T_{ε}} u (x, y) = u ({[\frac{x}{ε}]}_{ε Z^{n}} + ε y), \end{matrix}$ where ${[\cdot]}_{ε Z^{n}} : R^{n} \to ε Z^{n}$ is defined as ${[x]}_{ε Z^{n}} = x - ε {\frac{x}{ε}}_{□}$ . The operator $\tilde{T_{ε}}$ defined in this fashion is not surjective and therefore it does not satisfy the assumptions of our abstract setting. On the other hand, definition (38) involves an additional variable y for functions in the domain of $T_{ε}$ (in this respect we might call the notion defined in (38) periodic unfolding in the mean) and consequently it fulfills the requirements from Section 2.

The operator closure of $C^{\infty} (□) \subseteq L^{2} (□) \to L^{2} {(□)}^{n}$ , $φ \mapsto \nabla φ$ and its adjoint are denoted by $\begin{array}{l} {grad}_{y}^{#} : dom ({grad}_{y}^{#}) = H_{#}^{1} (□) \subseteq L^{2} (□) \to L^{2} {(□)}^{n} and - {div}_{y}^{#} = {({grad}_{y}^{#})}^{} . \end{array}$ Note that we may identify $C^{\infty} (□)$ with the space of $Z^{n}$ -periodic functions in $C^{\infty} (R^{n})$ , and $L^{2} (□)$ with $L^{2} ({(0, 1)}^{n})$ .

If we set $C_{d} = {\overset{o}{grad}}_{x}$ and $C_{s} = {grad}_{y}^{#}$ , it follows that the abstract conditions (12)–(14) are satisfied (Note that (10) is easy, by choosing $D_{1} = D_{2} = C_{c}^{\infty} (Q)$ and $E_{1} = E_{2} = C^{\infty} (□)$ ):
Lemma 5.2.
For* $ε \neq 0$ , we have $\begin{array}{l} ε {\overset{o}{grad}}_{x} T_{- ε} φ = ε T_{- ε} {\overset{o}{grad}}_{x} φ + {grad}_{y}^{#} T_{- ε} φ, for all φ \in dom ({\overset{o}{grad}}_{x}) \cap dom ({grad}_{y}^{#}), \\ T_{ε} {grad}_{y}^{#} \subseteq {grad}_{y}^{#} T_{ε}, \\ T_{ε} φ = φ, for all φ \in ker ({grad}_{y}^{#}) . \end{array}$

Note that $dom ({\overset{o}{grad}}_{x}) \cap dom ({grad}_{y}^{#}) = H_{0}^{1} (Q) \otimes L^{2} (□) \cap L^{2} (Q) \otimes H_{#}^{1} (□)$ and $ker ({grad}_{y}^{#}) ≃ L^{2} (Q) \otimes C$ .
Proof of Lemma 5.2.
Let $η \in C_{c}^{\infty} (Q) \overset{a}{\otimes} C^{\infty} {(□)}^{n}$ . Then $\begin{matrix} {⟨ ε T_{- ε} {\overset{o}{grad}}_{x} φ, η ⟩}_{L^{2} (Q) \otimes L^{2} {(□)}^{n}} = - {⟨ φ, ε {div}_{x} T_{ε} η ⟩}_{L^{2} (Q) \otimes L^{2} (□)} . \end{matrix}$ Using the smoothness of η and the chain rule, we compute $ε {div}_{x} T_{ε} η = ε T_{ε} {div}_{x} η - T_{ε} {div}_{y}^{#} η$ , where we use that ${div}_{x} T_{ε} η (x, y) = \partial_{x_{i}} (η_{i} (x, y - {\frac{x}{ε}}_{□}))$ a.e. Therefore, we obtain $\begin{matrix} {⟨ ε T_{- ε} {\overset{o}{grad}}_{x} φ, η ⟩}_{L^{2} (Q) \otimes L^{2} {(□)}^{n}} = {⟨ ε {\overset{o}{grad}}_{x} T_{- ε} φ - {grad}_{y}^{#} T_{- ε} φ, η ⟩}_{L^{2} (Q) \otimes L^{2} {(□)}^{n}} . \end{matrix}$ Since $C_{c}^{\infty} (Q) \overset{a}{\otimes} C^{\infty} (□)$ is dense in $dom ({\overset{o}{grad}}_{x}) \cap dom ({grad}_{y}^{#})$ , the first claim follows.

The second claim follows from the fact that $T_{ε} {grad}_{y}^{#} φ = {grad}_{y}^{#} T_{ε} φ$ for any $φ \in C^{\infty} (□)$ .

The last claim follows using that $ker ({grad}_{y}^{#}) ≃ L^{2} (Q) \otimes C$ , which can be obtained with the help of the Poincaré inequality: There exists $C > 0$ such that $\begin{matrix} {‖ φ - \int_{□} φ ‖}_{L^{2} (□)} ⩽ C {‖ {grad}_{y}^{#} φ ‖}_{L^{2} {(□)}^{n}} for all φ \in dom ({grad}_{y}^{#}) . \end{matrix}$ □

Quasi-periodic unfolding. Let $m > n$ , we consider the unit torus $□ : = R^{m} / Z^{m}$ equipped with the push-forward of the Lebesgue measure and let $Q \subseteq R^{n}$ . We consider the following choice for the abstract Hilbert spaces from Section 2: $H_{d} = L^{2} (Q)$ , $H_{s} = L^{2} (□)$ . For $ε \in R ∖ {0}$ and a fixed $λ \in R^{m \times n}$ , we define $T_{ε} : L^{2} (Q) \overset{a}{\otimes} L^{2} (□) \to L^{2} (Q) \otimes L^{2} (□)$ as $\begin{matrix} (39) & T_{ε} u (x, y) = u (x, y - {\frac{λ x}{ε}}_{□}), \end{matrix}$ where ${\cdot}_{□} : R^{m} \to □$ is the canonical quotient map. $T_{ε}$ is a linear isometry and we call its continuous extension $T_{ε} : L^{2} (Q) \otimes L^{2} (□) \to L^{2} (Q) \otimes L^{2} (□)$ quasi-periodic unfolding operator. This notion is suitable for the treatment of problems which feature rapidly oscillating quasi-periodic coefficients of the form $A (x, y + {\frac{λ x}{ε}}_{□})$ with some $A : Q \times □ \to R$ , cf. Section 5.3. Also, $T_{ε}$ is unitary and $T_{ε}^{- 1} = T_{- ε}$ . Choosing $C_{d} = {\overset{o}{grad}}_{x}$ and $C_{s} = \overline{λ {grad}_{y}^{#}}$ we have the following analogous result to Lemma 5.2. We do not present its proof since it follows completely analogously to the proof of the more general Lemma 5.5 (a).
Lemma 5.3.
For $ε \neq 0$ , we have $\begin{array}{l} ε {\overset{o}{grad}}_{x} T_{- ε} φ = ε T_{- ε} {\overset{o}{grad}}_{x} φ + \overline{λ {grad}_{y}^{#}} T_{- ε} φ, for all φ \in dom ({\overset{o}{grad}}_{x}) \cap dom (\overline{λ {grad}_{y}^{#}}), \\ T_{ε} \overline{λ {grad}_{y}^{#}} \subseteq \overline{λ {grad}_{y}^{#}} T_{ε}, \\ T_{ε} φ = φ, for all φ \in ker (\overline{λ {grad}_{y}^{#}}) . \end{array}$

Stochastic unfolding. Let $Q \subseteq R^{n}$ be open, $(Ω, Σ, μ, τ)$ be a probability space satisfying Assumption 1. In this section, we set $H_{d} = L^{2} (Q)$ and $H_{s} = L^{2} (Ω)$ and consider the stochastic unfolding operator $T_{ε} : L^{2} (Q) \otimes L^{2} (Ω) \to L^{2} (Q) \otimes L^{2} (Ω)$ defined in (7).
Remark 5.4.
The stochastic unfolding operator is a generalization of the periodic unfolding operator since in the case $(Ω, Σ, μ) = (□, L, d y)$ and $τ_{x} y = y + {x}_{□}$ the two operators coincide. Another instance of stochastic unfolding corresponds to the quasi periodic unfolding operator where we choose $Ω = □ = R^{m} / Z^{m}$ with $τ_{x} y = y + λ x$ for $x \in R^{n}$ and fixed $λ \in R^{m \times n}$ . The latter is well-suited for the treatment of problems involving quasi-periodic rapidly-oscillating coefficients.

