W2,2 Interior convergence for some class of elliptic anisotropic singular perturbation problems

TL;DR

This paper investigates the asymptotic behavior of solutions to a class of anisotropic elliptic singular perturbation problems within specific pseudo Sobolev spaces, providing insights into their interior convergence properties.

Contribution

It introduces new analysis of interior convergence for anisotropic elliptic problems with singular perturbations in pseudo Sobolev spaces.

Findings

Established interior convergence results for anisotropic elliptic problems.

Analyzed asymptotic behavior of solutions in pseudo Sobolev spaces.

Provided theoretical framework for singular perturbation analysis.

Abstract

In this paper, we deal with anisotropic singular perturbations of some class of elliptic problem. We study the asymptotic behavior of the solution in certain second order pseudo Sobolev space.

Equations

$$\left\{\begin{array}[]{cc}-\mathop{\mathrm{d}iv}(A_{\epsilon}\nabla u_{\epsilon})=f&\\ u_{\epsilon}\in W_{0}^{1,2}(\Omega)\text{ \ \ \ \ \ \ \ }&\end{array}\right.$$

$$X_1=(x_1,...,x_q)\text{ and }{X}_1=(x_{q+1},...,x_N),$$

$$\nabla=(\partial_{x_{1}},...,\partial_{x_{N}})^{T}=\left(\begin{array}[]{c}\nabla_{X_{1}}\\ \nabla_{X_{2}}\end{array}\right),$$

$${\nabla }_{X_1}=({\partial }_{x_1},...,{\partial }_{x_q}{)}^T\text{ and }{\nabla }_{X_2}=({\partial }_{x_{q+1}},...,{\partial }_{x_N}{)}^T$$

$$A_{\epsilon}=(a_{ij}^{\epsilon})=\left(\begin{array}[]{cc}\epsilon^{2}A_{11}&\epsilon A_{12}\\ \epsilon A_{21}&A_{22}\end{array}\right)\text{ with }A=(a_{ij})=\left(\begin{array}[]{cc}A_{11}&A_{12}\\ A_{21}&A_{22}\end{array}\right),$$

$$a_{ij}^{\epsilon}=\left\{\begin{array}[]{c}\epsilon^{2}a_{ij}\text{ for }i,j\in\left\{1,..,q\right\}\text{ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }\\ a_{ij}\text{ for }i,j\in\left\{q+1,..,N\right\}\text{ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }\\ \epsilon a_{ij}\text{ for }i\in\left\{1,..,q\right\}\text{, }j\in\left\{q+1,..,N\right\}\\ \epsilon a_{ij}^{\epsilon}\text{ for }i\in\left\{q+1,..,N\right\}\text{, }j\in\left\{1,..,q\right\}\end{array}\right.$$

$$A(x)\zeta \cdot \zeta \ge \lambda {\left|\zeta \right|}^2\text{,}\forall \zeta \in {\mathbb{R}}^N,\text{ a.e }x\in \Omega .$$

$$V^{1,2}=\left\{u\in L^2(\Omega )\mid {\nabla }_{X_2}u\in L^2(\Omega )\text{ and }u(X_1,\cdot )\in W_0^{1,2}({\Omega }_{X_1})\text{ a.e }{X}_1\in {\Omega }^1\right\},.$$

$${\Vert u\Vert }_{1,2}={\left({\Vert u\Vert }_{L^2(\Omega )}^2+{\Vert {\nabla }_{X_2}u\Vert }_{L^2(\Omega )}^2\right)}^{\frac{1}{2}}.$$

$$V_{loc}^{2,2}=\left\{u\in V^{1,2}\mid {\nabla }_{X_2}^{2}u\in L_{loc}^2(\Omega )\right\},$$

$${\Vert u\Vert }_{2,2}^{\omega }={\left({\Vert u\Vert }_{L^2(\Omega )}^2+{\Vert {\nabla }_{X_2}u\Vert }_{L^2(\Omega )}^2+{\Vert {\nabla }_{X_2}^{2}u\Vert }_{L^2(\omega )}^2\right)}^{\frac{1}{2}}\text{, }\omega \subset \subset \Omega \text{ open}$$

$${\Vert {\nabla }_{X_2}^{2}u\Vert }_{L^2(\omega )}^2=\sum _{i,j=q+1}^N{\Vert {\partial }_{ij}^{2}u\Vert }_{L^2(\omega )}^2.$$

$${\Vert {\nabla }_{X_1}^{2}u\Vert }_{L^2(\omega )}^2=\sum _{i,j=1}^q{\Vert {\partial }_{ij}^{2}u\Vert }_{L^2(\omega )}^2,$$

