2.1 The microscopic geometry

Let $\Omega \subset \mathbb{R}^n$ with Lipschitz boundary and $\epsilon>0 $ a sequence with $\epsilon^{-1} \in \mathbb{N}$ . We define the unit cube $Y:=(0,1)^n$ and $Y_2 \subset Y$ such that $\overline{Y_2} \subset Y$ , so $Y_2 $ strictly included in Y. Further, we define $Y_1:= Y\setminus \overline{Y_2}$ and $\Gamma := \partial Y_2$ , and we suppose that $\Gamma \in C^{1,1}$ . We assume that $Y_1$ is connected and for the sake of simplicity we also assume that $Y_2$ is connected. The general case of disconnected $Y_2$ is easily obtained by considering the connected components of $Y_2$ , see also Remark 2. Now, the microscopic domains ${\Omega_{\epsilon}^1}$ and ${\Omega_{\epsilon}^2}$ are defined by scaled and shifted reference elements $Y_j$ for $j=1,2$ . Let $K_{\epsilon}:= \{k\in \mathbb{Z}^n \, : \, \epsilon(k + Y) \subset \Omega\}$ and define

\begin{align*}{\Omega_{\epsilon}^2}:= \bigcup_{k\in K_{\epsilon}} \epsilon (Y_2 + k), \quad \quad {\Omega_{\epsilon}^1} := \Omega \setminus \overline{{\Omega_{\epsilon}^2}}, \quad \quad {\Gamma_{\epsilon}} := \partial {\Omega_{\epsilon}^2}.\end{align*}

Hence, ${\Gamma_{\epsilon}} $ denotes the oscillating interface between ${\Omega_{\epsilon}^1}$ and ${\Omega_{\epsilon}^2}$ . Due to the assumptions on $Y_1$ and $Y_2$ it holds that ${\Omega_{\epsilon}^1}$ is connected and ${\Omega_{\epsilon}^2}$ is disconnected, and ${\Gamma_{\epsilon}} \in C^{1,1}$ is not touching the outer boundary $\partial \Omega$ .

2.2 The microscopic model

We are looking for a solution $\left({u_{\epsilon}^1},{u_{\epsilon}^2}\right)$ with ${u_{\epsilon}^j}: (0,T)\times {\Omega_{\epsilon}^j} \rightarrow \mathbb{R}$ for $j=1,2$ , such that it holds that

(2.1) \begin{align}\begin{aligned}\partial_t {u_{\epsilon}^j} - \nabla \cdot \big({D^j_{\epsilon}} \nabla {u_{\epsilon}^j} \big) &= {f_{\epsilon}^j}\big({u_{\epsilon}^j}\big) &\mbox{ in }& (0,T)\times {\Omega_{\epsilon}^j},\\\epsilon \big( \partial_t {u_{\epsilon}^j} - {\nabla_{{\Gamma_{\epsilon}}}} \cdot \big( {D_{{\Gamma_{\epsilon}}}^j} {\nabla_{{\Gamma_{\epsilon}}}} {u_{\epsilon}^j}\big) - {h_{\epsilon}^j}\left({u_{\epsilon}^1} , {u_{\epsilon}^2}\right) \big) &= - {D^j_{\epsilon}} \nabla {u_{\epsilon}^j}\cdot \nu &\mbox{ on }& (0,T) \times {\Gamma_{\epsilon}},\\-D_{\epsilon}^1 \nabla {u_{\epsilon}^1} \cdot \nu &= 0 &\mbox{ on }& (0,T)\times \partial \Omega,\\{u_{\epsilon}^j}(0) &= u_{\epsilon,i}^j &\mbox{ in }& {\Omega_{\epsilon}^j},\\{u_{\epsilon}^j}|_{{\Gamma_{\epsilon}}}(0) &= u_{\epsilon,i,{\Gamma_{\epsilon}}}^j &\mbox{ on }& {\Gamma_{\epsilon}},\end{aligned}\end{align}

where $\nu$ denotes the outer unit normal (we neglect a subscript for the underlying domain, since this should be clear from the context), and ${u_{\epsilon}^j}|_{{\Gamma_{\epsilon}}}$ denotes the trace of ${u_{\epsilon}^j}$ on ${\Gamma_{\epsilon}}$ . If it is clear from the context, we use the same notation for a function and its trace, for example, we just write ${u_{\epsilon}^j} $ for ${u_{\epsilon}^j}|_{{\Gamma_{\epsilon}}}$ . The precise weak formulation of the micro-model above is stated in Section 3, see (3.2), after introducing the necessary function spaces.

In the following, with $T_y \Gamma$ and $T_x {\Gamma_{\epsilon}}$ for $y \in \Gamma $ and $x \in {\Gamma_{\epsilon}}$ we denote the tangent spaces of $\Gamma$ at y and ${\Gamma_{\epsilon}}$ at x, respectively. The orthogonal projection $P_{\Gamma}(y): \mathbb{R}^n \rightarrow T_y \Gamma $ for $y \in \Gamma$ is given by

\begin{align*}P_{\Gamma}(y) \xi = \xi - (\xi\cdot \nu(y))\nu(y) \quad \quad \mbox{for } \xi \in \mathbb{R}^n,\end{align*}

where $\nu(y) $ denotes the outer unit normal at $y\in \Gamma$ . Let us extend the unit normal Y-periodically. Then, the orthogonal projection $P_{{\Gamma_{\epsilon}}}(x): \mathbb{R}^n \rightarrow T_x {\Gamma_{\epsilon}} $ for $x \in {\Gamma_{\epsilon}}$ is given by

\begin{align*}P_{{\Gamma_{\epsilon}}}(x)\xi = \xi - \left(\xi \cdot \nu\left({\frac{x}{\epsilon}}\right)\right)\nu\left({\frac{x}{\epsilon}}\right) \quad \quad \mbox{for } \xi \in \mathbb{R}^n.\end{align*}

Assumptions on the data:

In the following let $j \in \{1,2\}$ .

1. (A1) For the bulk diffusion, we have ${D^j_{\epsilon}}(x):= D^j\left({\frac{x}{\epsilon}}\right)$ with $D^j \in L^{\infty}_{{\mathrm{per}}}\left(Y_j\right)^{n\times n}$ symmetric and coercive, that is, there exits $c_0 >0$ such that for almost every $y \in Y_j$

   \begin{align*}D^j(y)\xi \cdot \xi \geqslant c_0 |\xi|^2 \quad \mbox{ for all } \xi \in \mathbb{R}^n.\end{align*}

2. (A2) For the surface diffusion, we suppose ${D_{{\Gamma_{\epsilon}}}^j}(x):= D_{\Gamma}^j\left({\frac{x}{\epsilon}}\right) $ with $D_{\Gamma}^j \in L^{\infty}_{{\mathrm{per}}}(\Gamma)^{n\times n}$ symmetric and $D_{\Gamma}^j(y)|_{T_y\Gamma}: T_y \Gamma \rightarrow T_y \Gamma$ for almost every $y \in \Gamma$ . Further, we assume that $D_{\Gamma}^j$ is coercive, that is, there exists $c_0>0$ such that for almost every $y \in \Gamma$

   \begin{align*}D_{\Gamma}^j(y)\xi\cdot \xi \geqslant c_0 |\xi|^2 \quad \mbox{ for all } \xi \in T_y \Gamma.\end{align*}