With help of the dynamical system τ (for fixed $i \in {1, \dots, n}$ ) we introduce the group of unitary operators ( $h \in R$ ) $T_{h e_{i}} : L^{2} (Ω) \to L^{2} (Ω)$ given by $\begin{matrix} (40) & T_{h e_{i}} φ = φ \circ τ_{h e_{i}} . \end{matrix}$ This group is strongly continuous and we denote its infinitesimal generator by $\partial_{ω, i} : dom (\partial_{ω, i}) \subseteq L^{2} (Ω) \to L^{2} (Ω)$ , i.e., $\begin{matrix} \partial_{ω, i} φ = lim_{h \to 0} \frac{T_{h e_{i}} φ - φ}{h} . \end{matrix}$ Analogously to (40), we define $T_{x} : L^{2} (Ω) \to L^{2} (Ω)$ for $x \in R^{n}$ . Using the stochastic partial derivatives $\partial_{ω, i}$ , we define the stochastic gradient $\begin{matrix} {grad}_{ω} : H^{1} (Ω) : = ⋂_{i \in {1, \dots, n}} dom (\partial_{ω, i}) \subseteq L^{2} (Ω) \to L^{2} {(Ω)}^{n} \end{matrix}$ given by $\begin{matrix} {grad}_{ω} φ = (\partial_{ω, 1} φ, \dots, \partial_{ω, n} φ) . \end{matrix}$ It is a closed and densely defined linear operator and we let ${div}_{ω} = - {({grad}_{ω})}^{}$ . Also, we introduce the space of smooth* random variables by $\begin{matrix} H^{\infty} (Ω) = {φ \in L^{2} (Ω) : \partial_{ω, 1}^{α_{1}} \dots \partial_{ω, n}^{α_{n}} φ \in H^{1} (Ω) for all α_{1}, \dots, α_{n} \in N_{0}} . \end{matrix}$

In the case $n = 3$ , we define the stochastic counterpart of the ${\overset{o}{curl}}_{x}$ operator as follows. Let $\nabla_{ω} \times : H^{\infty} {(Ω)}^{3} \subseteq L^{2} {(Ω)}^{3} \to L^{2} {(Ω)}^{3}$ be given by $\begin{matrix} \nabla_{ω} \times φ = (\partial_{ω, 2} φ_{3} - \partial_{ω, 3} φ_{2}, \partial_{ω, 3} φ_{1} - \partial_{ω, 1} φ_{3}, \partial_{ω, 1} φ_{2} - \partial_{ω, 2} φ_{1}) . \end{matrix}$ We let ${curl}_{ω} = {(\nabla_{ω} \times)}^{* }$ . The choice $C_{d} = {\overset{o}{grad}}_{x}$ and $C_{s} = {grad}_{ω}$ satisfies assumptions (12)–(14), and if we set $C_{d} = {\overset{o}{curl}}_{x}$ and $C_{s} = {curl}_{ω}$ we merely obtain (12)–(13). (Again, note that (10) in both cases follows with the choice $D_{1} = D_{2} = C_{c}^{\infty} (Q)$ and $E_{1} = E_{2} = H^{\infty} (Ω)$ .)
Lemma 5.5.
Let* $ε \neq 0$ .
Then $\begin{array}{l} ε {\overset{o}{grad}}_{x} T_{- ε} φ = ε T_{- ε} {\overset{o}{grad}}_{x} φ + {grad}_{ω} T_{- ε} φ for all φ \in dom ({\overset{o}{grad}}_{x}) \cap dom ({grad}_{ω}), \\ T_{ε} {grad}_{ω} \subseteq {grad}_{ω} T_{ε}, \\ T_{ε} φ = φ for all φ \in ker ({grad}_{ω}) . \end{array}$

If $n = 3$ , then $\begin{array}{l} ε {\overset{o}{curl}}_{x} T_{- ε} φ = ε T_{- ε} {\overset{o}{curl}}_{x} φ + {curl}_{ω} T_{- ε} φ for all φ \in dom ({\overset{o}{curl}}_{x}) \cap dom ({curl}_{ω}), \\ T_{ε} {curl}_{ω} \subseteq {curl}_{ω} T_{ε} . \end{array}$

Proof.
(a) Let $η \in C_{c}^{\infty} (Q) \overset{a}{\otimes} H^{\infty} {(Ω)}^{n}$ . We have $\begin{matrix} {⟨ ε T_{- ε} {\overset{o}{grad}}_{x} φ, η ⟩}_{L^{2} (Q) \otimes L^{2} (Ω)} = {⟨ φ, ε {div}_{x} T_{ε} η ⟩}_{L^{2} (Q) \otimes L^{2} (Ω)} . \end{matrix}$ We compute $ε {div}_{x} T_{ε} η = ε T_{ε} {div}_{x} η - T_{ε} {div}_{ω} η$ , in order to obtain, $\begin{matrix} {⟨ ε T_{- ε} {\overset{o}{grad}}_{x} φ, η ⟩}_{L^{2} (Q) \otimes L^{2} (Ω)} = {⟨ ε {\overset{o}{grad}}_{x} T_{- ε} φ - {grad}_{ω} T_{- ε} φ, η ⟩}_{L^{2} (Q) \otimes L^{2} (Ω)} . \end{matrix}$ Note that $C_{c}^{\infty} (Q) \overset{a}{\otimes} H^{\infty} (Ω)$ is dense in $dom ({\overset{o}{grad}}_{x}) \cap dom ({grad}_{ω})$ (see (42) below) and therefore the first claim follows.

In order to obtain the second claim, it is sufficient to show that $T_{ε} \partial_{ω, i} φ = \partial_{ω, i} T_{ε} φ$ for any $φ \in H^{\infty} (Ω)$ (for any $i \in {1, \dots, n}$ ). Let $φ, η \in L^{2} (Q) \overset{a}{\otimes} H^{\infty} (Ω)$ . We have $\begin{array}{l} {⟨ T_{ε} \partial_{ω, i} φ, η ⟩}_{L^{2} (Q) \otimes L^{2} (Ω)} & = lim_{h \to 0} \frac{1}{h} {⟨ T_{ε} T_{h e_{i}} φ - T_{ε} φ, η ⟩}_{L^{2} (Q) \otimes L^{2} (Ω)} \\ = lim_{h \to 0} \frac{1}{h} {⟨ T_{ε} φ, T_{- h e_{i}} η - η ⟩}_{L^{2} (Q) \otimes L^{2} (Ω)} \\ = - {⟨ T_{ε} φ, \partial_{ω, i} η ⟩}_{L^{2} (Q) \otimes L^{2} (Ω)} \\ = {⟨ \partial_{ω, i} T_{ε} φ, η ⟩}_{L^{2} (Q) \otimes L^{2} (Ω)} . \end{array}$ As a result of this and by the density of $H^{\infty} (Ω)$ in $H^{1} (Ω)$ , the claim follows.

For $φ \in ker ({grad}_{ω})$ , we have $T_{x} φ = φ$ for all $x \in R^{n}$ (see, e.g., [16, Lemma 3.10]). As a result of this, the third claim follows.

(b) The proof follows analogously to part (a). □
Remark 5.6 (Boundary conditions).

The first commutation relation from Lemmas 5.2 and 5.5 (a) remains valid if the gradient operator ${\overset{o}{grad}}_{x}$ (which is defined on a domain which accounts for homogeneous Dirichlet boundary conditions) is replaced by gradient operators corresponding to other types of boundary conditions (e.g., with homogeneous Neumann condition or periodic boundary condition). In this respect, the unfolding procedure does not only apply to problems with certain boundary conditions.

Auxiliary results. We provide certain facts that will be helpful in the treatment of the stochastic homogenization problems considered in the following section (in particular for corrector type results). The following standard orthogonal decompositions hold (see, e.g., [7]) $\begin{matrix} (41) & L^{2} (Ω) = ker ({grad}_{ω}) \oplus^{⊥} \overline{ran} ({div}_{ω}), L^{2} {(Ω)}^{n} = ker ({div}_{ω}) \oplus^{⊥} \overline{ran} ({grad}_{ω}) . \end{matrix}$ $P_{inv}$ and $P_{pot}$ denote the orthogonal projections $P_{inv} : L^{2} (Ω) \to ker ({grad}_{ω})$ and $P_{pot} : L^{2} {(Ω)}^{n} \to \overline{ran} ({grad}_{ω}) = : L_{pot}^{2} (Ω)$ . We have $ker ({grad}_{ω}) = {u \in L^{2} (Ω) : T_{x} u = u for all x \in R^{n}} = : L_{inv}^{2} (Ω)$ (see, e.g., [16, Lemma 3.10]). If we additionally assume that the probability space is ergodic, we get $L_{inv}^{2} (Ω) ≃ C$ .

Lemma 5.7.
Let $n = 3$ . Then $\begin{matrix} ker ({curl}_{ω}) = L_{inv}^{2} {(Ω)}^{3} \oplus^{⊥} L_{pot}^{2} (Ω), ker ({curl}_{ω}^{}) = L_{inv}^{2} {(Ω)}^{3} \oplus^{⊥} L_{pot}^{2} (Ω) . \end{matrix}$

The following standard mollification procedure for random variables (see, e.g., [4,17]) is useful in the proof of the above lemma. For a sequence $δ \to 0$ , we consider a sequence of standard mollifiers $ρ_{δ} \in C_{c}^{\infty} (R^{n})$ (even and non-negative) and for $φ \in L^{2} (Ω)$ , we set $\begin{matrix} (42) & φ_{δ} = \int_{R^{n}} ρ_{δ} (y) T_{y} φ d y . \end{matrix}$ We obtain that $φ_{δ} \in H^{\infty} (Ω)$ (apply the definition of $\partial_{ω, i}$ , transform the integral to have the difference quotient on $ρ_{δ}$ , and then apply the dominated convergence theorem), $⟨ φ_{δ} ⟩ = ⟨ φ ⟩ = \int_{Ω} φ (ω) d μ (ω)$ and $φ_{δ} \to φ$ in $L^{2} (Ω)$ as $δ \to 0$ (see, e.g., [17, Section 7.2]). We collect some further useful properties of this mollification procedure: Lemma 5.8.
Let* $φ \in L^{2} (Ω)$ . Then $φ_{δ} \in H^{\infty} (Ω)$ ; and $φ_{δ} \to φ$ in $L^{2} (Ω)$ as $δ \to 0$ .