$${\Vert {\nabla }_{X_1{X}_2}^{2}u\Vert }_{L^2(\omega )}^2=\sum _{i=1}^q\sum _{j=q+1}^N{\Vert {\partial }_{ij}^{2}u\Vert }_{L^2(\omega )}^2.$$

$$\left\{\begin{array}[]{cc}-\mathop{\mathrm{d}iv}(A_{22}\nabla u_{0}(X_{1},\cdot)=f(X_{1},\cdot)\text{ \ \ \ \ \ \ \ \ \ \ \ \ \ }&\\ u_{0}(X_{1},\cdot)\in W_{0}^{1,2}(\Omega_{X_{1}})\text{ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }&\text{a.e }X_{1}\in\Omega^{1}\end{array}\right.$$

$$u_{ϵ}\to u_0\text{ in }{V}^{1,2}\text{ and }ϵ{\nabla }_{X_1}{u}_{ϵ}\to 0\text{ in }{L}^2(\Omega ).$$

$$d(u,v)=\sum _{n=0}^{\infty }{2}^{-n}\frac{{\Vert u-v\Vert }_{2,2}^{{\omega }_n}}{1+{\Vert u-v\Vert }_{2,2}^{{\omega }_n}}\text{, }u,v\in V_{loc}^{2,2}.$$

$$u_m\to u\text{, }{\nabla }_{X_2}{u}_m\to v\text{ in }{L}^2(\Omega )\text{,}$$

$${\nabla }_{X_2}^2{u}_m\to w_n\text{ in }{L}^2({\omega }_n).$$

$$\forall n\in \mathbb{N}:{\Vert u_m-u\Vert }_{2,2}^{{\omega }_n}\to 0\text{ as }m\to \infty .$$

$$d(u_m,u)\to 0\text{ as }m\to \infty ,$$

$$-{ϵ}^2{\Delta }_{X_1}{u}_{ϵ}(x)-{\Delta }_{X_2}{u}_{ϵ}(x)=f(x)\text{ a.e }x\in {\mathbb{R}}^N$$

$${\Vert {\nabla }_{X_2}^2{u}_{ϵ}\Vert }_{L^2({\mathbb{R}}^N)}$$

$${ϵ}^2{\Vert {\nabla }_{X_1}^2{u}_{ϵ}\Vert }_{L^2({\mathbb{R}}^N)}$$

$$\sqrt{2}ϵ{\Vert {\nabla }_{X_1{X}_2}^2{u}_{ϵ}\Vert }_{L^2({\mathbb{R}}^N)}$$

$$\mathcal{F}(u)(\xi )=(2\pi {)}^{-\frac{N}{2}}{\int }_{{\mathbb{R}}^N}u(x)e^{-ix\cdot \xi }dx\text{, }u\in \mathcal{S}({\mathbb{R}}^N)$$

$$\left({ϵ}^2\sum _{i=1}^q{\xi }_i^2+{\sum }_{i=q+1}^N{\xi }_i^2\right)\mathcal{F}(u_{ϵ})(\xi )=\mathcal{F}(f)(\xi ),$$

$$\left({ϵ}^4\sum _{i,j=1}^q{\xi }_i^2{\xi }_j^2+{\sum }_{i,j=q+1}^N{\xi }_i^2{\xi }_j^2+2{ϵ}^2{\sum }_{j=q+1}^N\sum _{i=1}^q{\xi }_i^2{\xi }_j^2\right){\left|\mathcal{F}(u_{ϵ})(\xi )\right|}^2={\left|\mathcal{F}(f)(\xi )\right|}^2,$$

$${\sum }_{i,j=q+1}^N{\xi }_i^2{\xi }_j^2{\left|\mathcal{F}(u_{ϵ})(\xi )\right|}^2\le {\left|\mathcal{F}(f)(\xi )\right|}^2,$$

$${\sum }_{i,j=q+1}^N{\left|\mathcal{F}({\partial }_{ij}^2{u}_{ϵ})(\xi )\right|}^2\le {\left|\mathcal{F}(f)(\xi )\right|}^2,$$

$${\sum }_{i,j=q+1}^N{\Vert \mathcal{F}({\partial }_{ij}^2{u}_{ϵ})\Vert }_{L^2({\mathbb{R}}^N)}^2\le {\Vert \mathcal{F}(f)\Vert }_{L^2({\mathbb{R}}^N)}^2,$$

Full text

1. Description of the problem

In this paper, we study diffusion problems when the diffusion coefficients in certain directions are going toward zero. More precisely we are interested in studying the asymptotic behavior of the solution in certain second order pseudo Sobolev space. We consider the following elliptic problem