3. (A3) For the reaction rates in the bulk domains, we suppose that ${f_{\epsilon}^j} (t,x,z):= f^j\left(t,{\frac{x}{\epsilon}},z\right)$ with $f^j \in L^{\infty}\big((0,T)\times Y_j \times \mathbb{R}\big)$ is Y-periodic with respect to the second variable and uniformly Lipschitz continuous with respect to the last variable, that is, there exists $C>0$ such that for all $z,w \in \mathbb{R}$ and almost every $(t,y) \in (0,T)\times Y_j$ it holds that

   \begin{align*}|f^j(t,y,z)-f^j(t,y,w)| \leqslant C |z - w|.\end{align*}

4. (A4) For the reaction rates on the surface, we suppose that ${h_{\epsilon}^j}(t,x,z_1,z_2) := h^j\left(t,{\frac{x}{\epsilon}},z_1,z_2\right)$ with $h^j \in L^{\infty}((0,T)\times \Gamma \times \mathbb{R}^2)$ is Y-periodic with respect to the second variable and uniformly Lipschitz continuous with respect to the last variable, that is, there exists $C>0$ such that for all $z_1,z_2,w_1,w_2 \in \mathbb{R}$ and almost every $(t,y)\in (0,T)\times \Gamma $ it holds that

   \begin{align*}|h^j(t,y,z_1,z_2)-h^j(t,y,w_1,w_2)| \leqslant C \big(|z_1 - w_1| + |z_2 - w_2|\big).\end{align*}

5. (A5) For the initial conditions, we assume $u_{\epsilon,i}^j \in L^2\big({\Omega_{\epsilon}^j}\big)$ and $u_{\epsilon,i,{\Gamma_{\epsilon}}}^j \in L^2({\Gamma_{\epsilon}})$ with

   \begin{align*}\|u_{\epsilon,i}^j\|_{L^2\big({\Omega_{\epsilon}^j}\big)} + \sqrt{\epsilon}\|u_{\epsilon,i,{\Gamma_{\epsilon}}}^j\|_{L^2({\Gamma_{\epsilon}})} \leqslant C.\end{align*}
   Further, there exist $u_{0,i}^j \in L^2(\Omega) $ and $u_{0,i,\Gamma}^j \in L^2(\Omega)$ such that
   \begin{align*}u_{\epsilon,i}^j &\rightarrow u_{0,i}^j &\mbox{ in the two-scale sense}&,\\u_{\epsilon,i,{\Gamma_{\epsilon}}}^j &\rightarrow u_{0,i,\Gamma}^j &\mbox{ in the two-scale sense on } {\Gamma_{\epsilon}} &.\end{align*}

Additionally, we assume that the sequences $u_{\epsilon,i}^2$ and $u_{\epsilon,i,{\Gamma_{\epsilon}}}^2$ converge strongly in the two-scale sense. For the definition of the two-scale convergence see Section 4.

3.1 Function spaces

Due to the Laplace–Beltrami operator in the boundary condition in (2.1), it is not enough to consider the usual Sobolev space $H^1\big({\Omega_{\epsilon}^j}\big)$ as a solution space, because we need more regular traces. This gives rise to deal with the following function spaces for $j=1,2$ :

(3.1) \begin{align}\begin{aligned}{\mathbb{H}_{j,\epsilon}}&:= \left\{{\phi_{\epsilon}^j} \in H^1\big({\Omega_{\epsilon}^j}\big)\, : \, {\phi_{\epsilon}^j}|_{{\Gamma_{\epsilon}}} \in H^1({\Gamma_{\epsilon}})\right\},\\[4pt]{\mathbb{H}_j} &:= \left\{ \phi \in H^1\left(Y_j\right) \, : \, \phi|_{\Gamma} \in H^1(\Gamma)\right\},\end{aligned}\end{align}

with the inner products

\begin{align*} \big({\phi_{\epsilon}^j},\psi_{\epsilon}^j\big)_{{\mathbb{H}_{j,\epsilon}}} &:= \big({\phi_{\epsilon}^j},\psi_{\epsilon}^j\big)_{H^1\big({\Omega_{\epsilon}^j}\big)} + \epsilon \big({\phi_{\epsilon}^j},\psi_{\epsilon}^j\big)_{H^1({\Gamma_{\epsilon}})}, \\ (\phi,\psi)_{{\mathbb{H}_j}} &:= (\phi,\psi)_{H^1\left(Y_j\right)} + (\phi,\psi)_{H^1(\Gamma)}.\end{align*}

The associated norms are denoted by $\|\cdot\|_{{\mathbb{H}_{j,\epsilon}}}$ and $\|\cdot\|_{{\mathbb{H}_j}}$ . Obviously, the spaces ${\mathbb{H}_{j,\epsilon}}$ and ${\mathbb{H}_j}$ are separable Hilbert spaces and we have the dense embeddings $C^{\infty}\big(\overline{{\Omega_{\epsilon}^j}}\big) \subset {\mathbb{H}_{j,\epsilon}}$ and $C^{\infty}\big(\overline{Y_j}\big)\subset {\mathbb{H}_j}$ , see [Reference Gahn17, Proposition 5]. We also define the space

\begin{align*}{\mathbb{L}_{j,\epsilon}} := L^2\big({\Omega_{\epsilon}^j}\big) \times L^2({\Gamma_{\epsilon}}), \quad \quad {\mathbb{L}_j}:= L^2\left(Y_j\right)\times L^2(\Gamma)\end{align*}

with inner products

\begin{align*}\big({\phi_{\epsilon}^j},\psi_{\epsilon}^j\big)_{{\mathbb{L}_{j,\epsilon}}} &:= \big({\phi_{\epsilon}^j},\psi_{\epsilon}^j\big)_{L^2\big({\Omega_{\epsilon}^j}\big)} + \epsilon \big({\phi_{\epsilon}^j},\psi_{\epsilon}^j\big)_{L^2({\Gamma_{\epsilon}})}, \\ (\phi,\psi)_{{\mathbb{L}_j}} &:= (\phi,\psi)_{L^2\left(Y_j\right)} + (\phi,\psi)_{L^2(\Gamma)},\end{align*}

and again denote the associated norms by $\|\cdot\|_{{\mathbb{L}_{j,\epsilon}}}$ and $\|\cdot \|_{{\mathbb{L}_j}}$ . Obviously, we have

\begin{align*}{\mathbb{H}_{j,\epsilon}} \overset{\sim}{=} \left\{ ({u_{\epsilon}},{v_{\epsilon}}) \in H^1\big({\Omega_{\epsilon}^j}\big)\times H^1({\Gamma_{\epsilon}}) \, : \, {u_{\epsilon}}|_{{\Gamma_{\epsilon}}} = {v_{\epsilon}}\right\},\end{align*}

and a similar result for ${\mathbb{H}_j}$ . Therefore, we have the following Gelfand triples:

\begin{align*}{\mathbb{H}_{j,\epsilon}} \hookrightarrow {\mathbb{L}_{j,\epsilon}} \hookrightarrow {\mathbb{H}^{\prime}_{j,\epsilon}},\quad \quad {\mathbb{H}_j} \hookrightarrow {\mathbb{L}_j} \hookrightarrow {\mathbb{H}^{\prime}_j}.\end{align*}

We will also make use for $\alpha \in \left(\frac12,1\right]$ of the function space

\begin{align*}{\mathbb{H}^{\alpha}_j}:= \left\{\phi \in H^{\alpha}\left(Y_j\right) \, : \, \phi|_{\Gamma} \in H^{\alpha}(\Gamma)\right\}\end{align*}

with inner product

\begin{align*}(\phi,\psi)_{{\mathbb{H}^{\alpha}_j}} := (\phi,\psi)_{H^{\alpha}\left(Y_j\right)} + (\phi,\psi)_{H^{\alpha}(\Gamma)}.\end{align*}

By definition, we have ${\mathbb{H}_j} = {\mathbb{H}^1_j}$ . Obviously, we have the compact embedding ${\mathbb{H}_j} \hookrightarrow {\mathbb{H}^{\alpha}_j}$ for $\alpha \in \left(\frac12,1\right)$ .