Moreover, the following statements hold:
If $φ \in H^{1} (Ω)$ , then ${grad}_{ω} φ_{δ} = {({grad}_{ω} φ)}_{δ}$ and ${grad}_{ω} φ_{δ} \to {grad}_{ω} φ$ .

If $φ \in dom ({div}_{ω})$ , then ${div}_{ω} φ_{δ} = {({div}_{ω} φ)}_{δ}$ and ${div}_{ω} φ_{δ} \to {div}_{ω} φ$ .

( $n = 3$ ) If $φ \in dom ({curl}_{ω})$ , then ${curl}_{ω} φ_{δ} = {({curl}_{ω} φ)}_{δ}$ and ${curl}_{ω} φ_{δ} \to {curl}_{ω} φ$ .

( $n = 3$ ) If $φ \in dom ({curl}_{ω}^{})$ , then* ${curl}_{ω}^{} φ_{δ} = {({curl}_{ω}^{} φ)}_{δ}$ and ${curl}_{ω}^{} φ_{δ} \to {curl}_{ω}^{} φ$ .

${curl}_{ω} = {curl}_{ω}^{}$ .

Proof.
The first statements of the lemma follow the same way as they follow in the deterministic case.

(a)* We have (for $i \in {1, \dots, n}$ ), $\begin{array}{l} {‖ \frac{1}{h} (T_{h e_{i}} φ_{δ} - φ_{δ}) - {(\partial_{ω, i} φ)}_{δ} ‖}_{L^{2} (Ω)} = {‖ \int_{R^{n}} ρ_{δ} (y) (\frac{T_{y + h e_{i}} φ - T_{y} φ}{h} - T_{y} \partial_{ω, i} φ) ‖}_{L^{2} (Ω)} . \end{array}$ Using the above and that $ρ_{δ}$ is bounded and compactly supported, we obtain that there exists $C (δ) ⩾ 0$ such that $\begin{matrix} {‖ \frac{1}{h} (T_{h e_{i}} φ_{δ} - φ_{δ}) - {(\partial_{ω, i} φ)}_{δ} ‖}_{L^{2} (Ω)} ⩽ C (δ) {‖ \frac{1}{h} (T_{h e_{i}} φ - φ) - \partial_{ω, i} φ ‖}_{L^{2} (Ω)} . \end{matrix}$ In the limit $h \to 0$ the right-hand side vanishes, and this implies that ${grad}_{ω} φ_{δ} = {({grad}_{ω} φ)}_{δ}$ . Consequently, ${grad}_{ω} φ_{δ} \to {grad}_{ω} φ$ .

(b) For $η \in H^{1} (Ω)$ , we have $\begin{array}{l} - {⟨ φ_{δ}, {grad}_{ω} η ⟩}_{L^{2} {(Ω)}^{n}} = - ⟨ \int_{R^{n}} ρ_{δ} (y) T_{y} φ {grad}_{ω} η d y ⟩ = - \int_{R^{n}} ⟨ ρ_{δ} (y) T_{y} φ {grad}_{ω} η ⟩ d y . \end{array}$ Using that $T_{- y}$ and (the infinitesimal generator) ${grad}_{ω}$ commute, we obtain that the last expression equals $\begin{matrix} - \int_{R^{n}} ρ_{δ} (y) ⟨ φ {grad}_{ω} T_{- y} η ⟩ d y = ⟨ \int_{R^{n}} ρ_{δ} (y) T_{y} {div}_{ω} φ d y η ⟩ = {⟨ {({div}_{ω} φ)}_{δ}, η ⟩}_{L^{2} (Ω)} . \end{matrix}$ As a result of this, we have ${div}_{ω} φ_{δ} = {({div}_{ω} φ)}_{δ}$ and ${div}_{ω} φ_{δ} \to {div}_{ω} φ$ .

(c) Let $η \in dom ({curl}_{ω}^{})$ . We have $\begin{matrix} {⟨ φ_{δ}, {curl}_{ω}^{} η ⟩}_{L^{2} {(Ω)}^{3}} = ⟨ \int_{R^{n}} ρ_{δ} (y) T_{y} φ {curl}_{ω}^{} η d y ⟩ = \int_{R^{n}} ρ_{δ} (y) ⟨ φ T_{- y} {curl}_{ω}^{} η ⟩ d y . \end{matrix}$ In order to obtain the claim of the lemma, it is sufficient to show that $T_{- y} {curl}_{ω}^{} η = {curl}_{ω}^{} T_{- y} η$ . Indeed, in that case the above expression equals $\begin{matrix} \int_{R^{n}} ρ_{δ} (y) ⟨ φ {curl}_{ω}^{} T_{- y} η ⟩ d y = ⟨ \int_{R^{n}} ρ_{δ} (y) T_{y} {curl}_{ω} φ d y η ⟩ = {⟨ {({curl}_{ω} φ)}_{δ}, η ⟩}_{L^{2} {(Ω)}^{3}}, \end{matrix}$ and therefore the claim follows. In the following we show that $T_{- y} {curl}_{ω}^{} η = {curl}_{ω}^{} T_{- y} η$ . Let $ψ \in H^{1} {(Ω)}^{3}$ . We have $\begin{matrix} {⟨ T_{- y} η, {curl}_{ω} ψ ⟩}_{L^{2} {(Ω)}^{3}} = {⟨ η, T_{y} {curl}_{ω} ψ ⟩}_{L^{2} {(Ω)}^{3}} = {⟨ η, {curl}_{ω} T_{y} ψ ⟩}_{L^{2} {(Ω)}^{3}} = {⟨ T_{- y} {curl}_{ω}^{} η, ψ ⟩}_{L^{2} {(Ω)}^{3}} . \end{matrix}$ By density of $H^{1} {(Ω)}^{3}$ in $dom ({curl}_{ω})$ , we obtain that for any $ψ \in dom ({curl}_{ω})$ , ${⟨ T_{- y} η, {curl}_{ω} ψ ⟩}_{L^{2} {(Ω)}^{3}} = {⟨ T_{- y} {curl}_{ω}^{} η, ψ ⟩}_{L^{2} {(Ω)}^{3}}$ that implies $T_{- y} η \in dom ({curl}_{ω}^{})$ and $T_{- y} {curl}_{ω}^{} η = {curl}_{ω}^{} T_{- y} η$ . The proof is done.

(d) The proof follows analogously to part (c) if we obtain that for $η \in dom ({curl}_{ω})$ , $T_{- y} {curl}_{ω} η = {curl}_{ω} T_{- y} η$ . In order to show this, we consider a sequence ${(η_{k})}_{k}$ in $H^{1} {(Ω)}^{3}$ such that $η_{k} \to η$ and ${curl}_{ω} η_{k} \to {curl}_{ω} η$ (by definition such a sequence exists). The operator $T_{- y}$ is unitary and therefore we have $T_{- y} η_{k} \to T_{- y} η$ and $T_{- y} {curl}_{ω} η_{k} \to T_{- y} {curl}_{ω} η$ . Moreover, since $T_{- y} {curl}_{ω} η_{k} = {curl}_{ω} T_{- y} η_{k}$ , and using that ${curl}_{ω}$ is closed, we get ${curl}_{ω} T_{- y} η = T_{- y} {curl}_{ω} η$ .

(e) It is elementary to show that ${curl}_{ω} \subseteq {curl}_{ω}^{}$ ; see also [4, p. 22]. The remaining inclusion follows from (d)* as this statement shows that $H^{1} (Ω)$ is a core for ${curl}_{ω}^{}$ . Hence, $\begin{matrix} {curl}_{ω} \subseteq {curl}_{ω}^{} = \overline{{curl}_{ω}^{} |_{H^{1} (Ω)}} = \overline{{curl}_{ω} |_{H^{1} (Ω)}} = {curl}_{ω} . \end{matrix}$ □
Proof of Lemma 5.7.
Let $φ \in ker ({curl}_{ω})$ . Then $φ = P_{inv} φ + Ψ$ where $Ψ = φ - P_{inv} φ$ . Using (41) it follows that Ψ may be decomposed as follows $\begin{matrix} Ψ = Ψ_{1} + Ψ_{2}, \end{matrix}$ where $Ψ_{1} \in ker ({div}_{ω})$ and $Ψ_{2} \in \overline{ran} ({grad}_{ω})$ . In the following we show that $Ψ_{1} = 0$ . Using Lemma 5.8 we find a sequence ${(φ_{k})}_{k}$ in $H^{\infty} (Ω)$ such that ${grad}_{ω} φ_{k} \to Ψ_{2}$ as $k \to \infty$ . Moreover, since ${curl}_{ω} {grad}_{ω} φ_{k} = 0$ for all $k \in N$ , we conclude that $Ψ_{2} \in ker ({curl}_{ω})$ and therefore it follows that $\begin{matrix} Ψ_{1} = Ψ - Ψ_{2} = φ - P_{inv} φ - Ψ_{2} \in ker ({curl}_{ω}) . \end{matrix}$ For the mollified functions $Ψ_{1, δ}$ (defined as in (42)), by a direct computation we obtain $\begin{matrix} \int_{Ω} | {curl}_{ω} Ψ_{1, δ} |^{2} + | {div}_{ω} Ψ_{1, δ} |^{2} = \int_{Ω} | {grad}_{ω} Ψ_{1, δ} |^{2} . \end{matrix}$ Using Lemma 5.8 we pass to the limit $δ \to 0$ and it follows that ${grad}_{ω} Ψ_{1, δ} \to 0$ . Moreover, since $Ψ_{1, δ} \to Ψ_{1}$ , we obtain that ${grad}_{ω} Ψ_{1} = 0$ and therefore $Ψ_{1} = 0$ (using that $P_{inv} Ψ_{1} = 0$ ). This concludes the proof of the first part. The second claim follows from Lemma 5.8 (e)*. □

5.2. Periodic homogenization of elliptic equations

We start by showing that the classical example of periodic homogenization of elliptic equations fits into the previously described abstract framework (see Section 3). We refer to [1,3] for the standard treatment of elliptic equations with periodic coefficients and to [4,27] for its stochastic counterpart. Let $Q \subset R^{n}$ be open and bounded. In this section, we consider the periodic unfolding operator $T_{ε} : L^{2} (Q) \otimes L^{2} (□) \to L^{2} (Q) \otimes L^{2} (□)$ defined in (38).