[TABLE]

where $0<\epsilon\leq 1$ and $\Omega$ is a bounded domain (i.e. open bounded connected subset) of $\mathbb{R}^{N}$ and $f\in L^{2}(\Omega).$ We denote by $x=(x_{1},...,x_{N})=(X_{1},X_{2})$ the points in $\mathbb{R}^{N}$ where

[TABLE]

with this notation we set

[TABLE]

where

[TABLE]

The diffusion matrix $A_{\epsilon}$ is given by

[TABLE]

where $A_{11}$ and $A_{22}$ are $q\times q$ and $(N-q)\times(N-q)$ matrices. The coefficients $a_{ij}^{\epsilon}$ are given by

[TABLE]

We assume that $A\in L^{\infty}(\Omega)$ and for some $\lambda>0$ we have

[TABLE]

Recall the Hilbert space introduced in [2]

[TABLE]

equipped with the norm

[TABLE]

Here $\Omega_{X_{1}}=\left\{X_{2}\in\mathbb{R}^{N-q}:(X_{1},X_{2})\in\Omega\right\}$ and $\Omega^{1}=P_{1}(\Omega)$ where $P_{1}$ is the natural projector $\mathbb{R}^{N}\rightarrow\mathbb{R}^{q}.$

We introduce the second order local pseudo Sobolev space

[TABLE]

equipped with the family of norms $(\left\|\cdot\right\|_{2,2}^{\omega})_{\omega}$ given by

[TABLE]

where $\nabla_{X_{2}}^{2}u$ is the Hessian matrix of $u$ taken in the $X_{2}$ direction, the term $\left\|\nabla_{X_{2}}^{2}u\right\|_{L^{2}(\omega)}^{2}$ is given by

[TABLE]

We can show that $V_{loc}^{2,2}$ is a Fréchet space (i.e. locally convex, metrizable and complete). We also define the following

[TABLE]

and

[TABLE]

As $\epsilon\rightarrow 0$, the Limit problem is given by

[TABLE]

The existence and the uniqueness of the $W_{0}^{1,2}$ weak solutions to (1) and (3) follow from the Lax-Milgram theorem. In [1] the authors studied the relationship between $u_{\epsilon}$ and $u_{0}$ and they proved that $u_{0}\in V^{1,2}$ and the following convergences (see Theorem 2.1 in the above reference)

[TABLE]

For the $L^{p}$ case we refer the reader to [6], and [2],[4], [5] for other related problems. In this paper, we deal with the asymptotic behavior of the second derivatives of $u_{\epsilon}$, in other words we show the convergence of $u_{\epsilon}$ in the space $V_{loc}^{2,2}$ introduced previously. The arguments are based on the Riesz-Fréchet-Kolmogorov compacity theorem in $L^{p}$ spaces. Let us give the main result

Theorem 1**.**

Assume that $A\in L^{\infty}(\Omega)\cap C^{1}(\Omega)$ with (2), suppose that $f\in L^{2}(\Omega)$ then $u_{0}\in V_{loc}^{2,2}$ and $u_{\epsilon}\rightarrow u_{0}$ in $V_{loc}^{2,2}$, where $u_{\epsilon}\in W_{0}^{1,2}(\Omega)\cap W_{loc}^{2,2}(\Omega)$ and $u_{0}$ are the unique weak solutions to (1) and (3) respectively. In addition, the convergences $\epsilon^{2}\nabla_{X_{1}}^{2}u_{\epsilon}\rightarrow 0$, $\epsilon\nabla_{X_{1}X_{2}}^{2}u_{\epsilon}\rightarrow 0$ hold in $L_{loc}^{2}(\Omega).$

2. Some useful tools

Proposition 1**.**

The vector space $V_{loc}^{2,2}$ equipped with the family of norms $(\left\|\cdot\right\|_{2,2}^{\omega})_{\omega}$ is a Fréchet space.

Proof.