3.2 Existence and uniqueness of a weak solution

A weak solution of Problem (2.1) is defined in the following way: The tuple $\left({u_{\epsilon}^1},{u_{\epsilon}^2}\right)$ is a weak solution of (2.1) if for $j=1,2$

\begin{align*}{u_{\epsilon}^j} \in L^2\big((0,T),{\mathbb{H}_{j,\epsilon}}\big)\cap H^1\big((0,T),{\mathbb{H}^{\prime}_{j,\epsilon}}\big),\end{align*}

${u_{\epsilon}^j}$ and ${u_{\epsilon}^j}|_{{\Gamma_{\epsilon}}}$ fulfil the initial condition ${u_{\epsilon}^j}(0) = u_{\epsilon,i}^j$ and ${u_{\epsilon}^j}|_{{\Gamma_{\epsilon}}}(0) = u_{\epsilon,i,{\Gamma_{\epsilon}}}^j$ , and for all ${\phi_{\epsilon}^j}\in {\mathbb{H}_{j,\epsilon}}$ it holds almost everywhere in (0,T)

(3.2) \begin{align}\begin{aligned} \big\langle \partial_t {u_{\epsilon}^j} , {\phi_{\epsilon}^j} & \big\rangle_{{\mathbb{H}^{\prime}_{j,\epsilon}},{\mathbb{H}_{j,\epsilon}}} + \big({D^j_{\epsilon}} \nabla {u_{\epsilon}^j} , \nabla {\phi_{\epsilon}^j} \big)_{{\Omega_{\epsilon}^j}} + \epsilon \big({D_{{\Gamma_{\epsilon}}}^j} {\nabla_{{\Gamma_{\epsilon}}}} {u_{\epsilon}^j} , {\nabla_{{\Gamma_{\epsilon}}}} {\phi_{\epsilon}^j} \big)_{{\Gamma_{\epsilon}}}\\&= \big({f_{\epsilon}^j} \big({u_{\epsilon}^j}\big) , {\phi_{\epsilon}^j}\big)_{{\Omega_{\epsilon}^j}} + \epsilon \big({h_{\epsilon}^j}\left({u_{\epsilon}^1},{u_{\epsilon}^2}\right),{\phi_{\epsilon}^j}\big)_{{\Gamma_{\epsilon}}}.\end{aligned}\end{align}

Here $(\cdot,\cdot)_U$ stands for the inner product on $L^2(U)$ , for a suitable set $U \subset \mathbb{R}^n$ and for a Banach space X and its dual X ^′ we write $\langle \cdot , \cdot \rangle_{X^{\prime},X}$ for the duality pairing between X ^′ and X. The scaling factor $\epsilon$ for the time derivative in (2.1) is included in the duality pairing $\langle \cdot,\cdot \rangle_{{\mathbb{H}^{\prime}_{j,\epsilon}},{\mathbb{H}_{j,\epsilon}}}$ . In fact, if additionally it holds that $\partial_t {u_{\epsilon}^j} \in L^2\big((0,T),H^1\big({\Omega_{\epsilon}^j}\big)^{\prime}\big)$ and $\partial_t {u_{\epsilon}^j}|_{{\Gamma_{\epsilon}}} \in L^2((0,T),H^1({\Gamma_{\epsilon}})^{\prime})$ with respect to the Gelfand triples $H^1\big({\Omega_{\epsilon}^j}\big) \hookrightarrow L^2\big({\Omega_{\epsilon}^j}\big) \hookrightarrow H^1\big({\Omega_{\epsilon}^j}\big)^{\prime}$ and $H^1({\Gamma_{\epsilon}}) \hookrightarrow L^2({\Gamma_{\epsilon}}) \hookrightarrow H^1({\Gamma_{\epsilon}})^{\prime}$ , we get for all ${\phi_{\epsilon}^j} \in {\mathbb{H}_{j,\epsilon}}$

\begin{align*}\big\langle \partial_t {u_{\epsilon}^j} , {\phi_{\epsilon}^j} \big\rangle_{{\mathbb{H}^{\prime}_{j,\epsilon}},{\mathbb{H}_{j,\epsilon}}} = \big\langle \partial_t {u_{\epsilon}^j}, {\phi_{\epsilon}^j}\big\rangle_{H^1\big({\Omega_{\epsilon}^j}\big)^{\prime},H^1\big({\Omega_{\epsilon}^j}\big)} + \epsilon \big\langle \partial_t {u_{\epsilon}^j}|_{{\Gamma_{\epsilon}}} , {\phi_{\epsilon}^j}|_{{\Gamma_{\epsilon}}} \big\rangle_{H^1({\Gamma_{\epsilon}})^{\prime},H^1({\Gamma_{\epsilon}})}.\end{align*}

3.3 A priori estimates

We derive a priori estimates for the microscopic solution depending explicitly on $\epsilon$ . These estimates are necessary for the application of the two-scale compactness results from Section 4 to derive the macroscopic model. In a first step, we give estimates in the spaces $L^2\big((0,T),{\mathbb{H}_{j,\epsilon}}\big)$ and $H^1\big((0,T),{\mathbb{H}^{\prime}_{j,\epsilon}}\big)$ . Such kind of estimates is also needed to establish the existence of a weak solution via the Galerkin method. In a second step, we derive estimates for the difference of shifted microscopic solution with respect to the macroscopic variable. These estimates are necessary for strong two-scale compactness results in the disconnected domain.

The following trace inequality for perforated domains will be used frequently throughout the paper and follows easily by a standard decomposition argument and the trace inequality on the reference element $Y_j$ , see also [Reference Galdi21, Theorem II.4.1 and Exercise II.4.1]: For every $\theta >0,$ there exists a $C(\theta)>0$ such that for every ${v_{\epsilon}} \in H^1\big({\Omega_{\epsilon}^j}\big)$ it holds that

(3.3) \begin{align}\sqrt{\epsilon}\|{v_{\epsilon}}\|_{L^2({\Gamma_{\epsilon}})} \leqslant C(\theta) \|{v_{\epsilon}}\|_{L^2\big({\Omega_{\epsilon}^j}\big)} + \theta \epsilon \|\nabla {v_{\epsilon}} \|_{L^2\big({\Omega_{\epsilon}^j}\big)}.\end{align}

Proposition 2 The weak solution ${u_{\epsilon}} = \left({u_{\epsilon}^1},{u_{\epsilon}^2}\right)$ of the microscopic problem (2.1) fulfils the following a priori estimate

\begin{align*}\big\|\partial_t {u_{\epsilon}^j}\big\|_{L^2\big((0,T),{\mathbb{H}^{\prime}_{j,\epsilon}}\big)} + \big\|{u_{\epsilon}^j}\big\|_{L^2\big((0,T),{\mathbb{H}_{j,\epsilon}}\big)} \leqslant C,\end{align*}

for a constant $C > 0$ independent of $\epsilon$ .

Proof We choose ${u_{\epsilon}^j}$ as a test function in (3) (for $j=1,2$ ) to obtain with the Assumptions (A3) and (A4) on $f^j$ and $h^j$

\begin{align*}\frac12 \frac{d}{dt} \big\Vert {u_{\epsilon}^j} \big\Vert_{{\mathbb{L}_{j,\epsilon}}}^2 + \big({D^j_{\epsilon}} & \nabla {u_{\epsilon}^j} , \nabla {u_{\epsilon}^j} \big)_{{\Omega_{\epsilon}^j}} + \epsilon \big({D_{{\Gamma_{\epsilon}}}^j} {\nabla_{{\Gamma_{\epsilon}}}} {u_{\epsilon}^j} , {\nabla_{{\Gamma_{\epsilon}}}} {u_{\epsilon}^j} \big)_{{\Gamma_{\epsilon}}}\\&= \big({f_{\epsilon}^j} \big({u_{\epsilon}^j}\big) , {u_{\epsilon}^j}\big)_{{\Omega_{\epsilon}^j}} + \epsilon \big({h_{\epsilon}^j}\left({u_{\epsilon}^1},{u_{\epsilon}^2}\right),{u_{\epsilon}^j}\big)_{{\Gamma_{\epsilon}}}\\&\leqslant C\left(1 + \big\Vert {u_{\epsilon}^j}\big\Vert_{L^2\big({\Omega_{\epsilon}^j}\big)}^2 + \epsilon\big\Vert {u_{\epsilon}^1}\big\Vert^2_{L^2({\Gamma_{\epsilon}})} + \epsilon \big\Vert {u_{\epsilon}^2}\big\Vert^2_{L^2({\Gamma_{\epsilon}})} \right)\\&\leqslant C\left(1 + \big\Vert {u_{\epsilon}^1}\big\Vert^2_{{\mathbb{L}_{1,\epsilon}}} + \big\Vert {u_{\epsilon}^2}\big\Vert^2_{{\mathbb{L}_{2,\epsilon}}}\right).\end{align*}

