Homogenization of linear parabolic equations with three spatial and three temporal scales for certain matchings between the microscopic scales

TL;DR

This paper develops compactness results for multiscale sequences and applies them to homogenize a parabolic PDE with three spatial and three temporal scales, revealing unique elliptic and shifted parabolic phenomena.

Contribution

It introduces new compactness results for multiscale sequences and applies them to homogenize complex parabolic equations with multiple scales, uncovering novel phenomena.

Findings

Homogenized problem is elliptic.

Matching for local parabolic problem is shifted by p.

New phenomena in multiscale homogenization.

Abstract

In this paper we establish compactness results of multiscale and very weak multiscale type for sequences bounded in $L^{2}(0,T;H_{0}^{1}(\Omega ))$, fulfilling a certain condition. We apply the results in the homogenization of the parabolic partial differential equation $\varepsilon ^{p}\partial_{t}u_{\varepsilon }\left( x,t\right) -\nabla \cdot \left( a\left( x/\varepsilon ,x/\varepsilon ^{2},t/\varepsilon^{q},t/\varepsilon ^{r}\right) \nabla u_{\varepsilon }\left( x,t\right)\right) = f\left( x,t\right) $, where $0<p<q<r$. The homogenization result reveals two special phenomena, namely that the homogenized problem is elliptic and that the matching for when the local problem is parabolic is shifted by $p$, compared to the standard matching that gives rise to local parabolic problems.

Equations

$${\epsilon }^p{\partial }_t{u}_{\epsilon }\left(x,t\right)-\nabla \cdot \left(a\left(\frac{x}{\epsilon },\frac{x}{{\epsilon }^2},\frac{t}{{\epsilon }^q},\frac{t}{{\epsilon }^r}\right)\nabla u_{\epsilon }\left(x,t\right)\right)$$

$$u_{\epsilon }\left(x,0\right)$$

$$u_{\epsilon }\left(x,t\right)$$

$$\underset{\epsilon \to 0}{\lim }{\int }_{{\Omega }_T}{u}_{\epsilon }\left(x,t\right)v\left(x,t,\frac{x}{{\epsilon }_1},\cdots ,\frac{x}{{\epsilon }_n},\frac{t}{{\epsilon }_1^{\prime }},\cdots ,\frac{t}{{\epsilon }_m^{\prime }}\right)dxdt$$

$$={\int }_{{\Omega }_T}{\int }_{{\mathcal{Y}}_{n,m}}{u}_0\left(x,t,y^n,s^m\right)v\left(x,t,y^n,s^m\right)d{y}^{n}d{s}^{m}dxdt$$

$$u_{\epsilon }\left(x,t\right)\stackrel{n+1,m+1}{\rightharpoonup }{u}_0\left(x,t,y^n,s^m\right)\text{.}$$

$$\underset{\epsilon \to 0}{\lim }\frac{{\epsilon }_{k+1}}{{\epsilon }_k}=0$$

$$\underset{\epsilon \to 0}{\lim }\frac{1}{{\epsilon }_k}{\left(\frac{{\epsilon }_{k+1}}{{\epsilon }_k}\right)}^{\ell }=0\text{,}$$

$$u_{\epsilon }\left(x,t\right)\stackrel{n+1,m+1}{\rightharpoonup }{u}_0\left(x,t,y^n,s^m\right)\text{,}$$

$$\underset{\epsilon \to 0}{\lim }{\int }_{{\Omega }_T}{u}_{\epsilon }\left(x,t\right)v\left(x\right){\partial }_t\left({\epsilon }^r{c}_1\left(t\right)c_2\left(\frac{t}{{\epsilon }^q}\right)c_3\left(\frac{t}{{\epsilon }^r}\right)\right)dxdt=0$$

$$\underset{\epsilon \to 0}{\lim }{\int }_{{\Omega }_T}{u}_{\epsilon }\left(x,t\right)v\left(x\right){\partial }_t\left({\epsilon }^q{c}_1\left(t\right)c_2\left(\frac{t}{{\epsilon }^q}\right)\right)dxdt=0\text{.}$$

$$u_{\epsilon }\left(x,t\right)\rightharpoonup u\left(x,t\right)\text{ in }{L}^2(0,T;H_0^1(\Omega ))\text{,}$$

$$u_{\epsilon }\left(x,t\right)\stackrel{3,3}{\rightharpoonup }u\left(x,t\right)\text{,}$$

$$\nabla u_{\epsilon }\left(x,t\right)\stackrel{2,3}{\rightharpoonup }\nabla u\left(x,t\right)+{\nabla }_{y_1}{u}_1\left(x,t,y_1,s^2\right)$$

$$\nabla u_{\epsilon }\left(x,t\right)\stackrel{3,3}{\rightharpoonup }\nabla u\left(x,t\right)+{\nabla }_{y_1}{u}_1\left(x,t,y_1,s^2\right)+{\nabla }_{y_2}{u}_2\left(x,t,y^2,s^2\right)\text{,}$$

$$u_{\epsilon }\left(x,t\right)\stackrel{3,3}{\rightharpoonup }{u}_0\left(x,t,y^2,s^2\right)$$

$$\nabla u_{\epsilon }\left(x,t\right)\stackrel{3,3}{\rightharpoonup }{\tau }_0\left(x,t,y^2,s^2\right)\text{,}$$

$${\int }_{{\Omega }_T}\nabla u_{\epsilon }\left(x,t\right){\epsilon }^2{v}_1\left(x\right)v_2\left(\frac{x}{\epsilon }\right)\cdot v_3\left(\frac{x}{{\epsilon }^2}\right)c_1\left(t\right)c_2\left(\frac{t}{{\epsilon }^q}\right)c_3\left(\frac{t}{{\epsilon }^r}\right)dxdt$$

$$=-{\int }_{{\Omega }_T}{u}_{\epsilon }\left(x,t\right)({\epsilon }^2\nabla v_1\left(x\right)v_2\left(\frac{x}{\epsilon }\right)\cdot v_3\left(\frac{x}{{\epsilon }^2}\right)+\epsilon v_1\left(x\right){\nabla }_{y_1}{v}_2\left(\frac{x}{\epsilon }\right)\cdot v_3\left(\frac{x}{{\epsilon }^2}\right)$$

$$+v_1\left(x\right)v_2\left(\frac{x}{\epsilon }\right){\nabla }_{y_2}\cdot v_3\left(\frac{x}{{\epsilon }^2}\right))c_1\left(t\right)c_2\left(\frac{t}{{\epsilon }^q}\right)c_3\left(\frac{t}{{\epsilon }^r}\right)dxdt\text{,}$$

$$\underset{\epsilon \to 0}{\lim }{\int }_{{\Omega }_T}-u_{\epsilon }\left(x,t\right)({\epsilon }^2\nabla v_1\left(x\right)v_2\left(\frac{x}{\epsilon }\right)\cdot v_3\left(\frac{x}{{\epsilon }^2}\right)+\epsilon v_1\left(x\right){\nabla }_{y_1}{v}_2\left(\frac{x}{\epsilon }\right)\cdot v_3\left(\frac{x}{{\epsilon }^2}\right)$$