In this setting the role of $C_{d}$ and $C_{s}$ is played by ${\overset{o}{grad}}_{x}$ and ${grad}_{y}^{#}$ , respectively. Let $A \in L^{\infty} {(Q \times □)}^{n \times n}$ be such that there exists $c > 0$ with $| A (x, y) | ⩽ \frac{1}{c}$ and $\frac{1}{2} (A (x, y) + A {(x, y)}^{*}) ⩾ c$ a.e. We interpret A as a multiplication operator in ${(L^{2} (Q) \otimes L^{2} (□))}^{n}$ . For $ε > 0$ , we consider the following equation $\begin{matrix} (43) & - {div}_{x} T_{- ε} A T_{ε} {\overset{o}{grad}}_{x} u_{ε} = f \end{matrix}$ with $f \in L^{2} (Q) \otimes C$ . In this case, the term $T_{- ε} A T_{ε} {\overset{o}{grad}}_{x} u_{ε}$ boils down to the familiar expression $(x, y) \mapsto A (x, y + {\frac{x}{ε}}_{□}) {\overset{o}{grad}}_{x} u_{ε} (x, y)$ . Since ${\overset{o}{grad}}_{x}$ is injective and its range is closed (due to the Poincaré inequality), a direct application of Theorem 3.1 and the fact that $T_{ε} u = u$ for $u \in ker ({grad}_{y}^{#})$ imply the following:

Corollary 5.9.
Let A be given as above and $u_{ε}$ be the unique solution of ( 43 ). Then $\begin{array}{l} T_{ε} u_{ε} \to u strongly in L^{2} (Q) \otimes L^{2} (□), \\ T_{ε} {\overset{o}{grad}}_{x} u_{ε} ⇀ {\overset{o}{grad}}_{x} u + v weakly in {(L^{2} (Q) \otimes L^{2} (□))}^{n}, \end{array}$ where $u \in dom ({\overset{o}{grad}}_{x}) \otimes C$ and $v \in \overline{ran} ({grad}_{y}^{#})$ are the unique solution to $\begin{array}{l} (44) & \begin{array}{l} - {div}_{x} P A ({\overset{o}{grad}}_{x} u + v) = f, \\ - {div}_{y}^{#} A ({\overset{o}{grad}}_{x} u + v) = 0 . \end{array} \end{array}$

Above, P is the projection to constant functions (in the y-variable), i.e., $P φ = \int_{□} φ d y$ . Remark 5.10.
In order to transform the above (two-scale) homogenized problem (44) into the usual ‘one-scale’ form (see [1] for detailed investigation of such two-scale effective equations), we might introduce the following (uniquely defined) correctors $φ_{i} \in dom ({grad}_{y}^{#})$ ( $i \in {1, \dots, n}$ ) by $\begin{matrix} - {div}_{y}^{#} A (e_{i} + {grad}_{y}^{#} φ_{i}) = 0, \int_{□} φ_{i} = 0 . \end{matrix}$ It follows that the choice $v = \sum_{i = 1}^{n} {({\overset{o}{grad}}_{x} u)}_{i} {grad}_{y}^{#} φ_{i}$ satisfies the second equation in (44). Consequently, u is the solution of the equation $\begin{matrix} - {div}_{x} A_{hom} {\overset{o}{grad}}_{x} u = f, \end{matrix}$ where $A_{hom} e_{i} \cdot e_{j} = \int_{□} A (e_{i} + {grad}_{y}^{#} φ_{i}) \cdot (e_{j} + {grad}_{y}^{#} φ_{j}) d y$ . The last equation is the classical form of the homogenized equation. Notice, however, that the result presented in Corollary 5.9 is slightly different from the classical result; as the notion of convergence is different. We also refer to [4, Remark on page 32] for a similar statement in the stochastic setting.

5.3. Quasi-periodic homogenization of mixed type equations

We refer to the textbooks [3,17] for standard references about homogenization of elliptic and parabolic equations. In the following section we consider equations that may be elliptic in some regions of the physical space but parabolic in other parts of the domain. This corresponds to the below assumption (45) which allows the coefficients η and σ to vanish but not at the same point. In this sense, we present a homogenization result for mixed type elliptic and parabolic equations, where the type of the equation may even change rapidly.

Let $m > n$ , $Q \subseteq R^{n}$ be open and $□ : = R^{m} ∖ Z^{m}$ . In this section we denote by $T_{ε} : L^{2} (Q) \otimes L^{2} (□) \to L^{2} (Q) \otimes L^{2} (□)$ the quasi-periodic unfolding operator defined in (39). The role of $C_{d}$ and $C_{s}$ in this setting is played by the operators ${\overset{o}{grad}}_{x}$ and $\overline{λ {grad}_{y}^{#}}$ , respectively.

Let $ν_{0} > 0$ , $A \in L^{\infty} {(Q \times □)}^{n \times n}$ be such that there exists $c > 0$ with A Hermitian a.e and $c ⩽ A (x, y) ⩽ \frac{1}{c}$ a.e. Also, let $η, σ \in L^{\infty} (Q \times □)$ be such that there exists $c > 0$ with $\begin{matrix} (45) & Re (z η + σ) ⩾ c (for all z \in C_{Re ⩾ ν_{0}}) . \end{matrix}$ Let $H_{0} = L^{2} (Q) \otimes L^{2} (□)$ , $H_{1} = L^{2} (Q) \otimes L^{2} {(□)}^{n}$ and $H = H_{0} \oplus H_{1}$ . For $ε > 0$ and $f \in L^{2} (Q)$ , we consider the following system $\begin{matrix} (46) & {\overline{B}}_{ε} (\begin{matrix} u_{ε} \\ q_{ε} \end{matrix}) = (\begin{matrix} f \\ 0 \end{matrix}), where B_{ε} = T_{- ε} (\begin{matrix} \partial_{t, ν} η + σ & 0 \\ 0 & A^{- 1} \end{matrix}) T_{ε} + (\begin{matrix} 0 & {div}_{x} \\ {\overset{o}{grad}}_{x} & 0 \end{matrix}) . \end{matrix}$ According to Theorem 4.1 the above system has a unique solution $(u_{ε}, q_{ε}) \in L_{ν}^{2} (R; H)$ .

Remark 5.11.
We may rewrite this system in the following form $\begin{matrix} η_{ε} \partial_{t, ν} u_{ε} + σ_{ε} u_{ε} + {div}_{x} q_{ε} = 0, q_{ε} = - A_{ε} {\overset{o}{grad}}_{x} u, \end{matrix}$ where $η_{ε} = T_{- ε} η T_{ε}$ , $σ_{ε} = T_{- ε} σ T_{ε}$ and $A_{ε} = T_{- ε} A T_{ε}$ . This implies that $u_{ε}$ solves $\begin{matrix} (47) & η_{ε} \partial_{t, ν} u_{ε} + σ_{ε} u_{ε} - {div}_{x} A_{ε} {\overset{o}{grad}}_{x} u_{ε} = f . \end{matrix}$ This equation is of mixed elliptic and parabolic type since the coefficients η and σ may vanish in some regions of $Q \times □$ (but not at the same time).