Using the coercivity of ${D^j_{\epsilon}}$ and ${D_{{\Gamma_{\epsilon}}}^j}$ from the Assumptions (A1) and (A2), we obtain for $j=1,2$

\begin{align*}\frac{d}{dt}\big\Vert {u_{\epsilon}^j}\big\Vert_{{\mathbb{L}_{j,\epsilon}}}^2 + \big\Vert \nabla {u_{\epsilon}^j}\big\Vert_{L^2\big({\Omega_{\epsilon}^j}\big)}^2 + \epsilon \big\Vert {\nabla_{{\Gamma_{\epsilon}}}} {u_{\epsilon}^j} \big\Vert_{L^2({\Gamma_{\epsilon}})}^2 \leqslant C\left(1 + \big\Vert {u_{\epsilon}^1}\big\Vert^2_{{\mathbb{L}_{1,\epsilon}}} + \big\Vert {u_{\epsilon}^2}\big\Vert^2_{{\mathbb{L}_{2,\epsilon}}}\right).\end{align*}

Summing over $j=1,2$ , integrating with respect to time, Assumption (A5) and the Gronwall inequality implies the boundedness of $\big\Vert {u_{\epsilon}^j}\big\Vert_{L^2\big((0,T),{\mathbb{H}_{j,\epsilon}}\big)}$ uniformly with respect to $\epsilon$ .

It remains to check the bound for the time derivative $\partial_t {u_{\epsilon}^j}$ . As a test function in (3.2) we choose ${\phi_{\epsilon}^j} \in {\mathbb{H}_{j,\epsilon}}$ with $\big\Vert {\phi_{\epsilon}^j} \big\Vert_{{\mathbb{H}_{j,\epsilon}}} \leqslant 1$ to obtain (using the boundedness of the diffusion tensors and again the growth condition for $h^j$ and $f^j$ ):

\begin{align*}\big\langle \partial_t {u_{\epsilon}^j} , {\phi_{\epsilon}^j} \big\rangle_{{\mathbb{H}^{\prime}_{j,\epsilon}},{\mathbb{H}_{j,\epsilon}}} \leqslant C\left( \big\Vert {u_{\epsilon}^1} \big\Vert_{{\mathbb{H}_{1,\epsilon}}} + \big\Vert {u_{\epsilon}^2} \big\Vert_{{\mathbb{H}_{2,\epsilon}}} \right)\big\Vert {\phi_{\epsilon}^j}\big\Vert_{{\mathbb{H}_{j,\epsilon}}}\leqslant C\left( \big\Vert {u_{\epsilon}^1} \big\Vert_{{\mathbb{H}_{1,\epsilon}}} + \big\Vert {u_{\epsilon}^2} \big\Vert_{{\mathbb{H}_{2,\epsilon}}} \right).\end{align*}

Squaring, integrating with respect to time and the boundedness of ${u_{\epsilon}^j}$ for $j=1,2$ already obtained above imply the desired result.

Next, we derive estimates for the difference of the shifted functions. First of all, we introduce some additional notations. For $h>0$ let us define

\begin{align*}\Omega_h &:= \{x\in \Omega \, : \, \mathrm{dist}(x,\partial \Omega) > h \} ,\\K_{\epsilon,h} &:= \{k\in \mathbb{Z}^n\, : \, \epsilon(Y + k) \subset \Omega_h\},\\\Omega_{\epsilon,h} &:= \mathrm{int} \bigcup_{k \in K_{\epsilon,h}} \epsilon \big(\overline{Y} + k \big),\end{align*}

and the related perforated domains and the related surface

\begin{align*}{\Omega_{\epsilon,h}^2}&:= \bigcup_{k \in K_{\epsilon,h}} \epsilon\big(Y_2 + k \big), \quad {\Omega_{\epsilon,h}^1} := \Omega_{\epsilon,h} \setminus \overline{{\Omega_{\epsilon,h}^2}},\quad {\Gamma_{\epsilon,h}} := \partial {\Omega_{\epsilon,h}^2}.\end{align*}

For $l \in \mathbb{Z}^n $ with $|l\epsilon|< h$ and $G_{\epsilon,h} \in \big\{\Omega_{\epsilon,h} , {\Omega_{\epsilon,h}^1},{\Omega_{\epsilon,h}^2}\big\}$ , we define for an arbitrary function ${v_{\epsilon}} : G_{\epsilon,h} \rightarrow \mathbb{R}$ the shifted function

\begin{align*}{v^l_{\epsilon}}(x):= {v_{\epsilon}}(x + l\epsilon),\end{align*}

and the difference between the shifted function and the function itself

(3.4) \begin{align} \delta_l {v_{\epsilon}}(x):=\delta {v_{\epsilon}}(x):= {v^l_{\epsilon}}(x) - {v_{\epsilon}}(x) = {v_{\epsilon}} (x+l\epsilon) - {v_{\epsilon}}(x).\end{align}

Here, in the writing $\delta {v_{\epsilon}}$ we neglect the dependence on l if it is clear from the context. Further, we define ${\mathbb{H}_{j,\epsilon,h}}$ in the same way as ${\mathbb{H}_{j,\epsilon}}$ in (3.1) by replacing ${\Omega_{\epsilon}^j}$ and ${\Gamma_{\epsilon}}$ with ${\Omega_{\epsilon,h}^j}$ and ${\Gamma_{\epsilon,h}}$ . In the same way, we define ${\mathbb{L}_{j,\epsilon,h}}$ . Further, for any function $\phi_{\epsilon,h}\in {\mathbb{H}_{2,\epsilon,h}}$ we write ${\overline{\phi}_{\epsilon,h}} $ for the zero extension to ${\Omega_{\epsilon}^2}$ . Especially it holds that ${\overline{\phi}_{\epsilon,h}} \in {\mathbb{H}_{2,\epsilon}}$ , since ${\Omega_{\epsilon}^2}$ is disconnected.

Proposition 3 Let $0 <h \ll 1$ , then for all $l \in \mathbb{Z}^n$ with $|\epsilon l|<h$ , it holds that

\begin{align*}\big\|\delta {u_{\epsilon}^2}\big\|_{L^{\infty}\left((0,T),{\mathbb{L}_{2,\epsilon,h}}\right)} + & \big\|\nabla \delta {u_{\epsilon}^2} \big\|_{L^2\left((0,T),{\mathbb{L}_{2,\epsilon,h}}\right)}\\&\leqslant C \left( \big\|\delta {u_{\epsilon}^1} \big\|_{L^2\big((0,T),L^2\big({\Omega_{\epsilon,h}^1}\big)\big)} + \big\|\delta \big(u_{\epsilon,i}^2,u_{\epsilon,i,{\Gamma_{\epsilon}}}^2\big)\big\|_{{\mathbb{L}_{2,\epsilon,h}}} + \epsilon \right),\end{align*}

for a constant $C>0$ independent of $h,\, \epsilon,$ and l.