$$+v_1\left(x\right)v_2\left(\frac{x}{\epsilon }\right){\nabla }_{y_2}\cdot v_3\left(\frac{x}{{\epsilon }^2}\right))c_1\left(t\right)c_2\left(\frac{t}{{\epsilon }^q}\right)c_3\left(\frac{t}{{\epsilon }^r}\right)dxdt=0$$

$${\int }_{{\Omega }_T}{\int }_{{\mathcal{Y}}_{2,2}}-u_0\left(x,t,y^2,s^2\right)v_1\left(x\right)v_2\left(y_1\right){\nabla }_{y_2}\cdot v_3\left(y_2\right)$$

$$\times c_1\left(t\right)c_2\left(s_1\right)c_3\left(s_2\right)d{y}^{2}d{s}^{2}dxdt=0\text{.}$$

$$-{\int }_{Y_2}{u}_0\left(x,t,y^2,s^2\right){\nabla }_{y_2}\cdot v_3\left(y_2\right)\text{d}{y}_2=0$$

$${\int }_{{\Omega }_T}\nabla u_{\epsilon }\left(x,t\right)\epsilon v_1\left(x\right)\cdot v_2\left(\frac{x}{\epsilon }\right)c_1\left(t\right)c_2\left(\frac{t}{{\epsilon }^q}\right)c_3\left(\frac{t}{{\epsilon }^r}\right)\text{d}x\text{d}t$$

$$=-{\int }_{{\Omega }_T}{u}_{\epsilon }\left(x,t\right)\left(\epsilon \nabla v_1\left(x\right)\cdot v_2\left(\frac{x}{\epsilon }\right)+v_1\left(x\right){\nabla }_{y_1}\cdot v_2\left(\frac{x}{\epsilon }\right)\right)$$

$$\times c_1\left(t\right)c_2\left(\frac{t}{{\epsilon }^q}\right)c_3\left(\frac{t}{{\epsilon }^r}\right)\text{d}x\text{d}t$$

$${\int }_{{\Omega }_T}{\int }_{{\mathcal{Y}}_{1,2}}-u_0\left(x,t,y_1,s^2\right)v_1\left(x\right){\nabla }_{y_1}\cdot v_2\left(y_1\right)c_1\left(t\right)c_2\left(s_1\right)c_3\left(s_2\right)\text{d}{y}_1\text{d}{s}^2\text{d}x\text{d}t=0\text{.}$$

$$-{\int }_{Y_1}{u}_0\left(x,t,y_1,s^2\right){\nabla }_{y_1}\cdot v_2\left(y_1\right)\text{d}{y}_1=0$$

$$\underset{\epsilon \to 0}{\lim }{\int }_{{\Omega }_T}{u}_{\epsilon }\left(x,t\right)v\left(x\right)({\epsilon }^r{\partial }_t{c}_1\left(t\right)c_2\left(\frac{t}{{\epsilon }^q}\right)c_3\left(\frac{t}{{\epsilon }^r}\right)$$

$$+{\epsilon }^{r-q}{c}_1\left(t\right){\partial }_{s_1}{c}_2\left(\frac{t}{{\epsilon }^q}\right)c_3\left(\frac{t}{{\epsilon }^r}\right)+{\epsilon }^{r-r}{c}_1\left(t\right)c_2\left(\frac{t}{{\epsilon }^q}\right){\partial }_{s_2}{c}_3\left(\frac{t}{{\epsilon }^r}\right))\text{d}x\text{d}t=0\text{.}$$

$${\int }_{{\Omega }_T}{\int }_{S^2}{u}_0\left(x,t,s^2\right)v\left(x\right)c_1\left(t\right)c_2\left(s_1\right){\partial }_{s_2}{c}_3\left(s_2\right)\text{d}{s}^2\text{d}x\text{d}t=0$$

$${\int }_{S_2}{u}_0\left(x,t,s^2\right){\partial }_{s_2}{c}_3\left(s_2\right)\text{d}{s}_2=0$$

$$\underset{\epsilon \to 0}{\lim }{\int }_{{\Omega }_T}{u}_{\epsilon }\left(x,t\right)v\left(x\right)\left({\epsilon }^q{\partial }_t{c}_1\left(t\right)c_2\left(\frac{t}{{\epsilon }^q}\right)+{\epsilon }^{q-q}{c}_1\left(t\right){\partial }_{s_1}{c}_2\left(\frac{t}{{\epsilon }^q}\right)\right)\text{d}x\text{d}t=0\text{.}$$

Full text

1 Introduction

Let $T>0$ and $\Omega_{T}=\Omega\times\left(0,T\right)$, where $\Omega$ is an open bounded subset of $\mathbb{R}^{N}$ with smooth boundary and $(0,T)$ is an open bounded interval in $\mathbb{R}$. We consider the homogenization of the linear parabolic equation

[TABLE]

where $0<p<q<r$ are real numbers, $f\in L^{2}(\Omega_{T})$ and $u_{0}\in L^{2}(\Omega)$. The coefficient $a$ is periodic with respect to the unit cube $Y=(0,1)^{N}$ in the first two variables and with respect to the unit interval $S=(0,1)$ in the third and fourth variable. More detailed information on the equation will be provided in Section 3.

Homogenization means that we study the limit behavior as $\varepsilon\rightarrow 0$ and search for a weak $L^{2}(0,T;H_{0}^{1}(\Omega))$-limit $u$ to $\left\{u_{\varepsilon}\right\}$ which is the solution to a so-called homogenized problem. This limit problem is governed by a coefficient $b$ that unlike $a\left(x/\varepsilon,x/\varepsilon^{2},t/\varepsilon^{q},t/\varepsilon^{r}\right)$ does not include rapid oscillations. In the homogenization procedure local problems are also extracted which include information about the microstructure and whose solutions are utilized to determine $b$.

The present paper is a further generalization of the work presented in [13]. In earlier works, like e.g. [11], boundedness in $W^{1,2}(0,T;H_{0}^{1}(\Omega),L^{2}(\Omega))$, meaning that $\left\{u_{\varepsilon}\right\}$ is bounded in $L^{2}(0,T;H_{0}^{1}(\Omega))$ and $\left\{\partial_{t}u_{\varepsilon}\right\}$ is bounded in $L^{2}(0,T;H^{-1}(\Omega))$, has been required when compactness results have been established. In [13], compactness results of $\left(2,2\right)$-scale and very weak $\left(2,2\right)$-scale convergence type were proven by requiring boundedness of the sequence $\left\{u_{\varepsilon}\right\}$ in $L^{2}(0,T;H_{0}^{1}(\Omega))$ but replacing the assumption of boundedness of the time derivative in $L^{2}(0,T;H^{-1}(\Omega))$ by a certain condition. This new approach originates, up to the authors’ knowledge, from [14] and will be used in the present work. Here we focus on establishing appropriate compactness results and a homogenization result for the parabolic partial differential equation (1). In particular, we generalize the result from [13] to the $(2,3)$-scale and $\left(3,3\right)$-scale convergence types, adapting to the problem (1), and present compactness results for both multiscale and very weak multiscale convergence.