Let $(\omega_{n})_{n\in\mathbb{N}}$ be a countable open covering of $\Omega$ with $\omega_{n}\subset\subset$ $\Omega$, $\omega_{n}\subset\omega_{n+1}$ for every $n\in\mathbb{N}.$ The countable family $(\left\|\cdot\right\|_{2,2}^{\omega_{n}})_{n\in\mathbb{N}}$ define a base of norms for the $V^{2,2}$ topology. The general theory of locally convex topological vector spaces shows that this topology is metrizable, explicitly a distance $d$ which define this topology is given by ( see for instance [8])

[TABLE]

Let $(u_{m})$ be a Cauchy sequence in $V_{loc}^{2,2}$ then $(u_{m})$ is a Cauchy sequence for each norm $\left\|\cdot\right\|_{2,2}^{\omega_{n}}$, $n\in\mathbb{N}.$ Whence, there exist $u,v\in L^{2}(\Omega)$ such that

[TABLE]

and for every $n\in\mathbb{N}$ fixed there exists $w_{n}\in L^{2}(\omega_{n})$ such that

[TABLE]

The continuity of $\nabla_{X_{2}}$ and $\nabla_{X_{2}}^{2}$ on $D^{\prime}(\Omega)$ and $D^{\prime}(\omega_{n})$ shows that $v=\nabla_{X_{2}}u$ and $\nabla_{X_{2}}^{2}u=w_{n}$ for every $n\in\mathbb{N}.$ Hence $u\in V_{loc}^{2,2}$ and

[TABLE]

Finally the normal convergence of the series (5) implies

[TABLE]

and therefore the completion of $V_{loc}^{2,2}$ follows.

Remark 1**.**

Notice that a sequence $(u_{m})$ in $V_{loc}^{2,2}$ converges to $u$ with respect to $d$ if and only if $\left\|u_{m}-u\right\|_{2,2}^{\omega}\rightarrow 0$ as $m\rightarrow\infty,$ for every $\omega\subset\subset$ $\Omega$ open.

Now, let us give two useful lemmas

Lemma 1**.**

Let $f\in L^{2}(\mathbb{R}^{N}),$ for every $\epsilon\in(0,1]$ let $u_{\epsilon}\in W^{2,2}(\mathbb{R}^{N})$ such that

[TABLE]

then for every $\epsilon\in(0,1]$ we have the bounds

[TABLE]

Proof.

Let $\mathcal{F}$ be the Fourier transform defined on $L^{2}(\mathbb{R}^{N})$ as the extension, by density, of the Fourier transform defined on the Schwartz space $\mathcal{S}(\mathbb{R}^{N})$ by

[TABLE]

where $\cdot$ is the standard scalar product of $\mathbb{R}^{N}$. Applying $\mathcal{F}$ on (6) we obtain

[TABLE]

then

[TABLE]

thus

[TABLE]

hence

[TABLE]

then

[TABLE]

and the Parseval identity gives

[TABLE]

Hence

[TABLE]

Similarly we obtain from (7) the bounds

[TABLE]

[TABLE]

Notation 1**.**

For a function $u\in L^{p}(\mathbb{R}^{N})$ and $h\in\mathbb{R}^{N}$ we denote $\tau_{h}u(x)=u(x+h),$ $x\in\mathbb{R}^{N}.$

Lemma 2**.**

Let $\Omega$ be an open bounded subset of $\mathbb{R}^{N}$ and let $(u_{k})_{k\in\mathbb{N}}$ be a converging sequence in $L^{p}(\Omega)$,$1\leq p<\infty$ and let $\omega\subset\subset\Omega$ open, then for every $\sigma>0$ there exists $0<\delta<dist(\partial\Omega,\omega)$ such that

[TABLE]

in other words we have $\underset{h\rightarrow 0}{\lim}$ $\underset{k\in\mathbb{N}}{\sup}\left\|\tau_{h}u_{k}-u_{k}\right\|_{L^{p}(\omega)}=0.$

Proof.

Let $\omega\subset\subset\Omega$ open. For a function $v\in L^{p}(\Omega),$ extend $v$ by [math] outside of $\Omega,$ since the translation $h\rightarrow\tau_{h}v$ is continuous from $\mathbb{R}^{N}$ to $L^{p}(\mathbb{R}^{N})$ (see for instance [8]) then for every $\sigma>0$ there exists $0<\delta<dist(\partial\Omega,\omega)$ such that

[TABLE]

We denote $\lim u_{k}=u\in L^{p}(\Omega)$, and let $\sigma>0$ then (8) shows that there exists $0<\delta<dist(\partial\Omega,\omega)$ such that

[TABLE]

By the triangular inequality and the invariance of the Lebesgue measure under translations we have for every $k\in\mathbb{N}$ and $\left|h\right|\leq\delta$

[TABLE]

Since $u_{k}\rightarrow u$ in $L^{p}(\Omega)$ then there exists $k_{0}\in\mathbb{N}$, such that

[TABLE]

Then from (9) we obtain

[TABLE]

Similarly (8) shows that for every $k\in\left\{0,1,2,...,k_{0}-1\right\}$ there exists $0<\delta_{k}<dist(\partial\Omega,\omega)$ such that

[TABLE]

Taking $\delta^{\prime}=\underset{k\in\left\{0,..,k_{0}-1\right\}}{\min}(\delta_{k},\delta)$ and combining (10) and (11) we obtain

[TABLE]

3.