Proof Let $0 <h \ll 1$ and $l \in \mathbb{Z}^n$ with $|\epsilon l|<h$ , and we shortly write ${u^{2,l}_{\epsilon}} := \left({u_{\epsilon}^2}\vert_{{\Omega_{\epsilon,h}^2}}\right)^l$ , that is, the shifts with respect to $l\epsilon$ of the restriction ${u_{\epsilon}^2}\vert_{{\Omega_{\epsilon,h}^2}}$ (we neglect the index h). In the same way, we define ${u^{1,l}_{\epsilon}}$ . Let $\phi_{\epsilon,h} \in {\mathbb{H}_{2,\epsilon,h}}$ . Then, for $x \in {\Omega_{\epsilon}^2} \setminus \big({\Omega_{\epsilon,h}^2} + \epsilon l\big) $ it holds that $x - l\epsilon \notin {\Omega_{\epsilon,h}^2}$ and therefore ${\overline{\phi}^{-l}_{\epsilon,h}}(x) = 0$ and similar from $x \in {\Gamma_{\epsilon}} \setminus\big({\Gamma_{\epsilon,h}} + \epsilon l\big)$ it follows ${\overline{\phi}^{-l}_{\epsilon,h}}(x) = 0$ . This implies for all $\psi \in C_0^{\infty}(0,T)$

\begin{align*}\int_0^T &\left({u^{2,l}_{\epsilon}},\phi_{\epsilon,h}\right)_{{\mathbb{L}_{2,\epsilon,h}}} \psi^{\prime}(t) dt\\&= \int_0^T \left[ \int_{{\Omega_{\epsilon,h}^2}} {u_{\epsilon}^2} (t, x + l\epsilon) \phi_{\epsilon,h}(x) dx + \epsilon \int_{{\Gamma_{\epsilon,h}}} {u_{\epsilon}^2}(t,x+l\epsilon) \phi_{\epsilon,h}(x) d\sigma\right] \psi^{\prime}(t) dt\\&= \int_0^T \left[ \int_{{\Omega_{\epsilon,h}^2} + l \epsilon } {u_{\epsilon}^2}(t,x) \phi_{\epsilon,h}^{-l}(x) dx + \epsilon \int_{{\Gamma_{\epsilon,h}} + l \epsilon} {u_{\epsilon}^2}(t,x) \phi_{\epsilon,h}^{-l}(x) d\sigma\right] \psi^{\prime}(t) dt\\&= \int_0^T \left[ \int_{{\Omega_{\epsilon}^2} } {u_{\epsilon}^2}(t,x) \phi_{\epsilon,h}^{-l}(x) dx + \epsilon \int_{{\Gamma_{\epsilon}} } {u_{\epsilon}^2}(t,x) \phi_{\epsilon,h}^{-l}(x) d\sigma\right] \psi^{\prime}(t) dt\\&= \int_0^T \left({u_{\epsilon}^2} , {\overline{\phi}^{-l}_{\epsilon,h}} \right)_{{\mathbb{L}_{2,\epsilon}}} \psi^{\prime}(t) dt = - \int_0^T \left\langle \partial_t {u_{\epsilon}^2} , {\overline{\phi}^{-l}_{\epsilon,h}} \right\rangle_{{\mathbb{H}^{\prime}_{2,\epsilon}},{\mathbb{H}_{2,\epsilon}}} \psi(t) dt .\end{align*}

Hence, we have $\partial_t {u^{2,l}_{\epsilon}} \in L^2\big((0,T),{\mathbb{H}^{\prime}_{2,\epsilon,h}}\big)$ with

\begin{align*}\big\langle \partial_t {u^{2,l}_{\epsilon}} , \phi_{\epsilon,h} \big\rangle_{{\mathbb{H}^{\prime}_{2,\epsilon,h}},{\mathbb{H}_{2,\epsilon,h}}} = \Big\langle \partial_t {u_{\epsilon}^2} , {\overline{\phi}^{-l}_{\epsilon,h}} \Big\rangle_{{\mathbb{H}^{\prime}_{2,\epsilon}},{\mathbb{H}_{2,\epsilon}}}\end{align*}

almost everywhere in (0,T). Using ${\overline{\phi}^{-l}_{\epsilon,h}} \in {\mathbb{H}_{2,\epsilon}}$ as a test function in (3.2), we obtain using the periodicity of $D^2$ , $D^2_{\Gamma}$ , $f^2$ and $h^2$ , by an elemental calculation

\begin{align*}\Big\langle \partial_t {u_{\epsilon}^2} , {\overline{\phi}^{-l}_{\epsilon,h}} \Big\rangle_{{\mathbb{H}^{\prime}_{2,\epsilon}},{\mathbb{H}_{2,\epsilon}}} =& -\big(D_{\epsilon}^2 \nabla {u^{2,l}_{\epsilon}} , \nabla \phi_{\epsilon,h} \big)_{{\Omega_{\epsilon,h}^2}} - \epsilon \Big(D_{{\Gamma_{\epsilon}}}^2 {\nabla_{{\Gamma_{\epsilon}}}} {u^{2,l}_{\epsilon}} , {\nabla_{{\Gamma_{\epsilon}}}} \phi_{\epsilon,h}\Big)_{{\Gamma_{\epsilon,h}}}\\&+ \Big(f_{\epsilon}^2\big({u^{2,l}_{\epsilon}}\big) , \phi_{\epsilon,h} \Big)_{{\Omega_{\epsilon,h}^2}} + \epsilon \Big(h_{\epsilon}^2\big({u^{1,l}_{\epsilon}},{u^{2,l}_{\epsilon}}\big) , \phi_{\epsilon,h} \Big)_{{\Gamma_{\epsilon,h}}}.\end{align*}

Subtracting the above equation for $l=0$ and arbitrary $l\in \mathbb{Z}^n$ with $|\epsilon l|<h$ we obtain

\begin{align*}\big\langle \partial_t& \delta {u_{\epsilon}^2},\phi_{\epsilon,h}\big\rangle_{{\mathbb{H}^{\prime}_{2,\epsilon,h}},{\mathbb{H}_{2,\epsilon,h}}} + \big(D_{\epsilon}^2 \nabla \delta {u_{\epsilon}^2}, \nabla \phi_{\epsilon,h}\big)_{{\Omega_{\epsilon,h}^2}} + \epsilon \Big(D_{{\Gamma_{\epsilon}}}^2 {\nabla_{{\Gamma_{\epsilon}}}} \delta {u_{\epsilon}^2} , {\nabla_{{\Gamma_{\epsilon}}}} \phi_{\epsilon,h} \Big)_{{\Gamma_{\epsilon,h}}}\\&= \Big(f_{\epsilon}^2\big(\big({u_{\epsilon}^2}\big)^l\big) - f_{\epsilon}^2\big({u_{\epsilon}^2}\big),\phi_{\epsilon,h}\Big)_{{\Omega_{\epsilon,h}^2}} + \epsilon \Big(h_{\epsilon}^2\big(\big({u_{\epsilon}^1}\big)^l,\big({u_{\epsilon}^2}\big)^l\big) - h_{\epsilon}^2\left({u_{\epsilon}^1},{u_{\epsilon}^2}\right) , \phi_{\epsilon,h}\Big)_{{\Gamma_{\epsilon,h}}}.\end{align*}

Choosing $\phi_{\epsilon,h}:=\delta {u_{\epsilon}^2}$ $\big($ more precisely we take the restriction of ${u_{\epsilon}^2}$ to ${\Omega_{\epsilon,h}^2}\big)$ we obtain with the coercivity of $D^2$ and $D^2_{\Gamma}$ , as well as the Lipschitz continuity of $f^2$ and $h^2$

\begin{align*}\frac12 \frac{d}{dt} \big\|\delta {u_{\epsilon}^2} \big\|^2_{{\mathbb{L}_{2,\epsilon,h}}} + c_0 \big\|\nabla \delta {u_{\epsilon}^2}\big\|_{{\mathbb{L}_{2,\epsilon,h}}}^2 &\leqslant C \left( \big\|\delta {u_{\epsilon}^2} \big\|^2_{L^2\big({\Omega_{\epsilon,h}^2}\big)} + \epsilon \sum_{j=1}^2 \big\|\delta {u_{\epsilon}^j} \big\|^2_{L^2\big({\Gamma_{\epsilon,h}}\big)}\right)\\&\leqslant \sum_{j=1}^2 \left( C(\theta)\big\Vert \delta {u_{\epsilon}^j} \big\Vert_{L^2\big({\Omega_{\epsilon,h}^j}\big)}^2 + \epsilon \theta \big\Vert \nabla {u_{\epsilon}^j} \big\Vert^2_{L^2\big({\Omega_{\epsilon,h}^j}\big)} \right),\end{align*}

for arbitrary $\theta >0$ , where we used the trace inequality (3.3). Choosing $\theta$ small enough the gradient term for $j=2$ can be absorbed from the left-hand side. Integrating with respect to time, using the a priori estimates from Proposition 2 for the gradients of ${u_{\epsilon}^1}$ , as well as the Gronwall inequality, we obtain the desired result.