For the homogenization part of this paper we apply the convergence results to establish a homogenization result for (1) with 13 different outcomes, depending on the choices of parameters $p$, $q$ and $r$. The homogenization result will reveal two phenomena, which also occurred in both [13] and the proceeding work [6], where the homogenization of parabolic problems of a similar kind, but with only one rapid scale in space and time each, was presented. The first phenomenon is that the homogenized problem is of elliptic type even though the original problem is a parabolic one and the second is that resonance occurs for different matchings between the microscopic scales than the standard ones. By resonance we mean that the local problem is parabolic, which only occurs for certain matchings between the microscopic scales. What we call the standard matching is when a temporal scale equals the square of a spatial one, as was the case in several other studies, see e.g. [4], [12], [18], [3], [9], [10], [21], [11] or [7] for more on this matter. However, in our case the matching for when we have resonance is shifted by $p$. Note that in our equation, (1), we would get resonance for the standard matching if $p=0$, cf. Section 5.3.1 in [19].

The paper is organized as follows. In Section 2 we recall some of the key definitions, namely evolution multiscale convergence and very weak evolution multiscale convergence. We prove the main convergence results (see Theorems 6 and 9), which lay the foundation to establish the homogenization result. Theorem 6 is where we find characterizations of the $(2,3)$-scale and $\left(3,3\right)$-scale limits for $\left\{\nabla u_{\varepsilon}\right\}$ under certain assumptions. In Theorem 9 we consider very weak $\left(2,3\right)$-scale and $\left(3,3\right)$-scale convergence for the sequences $\left\{\varepsilon^{-1}u_{\varepsilon}\right\}$ and $\left\{\varepsilon^{-2}u_{\varepsilon}\right\}$, respectively. In Section 3, we state a homogenization result presented in Theorem 10.

We end the introduction with some essential notations used throughout this paper.

Notation 1

We denote $\mathcal{Y}_{n,m}=Y^{n}\times S^{m}$ with $Y^{n}=Y_{1}\times Y_{2}\times\cdots\times Y_{n}$ and $S^{m}=S_{1}\times S_{2}\times\cdots\times S_{m}$, where $Y_{1}=Y_{2}=\ldots=Y_{n}=Y=\left(0,1\right)^{N}$ and $S_{1}=S_{2}=\ldots=S_{m}=S=\left(0,1\right)$. We let $y^{n}=y_{1},y_{2},\ldots,y_{n}$, $dy^{n}=dy_{1}dy_{2}\cdots dy_{n}$, $s^{m}=s_{1},s_{2},\ldots,s_{m}$ and $ds^{m}=ds_{1}ds_{2}\cdots ds_{m}$. We define the function space $\mathcal{W}_{i,j}=\left\{u\in L_{\sharp}^{2}(S_{j};H_{\sharp}^{1}(Y_{i})/\mathbb{R}):\partial_{s_{j}}u\in L_{\sharp}^{2}(S_{j};(H_{\sharp}^{1}(Y_{i})/\mathbb{R})^{\prime})\right\}$. The subscript ♯ is used to denote periodicity of the functions involved over the domain in question. Lastly, for $k=1,\ldots,n$ and $j=1,\ldots,m$, the scale functions $\varepsilon_{k}\left(\varepsilon\right)$ and $\varepsilon_{j}^{\prime}\left(\varepsilon\right)$ are strictly positive functions that tend to zero as $\varepsilon$ does and $\left\{\varepsilon_{1},\ldots,\varepsilon_{n}\right\}$ and $\left\{\varepsilon_{1}^{\prime},\ldots,\varepsilon_{m}^{\prime}\right\}$ denote lists of spatial and temporal scales, respectively.

2 Multiscale and very weak multiscale convergence

The concept of multiscale convergence is a generalization of the classical two-scale convergence, originating from [16] and [17]. Two-scale convergence is suitable for sequences having one microscopic spatial scale and it has been generalized, first to include multiple spatial scales by Allaire and Briane in [2], and later to also include multiple temporal scales.

Definition 2

A sequence $\left\{u_{\varepsilon}\right\}$ in $L^{2}(\Omega_{T})$ is said to $\left(n+1,m+1\right)$-scale converge to $u_{0}\in L^{2}(\Omega_{T}\times\mathcal{Y}_{n,m})$ if

[TABLE]

for all $v\in L^{2}(\Omega_{T};C_{\sharp}(\mathcal{Y}_{n,m})).$ This is denoted by

[TABLE]

We make some standard assumptions on the scales. We say that the scales in a list $\left\{\varepsilon_{1},\ldots,\varepsilon_{n}\right\}$ are separated if

[TABLE]

and well-separated if there exists a positive integer $\ell$ such that

[TABLE]

where $k=1,\ldots,n-1$. Following the definition by Persson, see e.g. [20], the generalization of separatedness and well-separatedness to include two lists of scales reads as follows.

Definition 3

Let $\left\{\varepsilon_{1},\ldots,\varepsilon_{n}\right\}$ and $\left\{\varepsilon_{1}^{\prime},\ldots,\varepsilon_{m}^{\prime}\right\}$ be lists of (well-)separated scales. Collect all elements from both lists in one common list. If from possible duplicates, where by duplicates we mean scales which tend to zero equally fast, one member of each pair is removed and the list in order of magnitude of all the remaining elements is (well-)separated, the lists $\left\{\varepsilon_{1},\ldots,\varepsilon_{n}\right\}$ and $\left\{\varepsilon_{1}^{\prime},\ldots,\varepsilon_{m}^{\prime}\right\}$ are said to be jointly (well-)separated.

We present a compactness result for evolution multiscale convergence.

Theorem 4

Let $\left\{u_{\varepsilon}\right\}$ be a bounded sequence in $L^{2}(\Omega_{T})$ and suppose that the lists $\left\{\varepsilon_{1},\ldots,\varepsilon_{n}\right\}$ and $\left\{\varepsilon_{1}^{\prime},\ldots,\varepsilon_{m}^{\prime}\right\}$ are jointly separated. Then, up to a subsequence,

[TABLE]

where $u_{0}\in L^{2}(\Omega_{T}\times\mathcal{Y}_{n,m})$.

Proof. See Theorem A.1 in [11].

As the next theorem states, the evolution multiscale limit is unique.

Theorem 5

The $\left(n+1,m+1\right)$-scale limit is unique.

Proof. The proof is analogous to the proof of the uniqueness of the two-scale limit given in the discussion below Definition 1 in [15].

We are now ready to give a compactness result for the gradient of a sequence $\left\{u_{\varepsilon}\right\}$. The following theorem will play a vital role in the homogenization of (1).