Using our abstract homogenization result Theorem 4.11 we obtain: Corollary 5.12.
Let A, η, σ be given as above and let $(u_{ε}, q_{ε}) \in L_{ν}^{2} (R; H)$ be the unique solution to ( 46 ). Then $\begin{matrix} T_{ε} (u_{ε}, q_{ε}) ⇀ (ι_{s} u, ι_{s^{}} q) weakly in L_{ν}^{2} (R; H) . \end{matrix}$ Above,* $ι_{s}$ and $ι_{s^{}}$ denote the canonical embeddings* $ι_{s} : ker (\overline{λ {grad}_{y}^{#}}) \to H_{0}$ and $ι_{s^{}} : ker ({div}_{y}^{#} λ^{}) \to H_{1}$ , and $(u, q) \in L_{ν}^{2} (R; ker (\overline{λ {grad}_{y}^{#}}) \oplus ker ({div}_{y}^{#} λ^{}))$ is the unique solution to* $\begin{array}{l} (48) & \begin{matrix} {\overline{B}}_{hom} (\begin{matrix} u \\ q \end{matrix}) = (\begin{matrix} ι_{s}^{} f \\ 0 \end{matrix}), \\ B_{hom} = (\begin{matrix} \partial_{t, ν} ι_{s}^{} η ι_{s} + ι_{s}^{} σ ι_{s} & 0 \\ 0 & ι_{s^{}}^{} A^{- 1} ι_{s^{}} \end{matrix}) + (\begin{matrix} 0 & \overline{ι_{s}^{} {div}_{x} ι_{s^{}}} \\ \overline{ι_{s^{}}^{} {\overset{o}{grad}}_{x} ι_{s}} & 0 \end{matrix}) . \end{matrix} \end{array}$
Remark 5.13.
Dropping the embeddings $ι_{s}$ and $ι_{s^{}}$ , and the closure bars from the notation, system (48) reads $\begin{array}{l} (49) & \begin{array}{l} ι_{s}^{} η \partial_{t, ν} u + ι_{s}^{} σ u + ι_{s}^{} {div}_{x} q = f, \\ ι_{s^{}}^{} A^{- 1} q = - ι_{s^{}}^{} {\overset{o}{grad}}_{x} u . \end{array} \end{array}$ It can be shown that $ker (\overline{λ {grad}_{y}^{#}}) \subseteq ker ({div}_{y}^{#} λ^{})$ , cf. proof of Lemma 5.5 (a). As a result of this and using the standard orthogonal decomposition $L^{2} (Q) \otimes L^{2} (□) = ker ({div}_{y}^{#} λ^{}) \oplus^{⊥} \overline{ran} (\overline{λ {grad}_{y}^{#}})$ , the above implies that $q = - A ({\overset{o}{grad}}_{x} u - χ)$ where $χ \in \overline{ran} (\overline{λ {grad}_{y}^{#}})$ solves the corrector equation $\begin{matrix} - {div}_{y}^{#} λ^{} A ({\overset{o}{grad}}_{x} u - χ) = 0 . \end{matrix}$ In this sense, the first equation in (49) boils down to $\begin{matrix} ι_{s}^{} η \partial_{t, ν} u + ι_{s}^{} σ u - {div}_{x} ι_{s}^{} A ({\overset{o}{grad}}_{x} u - χ) = f, \end{matrix}$ which is the homogenized counterpart of (47). Moreover, under the assumption that $f \in H_{ν}^{1} (R, H_{0})$ , Corollary 4.13 implies the corrector type result $\begin{matrix} u_{ε} \to u, {\overset{o}{grad}}_{x} u_{ε} - ({\overset{o}{grad}}_{x} u - T_{- ε} χ) \to 0 . \end{matrix}$

5.4. Stochastic homogenization of Maxwell’s equations

In this section we consider stochastic homogenization of Maxwell’s equations. We refer to [19,58] for the treatment of Maxwell’s equations in the periodic setting using two-scale convergence arguments. The stochastic-periodic case is treated in [39] that is based on the notion of stochastic two-scale convergence from [4]. However, our approach is different, it relies on the operator theoretic formulation and on the unfolding strategy, in fact, the homogenization result Corollary 5.14 readily follows from the abstract Theorem 4.11. Also, we treat a more general situation—in contrast to our case—in [39] the assumptions on the coefficients do not allow for jumps or for regions, where the conductivity or dielectricity vanish. This is to the best of our knowledge entirely new. The reason for these cases being possible to consider here is the variety and flexibility of evolutionary equations. Indeed, explicit examples for highly oscillatory mixed type problems and an accompanying error and numerical analysis have been treated in an (easier) deterministic setting in [13,14,53].

Here, we shall improve on yet available results in as much as we treat the full Maxwell system and provide additionally some corrector type result Corollary 4.13, which are not presented earlier in the stochastic setting. We refer to [58, Theorem 3.3] for a corrector results in the periodic framework additionally relying on a semi-group setting, which is only possible to use if the dielectricty does not vanish.

Let $Q \subseteq R^{3}$ be open and $(Ω, Σ, μ, τ)$ be a probability space satisfying Assumption 1 (with $n = 3$ ). In this section we consider the stochastic unfolding operator $T_{ε} : L^{2} (Q) \otimes L^{2} (Ω) \to L^{2} (Q) \otimes L^{2} (Ω)$ defined in (7). The role of $C_{d}$ and $C_{s}$ in this setting is played by the operators ${\overset{o}{curl}}_{x}$ and ${curl}_{ω}$ , respectively.

We set $H_{0} = H_{1} = {(L^{2} (Q) \otimes L^{2} (Ω))}^{3}$ and $H = H_{0} \oplus H_{1}$ . We consider $ν_{0} > 0$ , $η_{0}, σ_{0}, μ_{0} \in L^{\infty} {(Q \times Ω)}^{3 \times 3}$ such that $η_{0}$ and $μ_{0}$ are Hermitian a.e. and there exists $c > 0$ such that $\begin{matrix} (50) & ν η_{0} + Re (σ_{0}) ⩾ c (for all ν > ν_{0}), μ_{0} ⩾ c . \end{matrix}$ For $ε > 0$ and $(f, g) \in L_{ν}^{2} (R; L^{2} {(Q)}^{3} \oplus L^{2} {(Q)}^{3})$ , we consider the following system of equations $\begin{array}{l} (51) & {\overline{B}}_{ε} (\begin{matrix} u_{ε} \\ q_{ε} \end{matrix}) = (\begin{matrix} f \\ g \end{matrix}), where B_{ε} = T_{- ε} (\begin{matrix} \partial_{t, ν} η_{0} + σ_{0} & 0 \\ 0 & \partial_{t, ν} μ_{0} \end{matrix}) T_{ε} + (\begin{matrix} 0 & - {curl}_{x} \\ {\overset{o}{curl}}_{x} & 0 \end{matrix}) . \end{array}$ The above system represents a system of Maxwell’s equations with random and oscillating coefficients (the oscillating coefficients have the form $T_{- ε} η_{0} T_{ε}$ and similarly for $σ_{0}$ and $μ_{0}$ ). According to Theorem 4.1 there exists a unique solution of the above equation $(u_{ε}, q_{ε}) \in L_{ν}^{2} (R; H)$ . Moreover, a direct application of Theorem 4.11 yields the following:

Corollary 5.14.
Let $η_{0}$ , $σ_{0}$ , $μ_{0}$ be given as above and let $(u_{ε}, q_{ε}) \in L_{ν}^{2} (R; H)$ be the unique solution to ( 51 ). We have $\begin{matrix} T_{ε} (u_{ε}, q_{ε}) ⇀ (ι_{s} u, ι_{s} q) weakly in L_{ν}^{2} (R, H) . \end{matrix}$ Above, $ι_{s}$ denotes the canonical embedding $ι_{s} : ker ({curl}_{ω}) ↪ H_{0} = H_{1}$ and $\begin{matrix} (u, q) \in L_{ν}^{2} (R; ker ({curl}_{ω}) \oplus ker ({curl}_{ω})) \end{matrix}$ denotes the unique solution to $\begin{matrix} (52) & \begin{matrix} {\overline{B}}_{hom} (\begin{matrix} u \\ q \end{matrix}) = (\begin{matrix} ι_{s}^{} f \\ ι_{s}^{} g \end{matrix}), \\ where B_{hom} = (\begin{matrix} \partial_{t, ν} ι_{s}^{} η_{0} ι_{s} + ι_{s}^{} σ_{0} ι_{s} & 0 \\ 0 & \partial_{t, ν} ι_{s}^{} μ_{0} ι_{s} \end{matrix}) + (\begin{matrix} 0 & - \overline{ι_{s}^{} {curl}_{x} ι_{s}} \\ \overline{ι_{s}^{} {\overset{o}{curl}}_{x} ι_{s}} & 0 \end{matrix}) . \end{matrix} \end{matrix}$