4.1 The connected domain ${\Omega_{\epsilon}^1}$

Here we give a strong compactness result for a sequence in the connected domain ${\Omega_{\epsilon}^1}$ under suitable a priori estimates. The case of a connected perforated domain can be treated more easily than a disconnected domain, because we can extend a bounded sequence in $H^1(\Omega_{\epsilon})$ to a bounded sequence in $H^1(\Omega)$ , due to [Reference Acerbi, Chiadò, Maso and Percivale1, Reference Cioranescu and Paulin13]. Hence, we can work in fixed Bochner spaces (not depending on $\epsilon$ ) and use standard methods from functional analysis. For this we need control for the time variable, which can be obtained from the uniform bound of the time derivative $\partial_t {u_{\epsilon}^1}$ . However, since $\partial_t {u_{\epsilon}^1}$ is pointwise only an element in the space ${\mathbb{H}^{\prime}_{1,\epsilon}}$ it is not clear if the time derivative of the extension of ${u_{\epsilon}^1}$ exists and if it is bounded uniformly with respect to $\epsilon$ . The following lemma gives us an estimate for the difference of the shifts with respect to time for functions with generalised time derivative. It is just an easy generalisation of [Reference Gahn, Neuss-Radu and Knabner19, Lemma 9].

Lemma 2 Let V and H be Hilbert spaces and we assume that $(V,H,V^{\prime})$ is a Gelfand triple. Let $v \in L^2((0,T),V) \cap H^1((0,T),V^{\prime})$ . Then, for every $\phi \in V$ and almost every $t \in (0,T)$ , $s\in(-T,T)$ , such that $t + s \in (0,T)$ , we have

\begin{align*}\big|(v(t+s) - v(t), \phi)_H\big| \leqslant \sqrt{|s|} \|\phi\|_V \|\partial_t v \|_{L^2((t,t+s),V^{\prime})}.\end{align*}

Especially, it holds that

\begin{align*}\big\|v(t+s) - v(t) \big\|^2_H \leqslant \sqrt{|s|} \big\|v(t+s) - v(t)\big\|_V \|\partial_t v\|_{L^2((t,t+s),V^{\prime})}.\end{align*}

In the following, for ${v_{\epsilon}} \in H^1\big({\Omega_{\epsilon}^1}\big)$ we denote by ${\tilde{v}_{\epsilon}} \in H^1(\Omega)$ the extension from [Reference Acerbi, Chiadò, Maso and Percivale1, Reference Cioranescu and Paulin13] with

\begin{align*}\Vert {\tilde{v}_{\epsilon}} \Vert_{L^2(\Omega)} \leqslant C \Vert {v_{\epsilon}}\Vert_{L^2\big({\Omega_{\epsilon}^1}\big)}, \quad \Vert \nabla {\tilde{v}_{\epsilon}} \Vert_{L^2(\Omega)} \leqslant C \Vert \nabla {v_{\epsilon}} \Vert_{L^2\big({\Omega_{\epsilon}^1}\big)},\end{align*}

with a constant $C>0$ independent of $\epsilon$ .

Proposition 4 Let $({v_{\epsilon}})\subset L^2\big((0,T),{\mathbb{H}_{1,\epsilon}}\big)\cap H^1\big((0,T),{\mathbb{H}^{\prime}_{1,\epsilon}}\big)$ be a sequence with

(4.1) \begin{align}\Vert \partial_t {v_{\epsilon}} \Vert_{L^2\big((0,T),{\mathbb{H}^{\prime}_{1,\epsilon}}\big)} + \Vert {v_{\epsilon}} \Vert_{L^2\big((0,T),{\mathbb{H}_{1,\epsilon}}\big)} \leqslant C.\end{align}

There exists $v_0 \in L^2\big((0,T),H^1(\Omega)\big)$ such that for all $\beta \in \left(\frac12, 1\right)$ up to a subsequence it holds that

\begin{align*}{\tilde{v}_{\epsilon}} \rightarrow v_0 \quad \mbox{ in } L^2\big((0,T),H^{\beta}(\Omega)\big).\end{align*}

Further, it holds that (up to a subsequence)

\begin{align*}\mathcal{T}_{\epsilon} {v_{\epsilon}} \rightarrow v_0 \quad \mbox{ in } L^2\left((0,T)\times \Omega , \mathbb{H}_1\right).\end{align*}

Proof Since ${\tilde{v}_{\epsilon}}$ is bounded in $L^2\big((0,T),H^1(\Omega)\big)$ there exists $v_0 \in L^2\big((0,T),H^1(\Omega)\big)$ , such that up to a subsequence ${v_{\epsilon}} $ converges weakly to $v_0$ in $L^2\big((0,T),H^1(\Omega)\big)$ . Lemma 2 and inequality (4.1) imply for $0<h\ll 0$

(4.2) \begin{align}\begin{aligned}\int_0^{T-h}\|{v_{\epsilon}} (t + h ) - {v_{\epsilon}}\|^2_{{\mathbb{L}_{1,\epsilon}}} dt&\leqslant C \sqrt{h} \|\partial_t {v_{\epsilon}}\|_{L^2\big((0,T),{\mathbb{H}^{\prime}_{1,\epsilon}}\big)} \int_0^{T-h} \|{v_{\epsilon}} (t+h) - {v_{\epsilon}}\|_{{\mathbb{H}_{1,\epsilon}}} dt\\&\leqslant C\sqrt{h}.\end{aligned}\end{align}

Now, from the properties of the extension ${\tilde{v}_{\epsilon}}$ we obtain

\begin{align*}\int_0^{T-h} \Vert {\tilde{v}_{\epsilon}}(t+h) - {\tilde{v}_{\epsilon}}\Vert^2_{L^2(\Omega)} dt \leqslant C \int_0^{T-h}\Vert {v_{\epsilon}}(t+h) - {v_{\epsilon}}\Vert^2_{L^2\big({\Omega_{\epsilon}^1}\big)} dt \leqslant C\sqrt{h}.\end{align*}

Since $H^1(\Omega) \hookrightarrow H^{\beta}(\Omega)$ is compact for $\beta \in \left(\frac12, 1\right)$ we can apply [Reference Simon30, Theorem 1] to $({\tilde{v}_{\epsilon}})$ as a sequence in $L^2\big((0,T),H^{\beta}(\Omega)\big)$ and obtain the strong convergence of ${\tilde{v}_{\epsilon}}$ to $v_0$ in $L^2\big((0,T),H^{\beta}(\Omega)\big)$ .