Theorem 6

Let $\left\{u_{\varepsilon}\right\}$ be a bounded sequence in $L^{2}(0,T;H_{0}^{1}(\Omega))$ and, for any $v\in D(\Omega)$, $c_{1}\in D(0,T)$, $c_{2}\in C_{\sharp}^{\infty}(S_{1})$, $c_{3}\in C_{\sharp}^{\infty}(S_{2})$ and $r>q>0$,

[TABLE]

and

[TABLE]

Then, with $\varepsilon_{1}=\varepsilon$, $\varepsilon_{2}=\varepsilon^{2}$, $\varepsilon_{1}^{\prime}=\varepsilon^{q}$ and $\varepsilon_{2}^{\prime}=\varepsilon^{r}$, up to a subsequence,

[TABLE]

[TABLE]

[TABLE]

and

[TABLE]

where $u\in L^{2}(0,T;H_{0}^{1}(\Omega))$, $u_{1}\in L^{2}(\Omega_{T}\times S^{2};H_{\sharp}^{1}(Y_{1})/\mathbb{R})$ and $u_{2}\in L^{2}(\Omega_{T}\times\mathcal{Y}_{1,2};H_{\sharp}^{1}(Y_{2})/\mathbb{R})$.

Proof. From the boundedness of $\left\{u_{\varepsilon}\right\}$ in $L^{2}(0,T;H_{0}^{1}(\Omega))$, the weak convergence (4) follows immediately. It also implies that $\left\{\nabla u_{\varepsilon}\right\}$ is bounded in $L^{2}(\Omega_{T})^{N}$ and hence, according to Theorems 4 and 5, we have

[TABLE]

and

[TABLE]

up to a subsequence, for some unique $u_{0}\in L^{2}(\Omega_{T}\times\mathcal{Y}_{2,2})$ and $\tau_{0}\in L^{2}(\Omega_{T}\times\mathcal{Y}_{2,2})^{N}$.

We proceed by characterizing $u_{0}$, where we first show that $u_{0}$ is independent of the local space and time variables $y_{1}$, $y_{2}$, $s_{1}$ and $s_{2}$. Letting $v_{1}\in\emph{D}(\Omega)$, $v_{2}\in C_{\sharp}^{\infty}(Y_{1})$, $v_{3}\in C_{\sharp}^{\infty}(Y_{2})^{N}$, $c_{1}\in\emph{D}(0,T)$, $c_{2}\in C_{\sharp}^{\infty}(S_{1})$ and $c_{3}\in C_{\sharp}^{\infty}(S_{2})$, it holds that

[TABLE]

where we have applied integration by parts and carried out the differentiation process. As $\varepsilon\rightarrow 0$, $\left\{\varepsilon^{2}\nabla u_{\varepsilon}\right\}$ approaches [math] due to boundedness of $\left\{\nabla u_{\varepsilon}\right\}$ and we obtain

[TABLE]

and since all but the third term vanish, (8) gives

[TABLE]

Applying the Variational Lemma we have

[TABLE]

a.e. in $\Omega_{T}\times\mathcal{Y}_{1,2}$, showing that $u_{0}$ is independent of $y_{2}$. Next we let $v_{1}\in\emph{D}(\Omega)$, $v_{2}\in C_{\sharp}^{\infty}(Y_{1})^{N}$, $c_{1}\in\emph{D}(0,T),\ c_{2}\in C_{\sharp}^{\infty}(S_{1})$ and $c_{3}\in C_{\sharp}^{\infty}(S_{2}).$ By integration by parts and after differentiation we have that

[TABLE]

and as $\varepsilon\rightarrow 0$ we obtain

[TABLE]

By the Variational Lemma

[TABLE]

a.e. in $\Omega_{T}\times S^{2}$, which shows that $u_{0}$ is independent of $y_{1}$. To show independence of $s_{2}$ we carry out the differentiations in (2) and obtain

[TABLE]

Passing to the limit we arrive at

[TABLE]

and the Variational Lemma gives

[TABLE]

a.e. in $\Omega_{T}\times S_{1}$. We conclude that $u_{0}$ does not depend on the local time variable $s_{2}$. For showing independence of $s_{1}$ we carry out the differentiations in (3) and obtain

[TABLE]

As $\varepsilon$ tends to zero we have

[TABLE]

and by the Variational Lemma

[TABLE]

a.e. in $\Omega_{T}$, hence $u_{0}$ is independent of $s_{1}$. In conclusion, we have shown that

[TABLE]

where $u_{0}\in L^{2}(\Omega_{T})$, and the last step in the characterization of $u_{0}$ is to show that $u_{0}\in L^{2}(0,T;H_{0}^{1}(\Omega))$. Observe that (10) means

[TABLE]

for all $v\in L^{2}(\Omega_{T};C_{\sharp}(\mathcal{Y}_{2,2}))$ and since $L^{2}(\Omega_{T})\subset L^{2}(\Omega_{T};C_{\sharp}(\mathcal{Y}_{2,2}))$ it follows that

[TABLE]

for all $v\in L^{2}(\Omega_{T})$. Observing that the weak convergence (4) implies

[TABLE]

for the same $u\in L^{2}(0,T;H_{0}^{1}(\Omega))$ we see that $u_{0}$ coincides with the weak limit $u$, hence $u_{0}\in L^{2}(0,T;H_{0}^{1}(\Omega))$ and the proof of (5) is complete.

Now we will identify $\tau_{0}$. Let $H$ denote the space of generalized divergence-free functions in $L^{2}(\Omega;L_{\sharp}^{2}(Y^{2})^{N})$ defined as

[TABLE]

Using $vc$, where $v\in\emph{D}(\Omega;C_{\sharp}^{\infty}(Y^{2}))^{N}\cap H$ and $c\in\emph{D}(0,T;C_{\sharp}^{\infty}(S^{2}))$, as a test function in (9) we get, up to a subsequence,

[TABLE]

for some $\tau_{0}\in L^{2}(\Omega_{T}\times\mathcal{Y}_{2,2})^{N}$. By integration by parts in the left-hand side we obtain

[TABLE]

where the last term has vanished due to the fact that $\nabla_{y_{2}}\cdot v=0$. Since

[TABLE]

Theorem 3.3 in [2] gives that $\left\{\varepsilon^{-2}\nabla_{y_{1}}\cdot v\left(x,x/\varepsilon,x/\varepsilon^{2}\right)\right\}$ is bounded in $H^{-1}(\Omega)$. Passing to the limit while using this boundedness yields

[TABLE]

for all $v\in\emph{D}(\Omega;C_{\sharp}^{\infty}(Y^{2}))^{N}\cap H$ and $c\in\emph{D}(0,T;C_{\sharp}^{\infty}(S^{2}))$. We conclude that

[TABLE]

or equivalently

[TABLE]

By the Variational Lemma we obtain

[TABLE]

a.e. in $\left(0,T\right)\times S^{2}$. This means that $\tau_{0}-\nabla u$ belongs to the orthogonal of $\emph{D}(\Omega;C_{\sharp}^{\infty}(Y^{2}))^{N}\cap H$ and by density (see property $\left(i\right)$ of Lemma 3.7 in [2]) to the orthogonal of the whole space $H$. According to property $(ii)$ of Lemma 3.7 in [2], we deduce that

[TABLE]

where $u_{1}\in L^{2}(\Omega_{T}\times S^{2};H_{\sharp}^{1}(Y_{1})/\mathbb{R})$ and $u_{2}\in L^{2}(\Omega_{T}\times\mathcal{Y}_{1,2};H_{\sharp}^{1}(Y_{2})/\mathbb{R})$, which proves (7).