Note that the fact $(u, q) \in L_{ν}^{2} (R; ker ({curl}_{ω}) \oplus ker ({curl}_{ω}))$ does not* imply in general that the limit solution is deterministic (or shift-invariant), i.e., the solution still depends on the probability space variable ω. In the following we present an equivalent formulation of the above equation by decomposing the solution $(u, q)$ to its deterministic (shift-invariant) and random parts. In particular, the deterministic part satisfies an effective Maxwell system and the random part is given by a suitable corrector equation. We introduce the following canonical embeddings $\begin{array}{l} ι_{0} : L_{inv}^{2} {(Ω)}^{3} ↪ ker ({curl}_{ω}), ι_{1} : L_{pot}^{2} (Ω) ↪ ker ({curl}_{ω}) . \end{array}$ Moreover, we consider the transformation $\begin{matrix} T : = (\begin{matrix} ι_{0} & ι_{1} & 0 & 0 \\ 0 & 0 & ι_{0} & ι_{1} \end{matrix}) : L_{inv}^{2} {(Ω)}^{3} \oplus L_{pot}^{2} (Ω) \oplus L_{inv}^{2} {(Ω)}^{3} \oplus L_{pot}^{2} (Ω) \to ker ({curl}_{ω}) \oplus ker ({curl}_{ω}) \end{matrix}$ that is unitary (by Lemma 5.7). As a result of this, we obtain the following:
Corollary 5.15.
Let $(u, q) \in L_{ν}^{2} (R; ker ({curl}_{ω}) \oplus ker ({curl}_{ω}))$ be the solution of ( 52 ). Then $\begin{matrix} (\begin{matrix} u_{0} \\ χ_{1} \\ q_{0} \\ χ_{2} \end{matrix}) : = T^{- 1} (u, q) \in L_{ν}^{2} (R; {({(L^{2} (Q) \otimes L_{inv}^{2} (Ω))}^{3} \oplus (L^{2} (Q) \otimes L_{pot}^{2} (Ω)))}^{2}) \end{matrix}$ is the unique solution to $\begin{matrix} (53) & \overline{T^{- 1} B_{hom} T} (\begin{matrix} u_{0} \\ χ_{1} \\ q_{0} \\ χ_{2} \end{matrix}) = T^{- 1} (\begin{matrix} ι_{s}^{} f \\ ι_{s}^{} g \end{matrix}) . \end{matrix}$
Remark 5.16.
Dropping the notation for the embeddings $ι_{s}$ , T and closure bars, system (53) reads $\begin{array}{l} \partial_{t, ν} P_{inv} η_{0} (u_{0} + χ_{1}) + P_{inv} σ_{0} (u_{0} + χ_{1}) - {curl}_{x} q_{0} = f, \\ \partial_{t, ν} P_{pot} η_{0} (u_{0} + χ_{1}) + P_{pot} σ_{0} (u_{0} + χ_{1}) = 0, \\ \partial_{t, ν} P_{inv} μ_{0} (q_{0} + χ_{2}) + {\overset{o}{curl}}_{x} u_{0} = g, \\ \partial_{t, ν} P_{pot} μ_{0} (q_{0} + χ_{2}) = 0 . \end{array}$ Note that in the second equation we used that $P_{pot} {curl}_{x} (q_{0} + χ_{2}) = 0$ since $P_{pot} q_{0} = 0$ and $P_{pot} {curl}_{x} χ_{2} = 0$ (that can be obtained by a direct computation) and in the fourth equation we use $P_{pot} {\overset{o}{curl}}_{x} (u_{0} + χ_{1}) = 0$ (obtained similarly as the previous claim). We regard the first and third equations as the effective Maxwell system for the (averaged) variable $(u_{0}, q_{0})$ and the second and fourth equations are corrector equations which allow us to express $(χ_{1}, χ_{2})$ as functions of $(u_{0}, q_{0})$ . In particular, the second and fourth equations imply that (additionally assuming that $η_{0}$ is positive-definite) $\begin{array}{l} χ_{1} (t) = - χ_{η_{0}}^{i} u_{0}^{i} (t) + e^{- A t} (χ_{η_{0}}^{i} u_{0}^{i} (0) + χ_{1} (0)) + \int_{0}^{t} e^{- A (t - τ)} A (χ_{η_{0}}^{i} - χ_{σ_{0}}^{i}) u_{0}^{i} (τ) d τ, \\ χ_{2} (t) = - χ_{μ_{0}}^{i} q_{0}^{i} (t) + χ_{μ_{0}}^{i} q_{0}^{i} (0) + χ_{2} (0), \end{array}$ where $A = η_{0}^{- 1} σ_{0}$ and $χ_{η_{0}}^{i}, χ_{σ_{0}}^{i}, χ_{μ_{0}}^{i} \in L^{2} (Q) \otimes L_{pot}^{2} (Ω)$ ( $i \in {1, 2, 3}$ ) are the unique solutions to (respectively) $\begin{matrix} (54) & P_{pot} η_{0} (e_{i} - χ_{η_{0}}^{i}) = 0, P_{pot} σ_{0} (e_{i} - χ_{σ_{0}}^{i}) = 0, P_{pot} μ_{0} (e_{i} - χ_{μ_{0}}^{i}) = 0 . \end{matrix}$ The derivation of these formulas is analogous to the periodic case [58] (see also [39] for the periodic-stochastic case). In this sense, the effective equations for the averaged variables $(u_{0}, q_{0})$ are non-local in time. This so-called memory effect has also been observed in [52] and [54] for coefficients that are not necessarily periodic or stochastic.
Remark 5.17 (Corrector equations).

The equations in (54) are standard corrector equations in stochastic homogenization and might be brought into the form of an elliptic partial differential equation on $R^{3}$ . For simplicity let us consider the first equation in (54) and assume that $η_{0}$ does not depend on the physical space variable, i.e., $η_{0} (x, ω) = η_{0} (ω)$ . The equation is equivalent to $\begin{matrix} (55) & \int_{Ω} η_{0} (ω) (e_{i} - χ_{η_{0}}^{i} (ω)) \cdot χ (ω) d μ (ω) = 0 for all χ \in L_{pot}^{2} (Ω), \end{matrix}$ which is a variational problem in the $L^{2}$ -probability space. It turns out that for μ-almost every $ω \in Ω$ , the vector field $x \mapsto χ_{η_{0}}^{i} (τ_{x} ω)$ has a potential $φ (ω, \cdot) \in H_{loc}^{1} (R^{3})$ that is a distributional solution to $\begin{matrix} (56) & - \nabla \cdot (η_{0} (τ_{x} ω) (e_{i} - \nabla φ (ω, x))) = 0 in R^{3} . \end{matrix}$ On the other hand, by standard theory in stochastic homogenization there exists a unique random field $φ : Ω \times R^{3} \to R$ that is a distributional solution to (56) for μ-a.e. $ω \in Ω$ such that $\nabla φ$ is stationary (i.e., $\nabla φ (ω, x + z) = \nabla φ (τ_{z} ω, x)$ for μ-a.e. $ω \in Ω$ and a.e. $x, z \in R^{3}$ ), square integrable $\int_{Ω} | \nabla φ (ω, x) |^{2} d μ (ω) < \infty$ , mean-free $\int_{Ω} \nabla φ d μ = 0$ , and anchored in the sense of $\int_{{(0, 1)}^{d}} φ (ω, x) d x = 0$ , μ-a.s. With this solution we recover the random vector field $χ_{η_{0}}^{i} (ω) : = \nabla φ (ω, 0)$ solving (55), see [24, Section 2.2] for details.

Note that both formulations, (55) and (56), are not accessible to a direct numerical approximation, since in the case of (55) a typical example for the probability measure μ would be a product-measure of the form ${\hat{μ}}^{\otimes R^{3}}$ , while (56) is posed on the unbounded domain $R^{3}$ . A standard approximation scheme is the so-called periodization method, where in (56) the domain $R^{3}$ is replaced by a large torus, say $L □$ with $L ≫ 1$ , see, e.g., [5].

Corrector type result. In order to avoid clutter in notation, in the following we disregard the notation for the embeddings $ι_{s}$ and T. A direct consequence of Corollaries 4.13 and 5.15 is the following corrector type result:

Corollary 5.18.
Let $δ > 0$ , $(u_{ε}, q_{ε})$ be the solution of ( 51 ) and $(u_{0}, χ_{1}, q_{0}, χ_{2})$ be the solution of ( 53 ). Additionally, we assume that $(f, g) \in H_{ν}^{1} (R; H) (= dom (\partial_{t, ν}))$ . Then $\begin{matrix} ‖ u_{ε} - T_{- ε} χ_{1} - u_{0} ‖_{L_{ν}^{2} (R; H_{0})}^{2} + ‖ q_{ε} - T_{- ε} χ_{2} - q_{0} ‖_{L_{ν}^{2} (R; H_{1})}^{2} \to 0 as ε \to 0 . \end{matrix}$

Above, we assumed that the right-hand side is smooth, however, if this is not the case, we can obtain a similar result using a mollification procedure as in Proposition 4.12.
5.5. Stochastic homogenization of some mixed type equations

A standard reference for homogenization of the wave and heat equation is [6], where general equations are treated in the framework of H-convergence. In contrast to standard results for the wave or heat equation, the equation we consider features coefficients η and σ, cf. (58), that are allowed to be alternatingly vanishing in some regions of the considered physical domain. This means that our setting contains equations of mixed alternating hyperbolic-parabolic type.

Let $Q \subseteq R^{n}$ be open and $(Ω, Σ, μ, τ)$ be a probability space satisfying Assumption 1. We consider the stochastic unfolding operator $T_{ε} : L^{2} (Q) \otimes L^{2} (Ω) \to L^{2} (Q) \otimes L^{2} (Ω)$ defined in (7). The role of $C_{d}$ and $C_{s}$ in this setting is played by the operators ${\overset{o}{grad}}_{x}$ and ${grad}_{ω}$ (respectively).

Let $ν_{0} > 0$ , $A \in L^{\infty} {(Q \times Ω)}^{n \times n}$ be such that there exists $c > 0$ with A Hermitian a.e and $c ⩽ A (x, ω) ⩽ \frac{1}{c}$ a.e. Also, let $η, σ \in L^{\infty} (Q \times Ω)$ be such that there exists $c > 0$ with $\begin{matrix} Re (z η + σ) ⩾ c (for all z \in C_{Re ⩾ ν_{0}}) . \end{matrix}$

Let $H_{0} = L^{2} (Q) \otimes L^{2} (Ω)$ , $H_{1} = L^{2} (Q) \otimes L^{2} {(Ω)}^{n}$ and $H = H_{0} \oplus H_{1}$ . For $ε > 0$ and $f \in L^{2} (Q)$ , we consider the following system $\begin{matrix} (57) & {\overline{B}}_{ε} (\begin{matrix} u_{ε} \\ q_{ε} \end{matrix}) = (\begin{matrix} f \\ 0 \end{matrix}), where B_{ε} = T_{- ε} (\begin{matrix} \partial_{t, ν} η + σ & 0 \\ 0 & \partial_{t, ν} A^{- 1} \end{matrix}) T_{ε} + (\begin{matrix} 0 & {div}_{x} \\ {\overset{o}{grad}}_{x} & 0 \end{matrix}) . \end{matrix}$ According to Theorem 4.1 the above system has a unique solution $(u_{ε}, q_{ε}) \in L_{ν}^{2} (R; H)$ .