Now we prove the convergence of $\mathcal{T}_{\epsilon} {v_{\epsilon}}$ . It holds that

\begin{align*}\Vert \mathcal{T}_{\epsilon} {v_{\epsilon}} - v_0 \Vert_{L^2\left((0,T)\times \Omega , \mathbb{H}_1\right) }^2 = \Vert \mathcal{T}_{\epsilon} {v_{\epsilon}} - v_0 \Vert_{L^2\left((0,T)\times \Omega,H^1(Y_1)\right)}^2 + \Vert \mathcal{T}_{\epsilon} {v_{\epsilon}} - v_0 \Vert_{L^2\left((0,T)\times \Omega , H^1(\Gamma)\right)}^2.\end{align*}

We only prove the convergence to zero for the second term, since the first one can be treated in a similar way. We obtain from the properties of the unfolding operator from Lemma 5, the trace inequality and the inequality (4.1)

\begin{align*}\Vert &\mathcal{T}_{\epsilon} {v_{\epsilon}} - v_0 \Vert_{L^2\left((0,T)\times \Omega,H^1(\Gamma)\right)} \leqslant C\Vert \mathcal{T}_{\epsilon} {v_{\epsilon}} - v_0 \Vert_{L^2((0,T)\times \Omega \times \Gamma)} + C\Vert \nabla_{\Gamma,y} \mathcal{T}_{\epsilon} {v_{\epsilon}} \Vert_{L^2((0,T)\times \Omega \times \Gamma)}\\&\leqslant C\left( \Vert \mathcal{T}_{\epsilon} {v_{\epsilon}} - v_0 \Vert_{L^2\left((0,T)\times \Omega \times Y_1\right)} + \epsilon \Vert \nabla {v_{\epsilon}} \Vert_{L^2\left((0,T)\times {\Omega_{\epsilon}^1}\right)} + \epsilon^{\frac32} \Vert {\nabla_{{\Gamma_{\epsilon}}}} {v_{\epsilon}} \Vert_{L^2\left((0,T)\times {\Gamma_{\epsilon}}\right)} \right)\\&\leqslant C\left( \Vert {v_{\epsilon}} - v_0 \Vert_{L^2\left((0,T)\times {\Omega_{\epsilon}^1}\right)} + \Vert \mathcal{T}_{\epsilon} v_0 - v_0 \Vert_{L^2\left((0,T)\times \Omega \times Y_1\right)} + \epsilon \right).\end{align*}

The first term converges to zero for $\epsilon \to 0$ , due to the strong convergence of ${\tilde{v}_{\epsilon}} $ to $v_0$ , and the second term because of [Reference Cioranescu, Griso and Damlamian12, Proposition 4.4]. This gives the desired result.

4.2 The disconnected domain ${\Omega_{\epsilon}^2}$

In this section, we give a strong two-scale compactness result for the disconnected domain ${\Omega_{\epsilon}^2}$ of Kolmogorov–Simon-type, that is, it is based on a priori estimates for the difference of discrete shifts, see condition (ii) in Theorem 4.1. As already mentioned above, it is in general not possible to find an extension for a function in $H^1\left({\Omega_{\epsilon}^2}\right)$ to the whole domain $\Omega$ which preserves the a priori estimates. Hence, the method from Section 4.1 for the connected domain fails. Therefore, we consider the unfolded sequence in the Bochner space $L^p\Big(\Omega,L^2\big((0,T),\mathbb{H}_2^{\beta}\big)\Big)$ with $\beta \in \left(\frac12,1\right)$ and $p \in (1,2)$ and apply the Kolmogorov–Simon-compactness result from [Reference Gahn and Neuss-Radu18], which gives an extension of [Reference Simon30, Theorem 1] to higher-dimensional domains of definition. Here, a crucial point is the estimate for the shifts. An important reason to work here with general Bochner spaces, that is, Banach-valued functions spaces, is that we are dealing with manifolds and therefore linear shifts with respect to the space variable are not well defined.

In the following lemma, we estimate the shifts of the unfolded sequence with respect to the macroscopic variable by the shifts of the function itself, see again Section 3.3 for the notations.

Lemma 3 Let ${v_{\epsilon}} \in L^2\left((0,T)\times {\Omega_{\epsilon}^j}\right)$ for $j \in \{1,2\}$ and $w_{\epsilon} \in L^2\left((0,T)\times {\Gamma_{\epsilon}}\right)$ . Then, for $0 < h \ll 1$ , $|z|< h$ , and $\epsilon $ small enough it holds that

\begin{align*}\big\|\mathcal{T}_{\epsilon} {v_{\epsilon}} (t,x + z , y) - \mathcal{T}_{\epsilon} {v_{\epsilon}} \big\|_{L^2(0,T) \times \Omega_{2h}\times Y_j\big)}^2 &\leqslant \sum_{k \in \{0,1\}^n} \|\delta {v_{\epsilon}} \|_{L^2\left((0,T)\times {\Omega_{\epsilon,h}^j}\right)}^2,\\\big\|\mathcal{T}_{\epsilon} w_{\epsilon} (t,x + z , y) - \mathcal{T}_{\epsilon} w_{\epsilon} \big\|_{L^2(0,T) \times \Omega_{2h}\times \Gamma\big)}^2 &\leqslant \epsilon \sum_{k \in \{0,1\}^n} \|\delta w_{\epsilon} \|_{L^2\left((0,T)\times {\Gamma_{\epsilon,h}}\right)}^2,\end{align*}

with $l = l(\epsilon,z,k) = k + \left[\frac{z}{\epsilon}\right]$ .

Theorem 4.1 Let ${v_{\epsilon}} \in L^2\left((0,T),{\mathbb{H}_{2,\epsilon}}\right)\cap H^1\left((0,T),{\mathbb{H}^{\prime}_{2,\epsilon}}\right)$ with:

1. (i) It holds the estimate

   \begin{align*}\|{v_{\epsilon}} \|_{L^2\left((0,T),{\mathbb{H}_{2,\epsilon}}\right)} + \|\partial_t {v_{\epsilon}} \|_{L^2\left((0,T),{\mathbb{H}^{\prime}_{2,\epsilon}}\right)} \leqslant C.\end{align*}

2. (ii) For $0 < h \ll 1 $ and $l\in \mathbb{Z}^n$ with $|l\epsilon|< h$ it holds that

   \begin{align*}\|\delta {v_{\epsilon}} \|_{L^2\big((0,T),L^2\big({\Omega_{\epsilon,h}^2}\big)\big)}\overset{\epsilon l \to 0}{\longrightarrow} 0.\end{align*}

Then, there exists $v_0 \in L^2\big((0,T),L^2(\Omega)\big)$ , such that for $\beta \in \left(\frac12,1\right)$ and $p\in [1,2)$ it holds up to a subsequence that

\begin{align*}\mathcal{T}_{\epsilon} {v_{\epsilon}} \rightarrow v_0 \quad \mbox{ in } L^p\Big(\Omega,L^2\big((0,T),\mathbb{H}_2^{\beta}\big)\Big).\end{align*}

Especially, ${v_{\epsilon}}$ and ${v_{\epsilon}}|_{{\Gamma_{\epsilon}}}$ converge strongly in the two-scale sense to $v_0$ (with respect to $L^p$ ).

Proof Our aim is to apply [Reference Gahn and Neuss-Radu18, Corollary 2.5] to $(\mathcal{T}_{\epsilon} {v_{\epsilon}} )$ as a sequence in $ L^p\Big(\Omega, L^2\big((0,T),\mathbb{H}_2^{\beta}\big)\Big)$ for $p \in [1,2)$ and $\beta \in \left(\frac12,1\right)$ . Hence, we have to check the following three conditions:

1. (K1) For every measurable set $A\subset \Omega,$ the sequence $\left\{ \int_A {v_{\epsilon}} dx \right\}$ is relatively compact in $L^2\big((0,T),\mathbb{H}_2^{\beta}\big)$ ,

2. (K2) for all $0<h \ll 1$ and $|z|<h$ it holds that

   \begin{align*} \sup_{\epsilon} \| {v_{\epsilon}}(\cdot + z) - {v_{\epsilon}} \|_{L^p\big(\Omega_h,L^2\big((0,T),\mathbb{H}_2^{\beta}\big)\big)} \rightarrow 0 \quad \mbox{ for } z \to 0,\end{align*}

3. (K3) for $h>0$ it holds that $\, \sup_{ \epsilon} \int_{\Omega \setminus \Omega_h} |{v_{\epsilon}}(x)|^p dx \rightarrow 0 \, \mbox{ for } h \to 0$ .