Now, choosing a test function $v\in L^{2}(\Omega_{T};C_{\sharp}(\mathcal{Y}_{1,2}))$ in the left-hand side of (6), (7) gives

[TABLE]

Integrating over $Y_{2}$ while using the fact that

[TABLE]

we arrive at

[TABLE]

which proves (6).

In the case of appearance of sequences that are not bounded in any Lebesgue space, it might not be possible to obtain a multiscale limit. In [12], Holmbom introduced a concept of convergence that was improved by Nguetseng and Woukeng in [18] and further developed and named very weak multiscale convergence in [8]. The full generalization of the concept was given in [11], for which we provide the definition. This kind of convergence is crucial in the homogenization of (1), where unbounded sequences appear.

Definition 7

A sequence $\left\{w_{\varepsilon}\right\}$ in $L^{1}(\Omega_{T})$ is said to $(n+1,m+1)$-scale converge very weakly to $w_{0}\in L^{1}(\Omega_{T}\times\mathcal{Y}_{n,m})$ if

[TABLE]

for any $v_{1}\in D(\Omega;C_{\sharp}^{\infty}(Y^{n-1})),$ $v_{2}\in C_{\sharp}^{\infty}(Y_{n})/\mathbb{R}$ and $c\in D(0,T;C_{\sharp}^{\infty}(S^{m}))$, where

[TABLE]

We write

[TABLE]

Remark 8

Due to (11) the limit is unique.

In earlier works, see e.g. [19] or [11], compactness results for very weak evolution multiscale convergence for $\left\{u_{\varepsilon}\right\}$ bounded in $W^{1,2}(0,T;H_{0}^{1}(\Omega),L^{2}(\Omega))$ have been established. Here, we will prove analogous results without requiring boundedness of the time derivative in $L^{2}(0,T;H^{-1}(\Omega))$. Note that the conditions (12) and (13) are the same as (2) and (3) in Theorem 6.

Theorem 9

Let $\left\{u_{\varepsilon}\right\}$ be a bounded sequence in $L^{2}(0,T;H_{0}^{1}(\Omega))$ and, for any $v\in D(\Omega)$, $c_{1}\in D(0,T)$, $c_{2}\in C_{\sharp}^{\infty}(S_{1})$, $c_{3}\in C_{\sharp}^{\infty}(S_{2})$ and $r>q>0$,

[TABLE]

and

[TABLE]

Then, with $\varepsilon_{1}=\varepsilon$, $\varepsilon_{2}=\varepsilon^{2}$, $\varepsilon_{1}^{\prime}=\varepsilon^{q}$ and $\varepsilon_{2}^{\prime}=\varepsilon^{r}$, up to a subsequence

[TABLE]

and

[TABLE]

where $u_{1}\in L^{2}(\Omega_{T}\times S^{2};H_{\sharp}^{1}(Y_{1})/\mathbb{R})$ and $u_{2}\in L^{2}(\Omega_{T}\times\mathcal{Y}_{1,2};H_{\sharp}^{1}(Y_{2})/\mathbb{R})$ are the same as in (6) and (7) in Theorem 6.

Proof. We point out that the task to prove (14) and (15) is to show

[TABLE]

for any $v_{1}\in\emph{D}(\Omega)$, $v_{2}\in C_{\sharp}^{\infty}(Y_{1})/\mathbb{R}$ and $c\in\emph{D}(0,T;C_{\sharp}^{\infty}(S^{2}))$, and

[TABLE]

for any $v_{1}\in\emph{D}(\Omega;C_{\sharp}^{\infty}(Y_{1}))$, $v_{2}\in C_{\sharp}^{\infty}(Y_{2})/\mathbb{R}$ and $c\in\emph{D}(0,T;C_{\sharp}^{\infty}(S^{2}))$, respectively.

We start by proving (14). Note that any $v_{2}\in C_{\sharp}^{\infty}(Y_{1})/\mathbb{R}$ can be represented by

[TABLE]

for some $\rho\in C_{\sharp}^{\infty}(Y_{1})/\mathbb{R}$. The left-hand side of (16) can now be expressed as

[TABLE]

where we used antidifferentiation with respect to $y_{1}$ and integration by parts. By Theorem 6, as $\varepsilon$ tends to zero we obtain

[TABLE]

Integration by parts in the last term with respect to $x$ leaves us with

[TABLE]

and by integration by parts with respect to $y_{1}$ we arrive at

[TABLE]

which proves (14).

We continue by proving (15). Observing that any $v_{2}\in C_{\sharp}^{\infty}(Y_{2})/\mathbb{R}$ can be expressed as

[TABLE]

for some $\rho\in C_{\sharp}^{\infty}(Y_{2})/\mathbb{R}$, following the same steps as above the left-hand side of (17) can be written

[TABLE]

Since $\left\{\varepsilon^{-2}\nabla_{y_{1}}v_{1}\left(x,x/\varepsilon\right)\cdot\nabla_{y_{2}}\rho\left(x/\varepsilon^{2}\right)\right\}$ is bounded in $H^{-1}(\Omega)$, the last term in the second integral vanishes as we pass to the limit and, applying Theorem 6, we obtain

[TABLE]

By observing that

[TABLE]

all but the last term in the first integral vanish, leaving us with

[TABLE]

and integration by parts with respect to $y_{2}$ gives

[TABLE]

which proves (15).

3 Homogenization

This section is devoted to the homogenization of problem (1). We start by recalling the equation

[TABLE]

where $0<p<q<r$, $f\in L^{2}(\Omega_{T})$ and $u_{0}\in L^{2}(\Omega)$. Under the assumption that the coefficient $a\in C_{\sharp}(\mathcal{Y}_{2,2})^{N\times N}$ satisfies the coercivity condition

[TABLE]

for all $\left(y^{2},s^{2}\right)\in\mathbb{R}^{2N}\times\mathbb{R}^{2}$, all $\xi\in\mathbb{R}^{N}$ and some $C_{0}>0$, (18) possesses a unique solution $u_{\varepsilon}\in W^{1,2}(0,T;H_{0}^{1}(\Omega),L^{2}(\Omega))$ for every fixed $\varepsilon$, see Section 23.7 in [22]. Further, the a priori estimate

[TABLE]

holds for some $C_{1}>0$ independent of $\varepsilon$, according to the reasoning in Section 3 in [5].