Remark 5.19.
Setting $w_{ε} = \partial_{t, ν}^{- 1} u_{ε}$ , it follows that $w_{ε}$ solves the following wave equation $\begin{matrix} (58) & η_{ε} \partial_{t, ν}^{2} w_{ε} + σ_{ε} \partial_{t, ν} w_{ε} - {div}_{x} A_{ε} {\overset{o}{grad}}_{x} w_{ε} = f, \end{matrix}$ where the oscillating coefficients are given as the composition $A_{ε} : = T_{- ε} A T_{ε}$ , $η_{ε} : = T_{- ε} η T_{ε}$ and $σ_{ε} : = T_{- ε} σ T_{ε}$ .

Using our abstract homogenization result Theorem 4.11 we obtain:
Corollary 5.20.
Let A, η, σ be given as above and let $(u_{ε}, q_{ε}) \in L_{ν}^{2} (R; H)$ be the unique solution to ( 57 ). Then $\begin{matrix} T_{ε} (u_{ε}, q_{ε}) ⇀ (ι_{s} u, ι_{s^{}} q) weakly in L_{ν}^{2} (R; H) . \end{matrix}$ Above,* $ι_{s}$ and $ι_{s^{}}$ denote the canonical embeddings* $ι_{s} : ker ({grad}_{ω}) \to H_{0}$ and $ι_{s^{}} : ker ({div}_{ω}) \to H_{1}$ , and* $(u, q) \in L_{ν}^{2} (R; ker ({grad}_{ω}) \oplus ker ({div}_{ω}))$ is the unique solution to $\begin{array}{l} {\overline{B}}_{hom} (\begin{matrix} u \\ q \end{matrix}) = (\begin{matrix} ι_{s}^{} f \\ 0 \end{matrix}), \\ B_{hom} = (\begin{matrix} \partial_{t, ν} ι_{s}^{} η ι_{s} + ι_{s}^{} σ ι_{s} & 0 \\ 0 & \partial_{t, ν} ι_{s^{}}^{} A^{- 1} ι_{s^{}} \end{matrix}) + (\begin{matrix} 0 & \overline{ι_{s}^{} {div}_{x} ι_{s^{}}} \\ \overline{ι_{s^{}}^{} {\overset{o}{grad}}_{x} ι_{s}} & 0 \end{matrix}) . \end{array}$
Remark 5.21.
In order to recover the classical form of the homogenized wave equation, we set $w = \partial_{t, ν}^{- 1} u$ to obtain (with the usual abuse of notation) $\begin{matrix} P_{inv} η \partial_{t, ν}^{2} w + P_{inv} σ \partial_{t, ν} w + {div}_{x} P_{inv} q = f, \end{matrix}$ where q satisfies $P_{ker ({div}_{ω})} (\partial_{t, ν} A^{- 1} q + {\overset{o}{grad}}_{x} \partial_{t, ν} w) = 0$ (here $P_{ker ({div}_{ω})} = ι_{s^{}}^{}$ ). Applying $\partial_{t, ν}^{- 1}$ , it follows that $P_{ker ({div}_{ω})} A^{- 1} q = - {\overset{o}{grad}}_{x} w$ . As a result of this and using (41), we have that $A^{- 1} q = - {\overset{o}{grad}}_{x} w + χ$ where $χ \in L^{2} (Q) \otimes L_{pot}^{2} (Ω)$ (note that χ is uniquely determined). Also, since $q \in ker ({div}_{ω})$ , we have $- {div}_{ω} A ({\overset{o}{grad}}_{x} w - χ) = 0$ . Collecting these facts, we obtain that w satisfies $\begin{matrix} (59) & P_{inv} η \partial_{t, ν}^{2} w + P_{inv} σ \partial_{t, ν} w - {div}_{x} P_{inv} A ({\overset{o}{grad}}_{x} w - χ) = f, \end{matrix}$ where χ solves the usual corrector equation $- {div}_{ω} A ({\overset{o}{grad}}_{x} w - χ) = 0$ .

The above remark suggests the following corrector type statement $T_{ε} {\overset{o}{grad}}_{x} w_{ε} + χ \to {\overset{o}{grad}}_{x} w$ that is equivalent to $A^{- 1} T_{ε} q_{ε} - χ \to P_{ker ({div}_{ω})} A^{- 1} q = A^{- 1} q - χ$ . We formalize this in the following corollary, which is a direct consequence of Corollary 4.13.
Corollary 5.22 (Corrector type result).

Let $f \in H_{ν}^{1} (R; H_{0})$ , then $\begin{matrix} {‖ \partial_{t, ν}^{- 1} u_{ε} - \partial_{t, ν}^{- 1} u ‖}_{L_{ν}^{2} (R, H_{0})} + ‖ u_{ε} - u ‖_{L_{ν}^{2} (R; H_{0})} + ‖ T_{ε} q_{ε} - q ‖_{L_{ν}^{2} (R; H_{1})} \to 0 as ε \to 0 . \end{matrix}$

Above, we assumed that the right-hand side f is smooth for convenience, if this is not the case, we could use a mollification procedure as in Proposition 4.12. We briefly summarize the implications of the above proposition for the sequence $w_{ε}$ : $\begin{matrix} (60) & \begin{array}{l} w_{ε} \to w, \partial_{t, ν} w_{ε} \to \partial_{t, ν} w, \\ {\overset{o}{grad}}_{x} w_{ε} + T_{- ε} χ \to {\overset{o}{grad}}_{x} w (strongly in L_{ν}^{2} (R; L^{2} (Q) \otimes L^{2} (Ω))) . \end{array} \end{matrix}$

Remark 5.23.

It is well-known that the classical homogenized wave equation (59) is an unsatisfactory approximation for (58) on large time scales $t ≳ ε^{- 2}$ (see [12]). This fact is not reflected in our result ((60) is given in a norm accounting for the whole time line $t \in R$ ) since we use a norm which is weighted with an exponential weight $e^{- 2 ν t}$ ( $ν > 0$ ) that diminishes the effects on large time scales.

Footnotes

Acknowledgements

SN and MV acknowledge funding by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) – project number 405009441, and in the context of TU Dresden’s Institutional Strategy “The Synergetic University”.

References

Allaire, Homogenization and two-scale convergence, SIAM J. Math. Anal. 23(6) (1992), 1482–1518. doi:10.1137/0523084.

K.T.

Andrews and

Wright, Stochastic homogenization of elliptic boundary-value problems with

L^{p}

-data, Asymptotic Analysis 17(3) (1998), 165–184.

Bensoussan,

J.-L.

Lions and

Papanicolaou, Asymptotic Analysis for Periodic Structures, Studies in Mathematics and Its Applications, Vol. 5, North-Holland Publishing Co., Amsterdam–New York, 1978.

Bourgeat,

Mikelić and

Wright, Stochastic two-scale convergence in the mean and applications, J. Reine Angew. Math. 456 (1994), 19–51.

Bourgeat and

Piatnitski, Approximations of effective coefficients in stochastic homogenization, in: Annales de L’IHP Probabilités et Statistiques, Vol. 40, 2004, pp. 153–165.

Brahim-Otsmane,

G.A.

Francfort and

Murat, Correctors for the homogenization of the wave and heat equations, Journal de mathématiques pures et appliquées 71(3) (1992), 197–231.

Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Springer Science & Business Media, 2010.

Cioranescu,

Damlamian and

Griso, Periodic unfolding and homogenization, Comptes Rendus Mathematique 335(1) (2002), 99–104. doi:10.1016/S1631-073X(02)02429-9.

Cioranescu,

Damlamian and

Griso, The periodic unfolding method in homogenization, SIAM Journal on Mathematical Analysis 40(4) (2008), 1585–1620. doi:10.1137/080713148.

10.

Cioranescu and

Donato, An Introduction to Homogenization, Oxford Lecture Series in Mathematics and Its Applications, Vol. 17, The Clarendon Press, Oxford University Press, New York, 1999.

11.

Cooper and

Waurick, Fibre homogenisation, Journal of Functional Analysis 276(11) (2019), 3363–3405. doi:10.1016/j.jfa.2019.03.004.

12.

Dohnal,

Lamacz and

Schweizer, Bloch-wave homogenization on large time scales and dispersive effective wave equations, Multiscale Model. Simul. 12(2) (2014), 488–513. doi:10.1137/130935033.

13.

Franz and

Waurick, Resolvent estimates and numerical implementation for the homogenisation of one-dimensional periodic mixed type problems, Zeitschrift für Angewandte Mathematik und Mechanik 98(7) (2018), 1036–1294. doi:10.1002/zamm.201700329.

14.

Franz and

Waurick, Homogenisation of parabolic/hyperbolic media, in: Boundary and Interior Layers, Computational and Asymptotic Methods BAIL 2018, Birkhäuser, Berlin, 2020.

15.

Heida, An extension of the stochastic two-scale convergence method and application, Asymptotic Analysis 72(1–2) (2011), 1–30. doi:10.3233/ASY-2010-1022.

16.

Heida,

Neukamm and

Varga, Stochastic homogenization of Λ-convex gradient flows, Discrete & Continuous Dynamical Systems – S (2020).

17.

V.V.

Jikov,

S.M.

Kozlov and

O.A.