We start with the condition (K1). Let $A \subset \Omega$ be measurable and we define $V_{\epsilon}(t,y):= \int_A \mathcal{T}_{\epsilon} {v_{\epsilon}} (t,x,y)dx$ . To show the relative compactness of $(V_{\epsilon})$ , we use again [Reference Simon30, Theorem 1] as in the proof of Proposition 4. First of all, due to our assumptions on ${v_{\epsilon}}$ and the properties of the unfolding operator, for $t_1,t_2 \in (0,T)$ it holds that

\begin{align*}\left\| \int_{t_1}^{t_2} V_{\epsilon} dt \right\|_{\mathbb{H}_2} \leqslant \|\mathcal{T}_{\epsilon} {v_{\epsilon}} \|_{L^2((0,T)\times \Omega, \mathbb{H}_2)} \leqslant C.\end{align*}

Due to the compact embedding $\mathbb{H}_2 \hookrightarrow \mathbb{H}_2^{\beta}$ we obtain that $\int_{t_1}^{t_2} V_{\epsilon} dt$ is relatively compact in $\mathbb{H}_2^{\beta}$ . Further, for $0<s \ll 1$ we obtain with the estimates for ${v_{\epsilon}}$ and the trace inequality (3.3)

(4.3) \begin{align}\begin{aligned}\big\| &V_{\epsilon}(t + s,y) - V_{\epsilon}\big\|_{L^2((0,T-s),\mathbb{H}_2)} \leqslant \big\|\mathcal{T}_{\epsilon} {v_{\epsilon}} (t + s,x,y) - \mathcal{T}_{\epsilon} {v_{\epsilon}} \big\|_{L^2\left((0,T-s)\times \Omega, \mathbb{H}_2\right)}\\&\leqslant C \|{v_{\epsilon}} (t+s,x) - {v_{\epsilon}}\|_{L^2\left((0,T-s) \times {\Omega_{\epsilon}^2}\right)} + C\epsilon \|\nabla {v_{\epsilon}} (t+s,x) - \nabla {v_{\epsilon}} \|_{L^2\left((0,T-s) \times {\Omega_{\epsilon}^2}\right)}\\& \quad + C \epsilon^{\frac32}\Vert {\nabla_{{\Gamma_{\epsilon}}}} {v_{\epsilon}}(t+s,x) - {\nabla_{{\Gamma_{\epsilon}}}} {v_{\epsilon}} \Vert_{L^2\left((0,T-s)\times {\Gamma_{\epsilon}}\right)}\\&\leqslant C \left(s^{\frac14} + \epsilon\right),\end{aligned}\end{align}

where for the last inequality we used Lemma 2 to estimate the first term in the line before by using the same arguments as for the inequality (4.2) in the proof of Proposition 4. We show that inequality (4.3) implies the convergence of the difference of the shifts to zero for $s \to 0$ uniformly with respect to $\epsilon$ , see also [Reference Neuss-Radu and Jäger27, pp. 710–711] or [Reference Friesecke, James and Müller15, pp. 1476–1477] for the same argument. First of all, from (4.3) we obtain for every $0 < \rho $ the existence of $0 < \epsilon_0, \, \delta_0$ , such that for all $\epsilon \leqslant \epsilon_0 $ and $s\leqslant \delta_0$ it holds that

(4.4) \begin{align}\big\| V_{\epsilon}(t + s,y) - V_{\epsilon}\big\|_{L^2((0,T-s),\mathbb{H}_2)} \leqslant \rho.\end{align}

Since $\epsilon^{-1} \in \mathbb{N}$ , there are only finitely many elements $\epsilon$ , denoted by $\epsilon_i$ with $i = 1,\ldots, N$ , such that $\epsilon_0 < \epsilon_i$ . For every $\epsilon_i$ there exists a $0 < \delta_i$ , such that (4.4) is valid for $\epsilon = \epsilon_i$ and all $s \leqslant \delta_i$ . In fact, this follows directly from $V_{\epsilon_i} \in L^2((0,T),\mathbb{H}_2)$ , see for example [Reference Alt5, Theorem 4.15] or use the Kolmogorov–Simon-compactness theorem from [Reference Simon30, Theorem 1] applied to the function $V_{\epsilon_i}$ . Choosing $\delta:= \max_{i=0,\ldots,N} \{\delta_i\}$ , we obtain for all $s \leqslant \delta$

\begin{align*}\sup_{\epsilon} \big\| V_{\epsilon}(t + s,y) - V_{\epsilon}\big\|_{L^2((0,T-s),\mathbb{H}_2)} \leqslant \rho.\end{align*}

Hence, [Reference Simon30, Theorem 1] implies that $(V_{\epsilon})$ is relatively compact in $L^2\big((0,T),\mathbb{H}_2^{\beta}\big)$ , that is, condition (K1).

For (K2) we fix $0< h \ll 1$ and choose $|z|< h$ . Lemma 3 with $l= k + \left[\frac{z}{\epsilon}\right]$ (see the definition of the difference $\delta$ in (3.4)), the conditions (i) and (ii), as well as the trace inequality (3.3) imply

\begin{align*}\big\| &\mathcal{T}_{\epsilon} {v_{\epsilon}} (t, x+ z,y) - \mathcal{T}_{\epsilon} {v_{\epsilon}} \big\|_{L^2\left(\Omega_{2h} , L^2((0,T),\mathbb{H}_2)\right)}\\&\leqslant C \sum_{k \in \{0,1\}^n} \left( \|\delta {v_{\epsilon}} \|_{L^2\left((0,T)\times {\Omega_{\epsilon,h}^2}\right)} + \epsilon \|\delta \nabla {v_{\epsilon}} \|_{L^2\left((0,T)\times {\Omega_{\epsilon,h}^2}\right)} + \epsilon^{\frac32} \Vert \delta {\nabla_{{\Gamma_{\epsilon}}}} {v_{\epsilon}} \Vert_{L^2\left((0,T)\times {\Gamma_{\epsilon,h}}\right)} \right)\\&\leqslant C \left( \sum_{k \in \{0,1\}^n} \|\delta {v_{\epsilon}} \|_{L^2\left((0,T)\times {\Omega_{\epsilon,h}^2}\right)} + \epsilon \right) \overset{\epsilon,z \to 0}{\longrightarrow} 0.\end{align*}

Again we obtain for every $0 < \rho $ the existence of $0 < \epsilon_0, \, \delta_0$ , such that for all $\epsilon \leqslant \epsilon_0 $ and $\vert z\vert \leqslant \delta_0$ it holds that

(4.5) \begin{align}\big\| \mathcal{T}_{\epsilon} {v_{\epsilon}} (t, x+ z,y) - \mathcal{T}_{\epsilon} {v_{\epsilon}} \big\|_{L^2\left(\Omega_{2h} , L^2((0,T),\mathbb{H}_2)\right)} \leqslant \rho.\end{align}

Arguing as above, this inequality is also valid for all (finitely many) $\epsilon > \epsilon_0$ and $\vert z \vert $ small enough. This implies (K2). For the last condition (K3) we use the Hölder inequality to obtain for $p \in [1,2)$ and $0< h \ll 1$

\begin{align*}\big\|\mathcal{T}_{\epsilon} {v_{\epsilon}} \big\|_{L^p\left(\Omega \setminus \Omega_h, L^2((0,T),\mathbb{H}_2)\right)} \leqslant |\Omega \setminus \Omega_h|^{\frac{2-p}{2p}}\big\|\mathcal{T}_{\epsilon} {v_{\epsilon}} \big\|_{L^2\left((0,T)\times \Omega \setminus \Omega_h , \mathbb{H}_2\right)} \leqslant Ch^{\frac{2-p}{2p}} \overset{h\to 0}{\longrightarrow}0,\end{align*}

where we used again estimate (i). Now, [Reference Gahn and Neuss-Radu18, Corollary 2.5] implies the the strong convergence of $\mathcal{T}_{\epsilon} {v_{\epsilon}} $ up to a subsequence in $L^p\left(\Omega,L^2\Big((0,T),\mathbb{H}_2^{\beta}\Big)\right)$ to a limit function $v_0$ . Lemma 6 (see Appendix A) implies the strong two-scale convergence of ${v_{\epsilon}}$ to the same limit. The fact $v_0 \in L^2\big((0,T),L^2(\Omega)\big)$ follows from standard two-scale compactness results, see [Reference Allaire2], based on the estimate (i).