Before we are ready to give the homogenization result we show that the assumptions (2) and (3) in Theorems 6 and 9 are satisfied, i.e. that for $v\in\emph{D}(\Omega)$, $c_{1}\in\emph{D}(0,T)$,$\ c_{2}\in C_{\sharp}^{\infty}(S_{1})$, $c_{3}\in C_{\sharp}^{\infty}(S_{2})$ and $r>q>0$

[TABLE]

and

[TABLE]

The weak form of (18) is

[TABLE]

where $0<p<q<r$, for all $v\in H_{0}^{1}(\Omega)$ and $c\in\emph{D}(0,T)$. Taking the test function

[TABLE]

with $v_{1}\in\emph{D}(\Omega)$, $c_{1}\in\emph{D}(0,T)$,$\ c_{2}\in C_{\sharp}^{\infty}(S_{1})$ and $c_{3}\in C_{\sharp}^{\infty}(S_{2})$, we get, after rearranging,

[TABLE]

Passing to the limit while recalling that $\left\{u_{\varepsilon}\right\}$ is bounded in $L^{2}(0,T;H_{0}^{1}(\Omega))$, which implies boundedness of $\left\{\nabla u_{\varepsilon}\right\}$ in $L^{2}(\Omega_{T})^{N}$, we obtain

[TABLE]

and (20) is fulfilled. Following the same steps again but taking the test function

[TABLE]

where $v_{1}\in\emph{D}(\Omega)$, $c_{1}\in\emph{D}(0,T)$ and$\ c_{2}\in C_{\sharp}^{\infty}(S_{1})$, in the weak form (22) yields that (21) is fulfilled.

We are now prepared to prove the homogenization result. Depending on the choices of $p$, $q$ and $r$ $(0<p<q<r)$ in (18), we get different outcomes. In Theorem 10 we present the 13 possible cases, arising from different combinations of $p$, $q$ and $r$. Here we will see that the local problems are parabolic when the matching between the microscopic scales that give resonance is shifted by $p$ compared to the standard case (cf. Section 5.3.1 in [19]). This means that resonance appears when the temporal scale multiplied by $\varepsilon^{-p}$ is the square of a spatial scale.

Theorem 10

Let $\left\{u_{\varepsilon}\right\}$ be a sequence of solutions to (18) in $W^{1,2}(0,T;H_{0}^{1}(\Omega),L^{2}(\Omega))$. Then it holds that

[TABLE]

[TABLE]

and

[TABLE]

where $u\in L^{2}(0,T;H_{0}^{1}(\Omega))$ is the unique solution to the homogenized problem

[TABLE]

where the coefficient $b$ is characterized by the formulas below. For all 13 cases we assume that $0<p<q<r$.

Letting $r<2+p$, the homogenized coefficient is given by

[TABLE]

and $u_{1}\in L^{2}(\Omega_{T}\times S^{2};H_{\sharp}^{1}(Y_{1})/\mathbb{R})$ and $u_{2}\in L^{2}(\Omega_{T}\times\mathcal{Y}_{1,2};H_{\sharp}^{1}(Y_{2})/\mathbb{R})$ are given by the local problems

[TABLE]

and

[TABLE] 2. 2.

Choosing $r=2+p$, the coefficient $b$ is determined by (27) while $u_{1}\in L^{2}(\Omega_{T}\times S_{1};\mathcal{W}_{1,2})$ and $u_{2}\in L^{2}(\Omega_{T}\times\mathcal{Y}_{1,2};H_{\sharp}^{1}(Y_{2})/\mathbb{R})$ are the solutions to the local problems

[TABLE]

and

[TABLE] 3. 3.

If $2+p<r<4+p$ while $q<2+p$, we have

[TABLE]

where $u_{1}\in L^{2}(\Omega_{T}\times S_{1};H_{\sharp}^{1}(Y_{1})/\mathbb{R})$ and $u_{2}\in L^{2}(\Omega_{T}\times\mathcal{Y}_{1,2};H_{\sharp}^{1}(Y_{2})/\mathbb{R})$ are given by the system

[TABLE]

and

[TABLE] 4. 4.

Taking $r<4+p$ and $q=2+p$, the homogenized coefficient is given by (32) and $u_{1}\in L^{2}(\Omega_{T};\mathcal{W}_{1,1})$ and $u_{2}\in L^{2}(\Omega_{T}\times\mathcal{Y}_{1,2};H_{\sharp}^{1}(Y_{2})/\mathbb{R})$ are determined by

[TABLE]

and

[TABLE] 5. 5.

When $r<4+p$ and $q>2+p$ the coefficient $b$ is determined by

[TABLE]

and the local problems are

[TABLE]

and

[TABLE]

where $u_{1}\in L^{2}(\Omega_{T};H_{\sharp}^{1}(Y_{1})/\mathbb{R})$ and $u_{2}\in L^{2}(\Omega_{T}\times\mathcal{Y}_{1,2};H_{\sharp}^{1}(Y_{2})/\mathbb{R})$. 6. 6.

In the case when $r=4+p$ while $q<2+p$, the homogenized coefficient is characterized by (32) while $u_{1}\in L^{2}(\Omega_{T}\times S_{1};H_{\sharp}^{1}(Y_{1})/\mathbb{R})$ and $u_{2}\in L^{2}(\Omega_{T}\times\mathcal{Y}_{1,1};\mathcal{W}_{2,2})$ are given by the system of local problems

[TABLE]

and

[TABLE] 7. 7.

When $r=4+p$ and $q=2+p$, the coefficient $b$ is given by (32) where $u_{1}\in L^{2}(\Omega_{T};\mathcal{W}_{1,1})$ and $u_{2}\in L^{2}(\Omega_{T}\times\mathcal{Y}_{1,1};\mathcal{W}_{2,2})$ are the solutions to

[TABLE]

and

[TABLE] 8. 8.

Letting $r=4+p$ while $q>2+p$ gives us the homogenized coefficient (37) defined by the system of local problems

[TABLE]

and

[TABLE]

where $u_{1}\in L^{2}(\Omega_{T};H_{\sharp}^{1}(Y_{1})/\mathbb{R})$ and $u_{2}\in L^{2}(\Omega_{T}\times\mathcal{Y}_{1,1};\mathcal{W}_{2,2})$. 9. 9.

Choosing $r>4+p$ and $q<2+p$, we have the homogenized coefficient

[TABLE]

where $u_{1}\in L^{2}(\Omega_{T}\times S_{1};H_{\sharp}^{1}(Y_{1})/\mathbb{R})$ and $u_{2}\in L^{2}(\Omega_{T}\times\mathcal{Y}_{1,1};H_{\sharp}^{1}(Y_{2})/\mathbb{R})$ are the solutions to the local problems

[TABLE]

and

[TABLE] 10. 10.

When $r>4+p$ while $q=2+p$, the homogenized coefficient is given by (46) and the local problems are

[TABLE]

and

[TABLE]

with $u_{1}\in L^{2}(\Omega_{T};\mathcal{W}_{1,1})$ and $u_{2}\in L^{2}(\Omega_{T}\times\mathcal{Y}_{1,1};H_{\sharp}^{1}(Y_{2})/\mathbb{R})$. 11. 11.

When $r>4+p$ and $2+p<q<4+p$, we have

[TABLE]

together with the local problems

[TABLE]

and

[TABLE]

where $u_{1}\in L^{2}(\Omega_{T};H_{\sharp}^{1}(Y_{1})/\mathbb{R})$ and $u_{2}\in L^{2}(\Omega_{T}\times\mathcal{Y}_{1,1};H_{\sharp}^{1}(Y_{2})/\mathbb{R})$. 12. 12.