Oleinik, Homogenization of Differential Operators and Integral Functionals, Springer Science & Business Media, 2012.

18.

Kalauch,

Picard,

Siegmund,

Trostorff and

M.M.

Waurick, A Hilbert space perspective on ordinary differential equations with memory term, Journal of Dynamics and Differential Equations 26(2) (2014), 369–399. doi:10.1007/s10884-014-9353-6.

19.

Kristensson and

Wellander, Homogenization of the Maxwell equations at fixed frequency, SIAM Journal on Applied Mathematics 64(1) (2003), 170–195. doi:10.1137/S0036139902403366.

20.

Lukkassen,

Nguetseng and

Wall, Two-scale convergence, International Journal of Pure and Applied Mathematics 2(1) (2002), 35–86.

21.

McGhee,

Picard,

Trostorff and

Waurick, Partial differential equations, in: Wiley, 2014, pp. 527–550, chapter 16.

22.

Mielke,

Roubiček and

Stefanelli, Γ-Limits and relaxations for rate-independent evolutionary problems, Calc. Var. Partial Differential Equations 31(3) (2008), 387–416. doi:10.1007/s00526-007-0119-4.

23.

Murat and

Tartar, H-convergence, in: Topics in the Mathematical Modelling of Composite Materials, Springer, 2018, pp. 21–43. doi:10.1007/978-3-319-97184-1_3.

24.

Neukamm, An introduction to the qualitative and quantitative theory of homogenization, Interdisciplinary Information Sciences 24(1) (2018), 1–48.

25.

Neukamm and

Varga, Stochastic unfolding and homogenization of spring network models, Multiscale Modeling & Simulation 16(2) (2018), 857–899. doi:10.1137/17M1141230.

26.

Nguetseng, A general convergence result for a functional related to the theory of homogenization, SIAM J. Math. Anal. 20(3) (1989), 608–623. doi:10.1137/0520043.

27.

G.C.

Papanicolaou and

S.S.

Varadhan, Boundary value problems with rapidly oscillating random coefficients, Random Fields 1 (1979), 835–873.

28.

Picard, A structural observation for linear material laws in classical mathematical physics, Mathematical Methods in the Applied Sciences 32 (2009), 1768–1803. doi:10.1002/mma.1110.

29.

Picard, Mother operators and their descendants, J. Math. Anal. Appl. 403(1) (2013), 54–62. Corrected version in

arXiv:1203.6762

by R. Picard, S. Trostorff, M. Waurick. doi:10.1016/j.jmaa.2013.02.004.

30.

Picard, On an elasto-acoustic transmission problem in anisotropic, inhomogeneous media, Adv. Oper. Theory 3(4) (2018), 816–828. doi:10.15352/aot.1803-1323.

31.

Picard, On an electro-magneto-elasto-dynamic transmission problem, in: Maxwell’s Equations,

Pauly,

Langer and

Repin, eds, De Gryuter, 2019, pp. 383–402. doi:10.1515/9783110543612-011.

32.

Picard and

McGhee, Partial Differential Equations: A Unified Hilbert Space Approach, Expositions in Mathematics, Vol. 55, De Gruyter, Berlin, 2011.

33.

Picard,

Trostorff and

Waurick, On evolutionary equations with material laws containing fractional integrals, Math. Meth. Appl. Sci. 38(15) (2015), 3141–3154. doi:10.1002/mma.3286.

34.

Picard,

Trostorff and

Waurick, Well-posedness via monotonicity. An overview, in: Operator Semigroups Meet Complex Analysis, Harmonic Analysis and Mathematical Physics, Operator Theory: Advances and Applications, Vol. 250, 2015, pp. 397–452.

35.

Picard,

Trostorff and

Waurick, On maximal regularity for a class of evolutionary equations, Journal of Mathematical Analysis and Applications 449(2) (2017), 1368–1381. doi:10.1016/j.jmaa.2016.12.057.

36.

Picard,

Trostorff,

Waurick and

Wehowski, On non-autonomous evolutionary problems, Journal of Evolution Equations 13 (2013), 751–776. doi:10.1007/s00028-013-0201-7.

37.

Seifert,

Trostorff and

Waurick, Evolutionary equations, ISem23 Lecture Notes, 2020, arXiv:2003.12403.

38.

Süß and

Waurick, A solution theory for a general class of SPDEs, Stochastics and Partial Differential Equations: Analysis and Computations 5(2) (2017), 278–318. doi:10.1007/s40072-016-0088-8.

39.

J.F.

Tachago and

Nnang, Stochastic-periodic homogenization of Maxwell’s equations with linear and periodic conductivity, Acta Mathematica Sinica, English Series 33(1) (2017), 117–152. doi:10.1007/s10114-016-5597-x.

40.

ter Elst,

Gorden and

Waurick, The Dirichlet-to-Neumann operator for divergence form problems, Annali di Matematica Pura ed Applicata 198(1) (2019), 177–203. doi:10.1007/s10231-018-0768-2.

41.

Trostorff, Well-posedness and causality for a class of evolutionary inclusions, Dissertation, TU Dresden, 2011.

42.

Trostorff, Exponential stability for linear evolutionary equations, Asymptot. Anal. 85(3–4) (2013), 179–197. doi:10.3233/ASY-131181.

43.

Trostorff, Exponential stability for second order evolutionary problems, J. Math. Anal. Appl. 429(2) (2015), 1007–1032. doi:10.1016/j.jmaa.2015.04.046.

44.

Trostorff and

Waurick, A note on elliptic type boundary value problems with maximal monotone relations, Mathematische Nachrichten 287(13) (2014), 1545–1558. doi:10.1002/mana.201200242.

45.

Varga, Stochastic unfolding and homogenization of evolutionary gradient systems, Dissertation, Technische Universität Dresden, 2019.

46.

Visintin, Towards a two-scale calculus, ESAIM: Control, Optimisation and Calculus of Variations 12(3) (2006), 371–397.

47.

Waurick, A Hilbert space approach to homogenization of linear ordinary differential equations including delay and memory terms, Mathematical Methods in the Applied Sciences 35(9) (2012), 1067–1077. doi:10.1002/mma.2515.

48.

Waurick, Homogenization of a class of linear partial differential equations, Asymptotic Analysis 82 (2013), 271–294. doi:10.3233/ASY-2012-1145.

49.

Waurick, G-convergence of linear differential equations, Journal of Analysis and Its Applications 33(4) (2014), 385–415.

50.

Waurick, Homogenization in fractional elasticity, SIAM J. Math. Anal. 46(2) (2014), 1551–1576. doi:10.1137/130941596.

51.

Waurick, On non-autonomous integro-differential-algebraic evolutionary problems, Mathematical Methods in the Applied Sciences 38(4) (2015), 665–676. doi:10.1002/mma.3097.

52.

Waurick, On the homogenization of partial integro-differential-algebraic equations, Operators and Matrices 10(2) (2016), 247–283. doi:10.7153/oam-10-15.

53.

Waurick, Stabilization via homogenization, Applied Mathematics Letters 60 (2016), 101–107. doi:10.1016/j.aml.2016.04.004.

54.

Waurick, Nonlocal H-convergence, Calculus of Variations and Partial Differential Equations 57(6) (2018), 46 pages.

55.

Waurick, Continuous dependence on the coefficients for a class of non-autonomous evolutionary equations, in: Maxwell’s Equations Analysis and Numerics, Radon Series on Computational and Applied Mathematics, Vol. 24, Birkhäuser/Springer Basel AG, Basel, 2019.

56.

Waurick, Limiting processes in evolutionary equations – a Hilbert space approach to homogenization, Dissertation, Technische Universität Dresden, 2011, http://nbn-resolving.de/urn:nbn:de:bsz:14-qucosa-67442.

57.

Waurick, On the continuous dependence on the coefficients of evolutionary equations, Habilitation, Technische Universität Dresden, 2016, http://arxiv.org/abs/1606.07731.

58.

Wellander, Homogenization of the Maxwell equations. Case I. Linear theory, Appl. Math. 46(1) (2001), 29–51. doi:10.1023/A:1013727504393.

59.

V.V.

Zhikov and

Pyatnitskii, Homogenization of random singular structures and random measures, Izvestiya: Mathematics 70(1) (2006), 19–67. doi:10.1070/IM2006v070n01ABEH002302.

Two-scale homogenization of abstract linear time-dependent PDEs

Abstract

Keywords

1. Introduction

1.1. A brief reminder on two-scale homogenization

2. The operator theoretic setting for unfolding

4.1. A Hilbert space framework for evolutionary equations

Theorem 4.1 ([28, Solution Theory]).

Proof. For z ∈ C Re > ν 0 it follows from [40, Lemma 2.12] that M ( z ) + A is continuously invertible in H with inverse satisfying ‖ ( M ( z ) + A ) − 1 ‖ ⩽ 1 / c . Thus, [43, Remark 2.3(a)] applies and we obtain the assertion. □ Remark 4.2 (Initial Value Problems).

5.1. Examples of unfolding operators

Footnotes

Acknowledgements

References

Proof.
For $z \in C_{Re > ν_{0}}$ it follows from [40, Lemma 2.12] that $M (z) + A$ is continuously invertible in H with inverse satisfying $‖ {(M (z) + A)}^{- 1} ‖ ⩽ 1 / c$ . Thus, [43, Remark 2.3(a)] applies and we obtain the assertion. □
Remark 4.2 (Initial Value Problems).