Taking $q=4+p$, the coefficient in the homogenized problem is given by (51) and $u_{1}\in L^{2}(\Omega_{T};H_{\sharp}^{1}(Y_{1})/\mathbb{R})$ and $u_{2}\in L^{2}(\Omega_{T}\times Y_{1};\mathcal{W}_{2,1})$ are determined by

[TABLE]

and

[TABLE] 13. 13.

In the case when $q>4+p$, the coefficient is characterized by

[TABLE]

and the local problems are given by

[TABLE]

and

[TABLE]

where $u_{1}\in L^{2}(\Omega_{T};H_{\sharp}^{1}(Y_{1})/\mathbb{R})$ and $u_{2}\in L^{2}(\Omega_{T}\times Y_{1};H_{\sharp}^{1}(Y_{2})/\mathbb{R})$.

Proof. Since $\left\{u_{\varepsilon}\right\}$ satisfies the a priori estimate (19) and the conditions (20) and (21), Theorem 6 gives us (23), (24) and (25). The continuation of this proof will be divided into three parts. We start by finding the homogenized problem (26) followed by proving independencies of local time variables and determining the local problems, which together will provide us with the characterizations of the homogenized coefficient for all 13 cases.

Taking the test function

[TABLE]

where $v_{1}\in H_{0}^{1}(\Omega)$ and $c_{1}\in\emph{D}(0,T)$, in the weak form (22) and letting $\varepsilon$ tend to zero, Theorem 6 yields

[TABLE]

By the Variational Lemma we arrive at

[TABLE]

a.e. in $(0,T)$, which is the weak form of (26).

We start by deriving a common ground, divided into two paths, for the reasoning about independencies and the local problems. For the first path, in the weak form (22), we choose a test function which captures the oscillations from the second microscopic scale $\varepsilon_{2}=\varepsilon^{2}$, more precise we choose

[TABLE]

where $k>0$, $v_{1}\in\emph{D}(\Omega)$, $v_{2}\in C_{\sharp}^{\infty}(Y_{1})$, $v_{3}\in C_{\sharp}^{\infty}(Y_{2})/\mathbb{R}$, $c_{1}\in\emph{D}(0,T)$,$\ c_{2}\in C_{\sharp}^{\infty}(S_{1})$ and $c_{3}\in C_{\sharp}^{\infty}(S_{2})$. After differentiations we arrive at

[TABLE]

Passing to the limit, omitting terms that obviously tend to zero, we have

[TABLE]

For the second path, i.e. the one with respect to the first spatial microscopic scale $\varepsilon_{1}=\varepsilon$, we let

[TABLE]

where $k>0$, $v_{1}\in\emph{D}(\Omega)$, $v_{2}\in C_{\sharp}^{\infty}(Y_{1})/\mathbb{R}$, $c_{1}\in\emph{D}(0,T)$,$\ c_{2}\in C_{\sharp}^{\infty}(S_{1})$ and $c_{3}\in C_{\sharp}^{\infty}(S_{2})$, act as a test function in the weak form (22). Differentiating leads to

[TABLE]

and as $\varepsilon\rightarrow 0$, after omitting terms that vanish, we have

[TABLE]

Now we are ready to prove the independencies of local time variables and we start by showing when $u_{2}$ is independent of $s_{2}$. Let $r>4+p$ and choose $k=r-p-2$ in (59). As $\varepsilon\rightarrow 0$, applying Theorems 6 and 9, the limit of (60) becomes

[TABLE]

and by the Variational Lemma

[TABLE]

a.e. in $\Omega_{T}\times\mathcal{Y}_{2,1}$, which indicates that $u_{2}$ is independent of $s_{2}$.

Now we show independence of $s_{1}$ in $u_{2}$. Let $q>4+p$ and since $r>q$ this implies that $u_{2}$ is independent of $s_{2}$. Therefore we let $c_{3}\equiv 1$ in (59) and we choose $k=q-p-2$. Passing to the limit in (60), Theorems 6 and 9 yield

[TABLE]

and integrating over $S_{2}$ and applying the Variational Lemma on $\Omega_{T}\times Y^{2}$, we obtain that $u_{2}$ is independent of $s_{1}$.

Next we show independence of $s_{2}$ in $u_{1}$. Let $r>2+p$ and choose $k=r-p-1$ in (61). Letting $\varepsilon$ tend to zero in (62), applying Theorems 6 and 9, we have

[TABLE]

and the Variational Lemma on $\Omega_{T}\times\mathcal{Y}_{1,1}$ shows that $u_{1}$ is independent of $s_{2}$.

The last independence to show is when $u_{1}$ is independent of $s_{1}$. Here we let $q>2+p$ and recalling that since $r>q$, $u_{1}$ is independent of $s_{2}$. In (61) we choose $k=q-p-1$ and set $c_{3}\equiv 1$. As $\varepsilon\rightarrow 0$ in (62), Theorems 6 and 9 give

[TABLE]

Integrating over $S_{2}$ and using the Variational Lemma on $\Omega_{T}\times Y_{1}$ we have that $u_{1}$ is independent of $s_{1}$.

To sum up, we know that $u_{1}$ is independent of $s_{2}$ whenever $r>2+p$ and that $u_{1}$ is independent of both $s_{1}$ and $s_{2}$, when $q>2+p$. In the case when $r>4+p$, $u_{2}$ (and of course also $u_{1}$) is independent of $s_{2}$ and if $q>4+p$ we have that $u_{2}$ (and $u_{1}$) is independent of both $s_{1}$ and $s_{2}$. These independencies together with (58) give the formulas for the homogenized coefficient in the cases 1-13.

Now we are going to derive the system of local problems for each of the 13 cases. Each case has a system consisting of two local problems. The first local problem is with respect to the faster microscopic scale $\varepsilon_{2}=\varepsilon^{2}$ and our point of departure is always (60) where we have chosen $k=2$ in (59). The second local problem is with respect to the slower microscopic scale $\varepsilon_{1}=\varepsilon$ and the point of departure here is (62) where we have taken $k=1$ in (61).

Case 1: $r<2+p$. To obtain the first local problem we let $\varepsilon\rightarrow 0$ in (60) and applying Theorem 6 we have

[TABLE]

By the Variational Lemma on $\Omega_{T}\times\mathcal{Y}_{1,2}$, we obtain the weak form of (28).

For the second local problem, passing to the limit in (62), using Theorems 6 and 9, we obtain

[TABLE]

and the Variational Lemma on $\Omega_{T}\times S^{2}$ gives us the weak form of (29).

Case 2: $r=2+p$. Passing to the limit in (60) yields the same result as for the first local problem in case 1, which is the weak form of (30).

For the second local problem, we apply Theorems 6 and 9 as we pass to the limit in (62) to get

[TABLE]

Using the Variational Lemma on $\Omega_{T}\times S_{1}$, we get the weak form of (31).

Case 3: $2+p<r<4+p$ and $q<2+p$. Passing to the limit in (60) and applying Theorems 6 and 9, recalling that $u_{1}$ is independent of $s_{2}$, we arrive at

[TABLE]

Applying the Variational Lemma on $\Omega_{T}\times\mathcal{Y}_{1,2}$ we have the weak form of (33).

Because of the independence of $s_{2}$ in $u_{1}$, we can let $c_{3}\equiv 1$ in (61). As $\varepsilon\rightarrow 0$ in (62), by Theorems 6 and 9 we obtain

[TABLE]

and the Variational Lemma on $\Omega_{T}\times S_{1}$ gives the weak form of (34).

Case 4: $r<4+p$ and $q=2+p$. Passing to the limit in (60), remembering that $u_{1}$ is independent of $s_{2}$, by Theorems 6 and 9 we arrive at the same local problem as the first one in case 3, which is the weak form of (35).

Letting $c_{3}\equiv 1$ in (61) and passing to the limit in (62), applying Theorems 6 and 9, we get

[TABLE]

Integrating over $S_{2}$ in the first integral and applying the Variational Lemma on $\Omega_{T}$ we get the weak form of (36).

Case 5: $r<4+p$ and $q>2+p$. Remembering that $u_{1}$ is independent of both $s_{1}$ and $s_{2}$, when $\varepsilon\rightarrow 0$ in (60) we apply Theorems 6 and 9 and have

[TABLE]

By using the Variational Lemma on $\Omega_{T}\times\mathcal{Y}_{1,2}$ we arrive at the weak form of (38).

Because of the independencies, we can let $c_{2}\equiv 1$ and $c_{3}\equiv 1$ in (61). Applying Theorem 6 as $\varepsilon$ tends to zero in (62) yields

[TABLE]

and by the Variational Lemma on $\Omega_{T}$ we get the weak form of (39).

Case 6: $r=4+p$ and $q<2+p$. Noting that $u_{1}$ is independent of $s_{2}$, passing to the limit in (60), Theorems 6 and 9 give us

[TABLE]

Applying the Variational Lemma on $\Omega_{T}\times\mathcal{Y}_{1,1}$ we have the weak form of (40).

Because of the independence in $u_{1}$, we can let $c_{3}\equiv 1$ in (61) and as $\varepsilon\rightarrow 0$ (62) becomes, due to Theorems 6 and 9,

[TABLE]

Using the Variational Lemma on $\Omega_{T}\times S_{1}$ we obtain the weak form of (41).

Case 7: $r=4+p$ and $q=2+p$. As $\varepsilon\rightarrow 0$ in (60), we end up with the same local problem as the first one in case 6, which is the weak form of (42).

Letting $\varepsilon$ tend to zero in (62), recalling that $u_{1}$ is independent of $s_{2}$ so that $c_{3}\equiv 1$, Theorems 6 and 9 yield

[TABLE]

Integrating over $S_{2}$ in the first integral and taking the Variational Lemma on $\Omega_{T}$ gives us the weak form of (43).

Case 8: $r=4+p$ and $q>2+p$. Letting $\varepsilon$ tend to zero in (60), observing that $u_{1}$ is independent of both $s_{1}$ and $s_{2}$, Theorems 6 and 9 give

[TABLE]

and by applying the Variational Lemma on $\Omega_{T}\times\mathcal{Y}_{1,1}$ we get the weak form of (44).

For the second local problem, due to independencies in $u_{1}$, we can let both $c_{2}\equiv 1$ and $c_{3}\equiv 1$ in (61). Letting $\varepsilon\rightarrow 0$ in (62), from Theorem 6 we obtain

[TABLE]

and the Variational Lemma on $\Omega_{T}$ gives us the weak form of (45).

Case 9: $r>4+p$ and $q<2+p$. Recalling that $u_{2}$ (and $u_{1}$) is independent of $s_{2}$, we can let $c_{3}\equiv 1$ in (59). Passing to the limit in (60), Theorem 6 gives us

[TABLE]

and using the Variational Lemma on $\Omega_{T}\times\mathcal{Y}_{1,1}$ we obtain the weak form of (47).

Due to the independence in $u_{1}$ we can let $c_{3}\equiv 1$ in (61) and as $\varepsilon\rightarrow 0$ in (62), Theorems 6 and 9 yield

[TABLE]

By the Variational Lemma on $\Omega_{T}\times S_{1}$ we have the weak form of (48).

Case 10: $r>4+p$ and $q=2+p$. Because of the independence of $s_{2}$ in $u_{2}$ we let $c_{3}\equiv 1$ in (59) and as $\varepsilon$ tends to zero in (60), recalling that also $u_{1}$ is independent of $s_{2}$, Theorems 6 and 9 give the same first local problem as in case 9, which is the weak form of (49).

Again we can let $c_{3}\equiv 1$ in (61), due to independence in $u_{1}$. Letting $\varepsilon\rightarrow 0$ in (62), from Theorems 6 and 9 we have

[TABLE]

Integrating over $S_{2}$ in the first integral and using the Variational Lemma on $\Omega_{T}$ we get the weak form of (50).

Case 11: $r>4+p$ and $2+p<q<4+p$. Since $u_{2}$ is independent of $s_{2}$, we let $c_{3}\equiv 1$ in (59). We also have independence of $s_{1}$ and $s_{2}$ in $u_{1}$, so as $\varepsilon\rightarrow 0$ in (60), Theorems 6 and 9 give

[TABLE]

Applying the Variational Lemma on $\Omega_{T}\times\mathcal{Y}_{1,1}$ we get the weak form of (52).

Because of the independencies in $u_{1}$, for the second local problem, we can let both $c_{2}\equiv 1$ and $c_{3}\equiv 1$ in (61). Passing to the limit in (62), applying Theorem 6, we end up with

[TABLE]

and from the Variational Lemma on $\Omega_{T}$ we obtain the weak form of (53).

Case 12: $q=4+p$. Since $u_{2}$ is independent of $s_{2}$ we can take $c_{3}\equiv 1$ in (59). Recalling that $u_{1}$ is independent of $s_{1}$ and $s_{2}$, passing to the limit in (60), from Theorems 6 and 9 we have

[TABLE]

Integrating over $S_{2}$ in the first integral and applying the Variational Lemma on $\Omega_{T}\times Y_{1}$ we have the weak form of (54).

Because of the independencies in $u_{1}$ we can let $c_{2}\equiv 1$ and $c_{3}\equiv 1$ in (61) and as $\varepsilon$ tends to zero in (62), we get the same result as for the second local problem in case 11, sharing the weak form of (55).

Case 13: $q>4+p$. Recalling that $u_{2}$ is independent of $s_{1}$ and $s_{2}$, we can set $c_{2}\equiv 1$ and $c_{3}\equiv 1$ in (59). Noting that also $u_{1}$ is independent of both $s_{1}$ and $s_{2}$, letting $\varepsilon\rightarrow 0$ in (60), Theorem 6 yields

[TABLE]

and applying the Variational Lemma on $\Omega_{T}\times Y_{1}$ gives the weak form of (56).

For the second local problem, we again let $c_{2}\equiv 1$ and $c_{3}\equiv 1$ in (61) and as $\varepsilon\rightarrow 0$ in (62), Theorem 6 gives

[TABLE]

From the Variational Lemma on $\Omega_{T}$ we get the weak form of (57).