The perturbed Laplace equation

In this section we will prove Theorem 1 for the perturbed Laplace equation. We suppose that $A=Id$, and let $u_{\epsilon}\in W_{0}^{1,2}(\Omega)$ be the unique solution to

[TABLE]

Notice that the elliptic regularity [7] shows that $u_{\epsilon}\in W_{loc}^{2,2}(\Omega)$. Now, let $(\epsilon_{k})_{k\in\mathbb{N}}$ be a sequence in $(0,1]$ with $\lim\epsilon_{k}=0,$ and let $u_{k}=u_{\epsilon_{k}}$ be the solution of (12) with $\epsilon$ replaced by $\epsilon_{k}$. then one can prove the following

Proposition 2**.**

1) Let $\omega\subset\subset\Omega$ open then

[TABLE]

2) The sequences $\left(\nabla_{X_{2}}^{2}u_{k}\right)$, $(\epsilon_{k}^{2}\nabla_{X_{1}}^{2}u_{k})$, $(\epsilon_{k}\nabla_{X_{1}X_{2}}^{2}u_{k})$ are bounded in $L_{loc}^{2}(\Omega)$ i.e. for every $\omega\subset\subset\Omega$ open there exists $M\geq 0$ such that

[TABLE]

Proof.

1. Let $\omega\subset\subset\Omega$ open, then one can choose $\omega^{\prime}$ open such that $\omega\subset\subset\omega^{\prime}\subset\subset\Omega,$ let $\rho\in\mathcal{D}(\mathbb{R}^{N})$ with $\rho=1$ on $\omega$, $0\leq\rho\leq 1$ and $Supp(\rho)\subset\omega^{\prime}$. Let $0<h<dist(\omega^{\prime},\partial\Omega),$ to make the notations less heavy we set $U_{k}^{h}=\tau_{h}u_{k}-u_{k}$, then $U_{k}^{h}\in W^{2,2}(\omega^{\prime}).$ Notice that translation and derivation commute then we have

[TABLE]

with $F^{h}=\tau_{h}f-f$.

We set $\mathcal{W}_{k}^{h}=\rho U_{k}^{h}$ then we get

[TABLE]

for a.e $x\in\omega^{\prime}$.

Since $U_{k}^{h}\in W^{2,2}(\omega^{\prime})$ then $\mathcal{W}_{k}^{h}\in W_{0}^{2,2}(\omega^{\prime})$, so we can extend $\mathcal{W}_{k}^{h}$ by [math] outside of $\omega^{\prime}$ then $\mathcal{W}_{k}^{h}\in W^{2}(\mathbb{R}^{N}).$ The right hand side of the above equality is extended by [math] outside of $\omega^{\prime}$, hence the equation is satisfied in the whole space, and thus by Lemma 1 we get

[TABLE]

Then

[TABLE]

Notice that by (4) we have $u_{k}\rightarrow u$ in $V^{1,2}$ and $\epsilon_{k}\nabla_{X_{1}}u_{k}\rightarrow 0$ in $L^{2}(\Omega)$, then by Lemma 2 we deduce

[TABLE]

and similarly we obtain

[TABLE]

and hence

[TABLE]

Similarly we obtain

[TABLE]

and

[TABLE]

2. Following the same arguments, we get the estimation

[TABLE]

The convergences $u_{k}\rightarrow u$ in $V^{1,2}$, $\epsilon_{k}\nabla_{X_{1}}u_{k}\rightarrow 0$ in $L^{2}(\Omega)$ and boundedness of $\rho$ and its derivatives show that the right hand side of the above inequality is uniformly bounded in $k$, i.e. for some $M\geq 0$ independent of $k$ we have

[TABLE]

and therefore, the sequences $\left(\nabla_{X_{2}}^{2}u_{k}\right)$, $(\epsilon_{k}^{2}\nabla_{X_{1}}^{2}u_{k})$, $(\epsilon_{k}\nabla_{X_{1}X_{2}}^{2}u_{k})$ are bounded in $L_{loc}^{2}(\Omega).$

Now, we are ready to prove the following

Theorem 2**.**

Let $u_{\epsilon}\in W_{0}^{1,2}(\Omega)\cap W_{loc}^{2,2}(\Omega)$ be the solution of (12) then $u_{\epsilon}\rightarrow u_{0}$ strongly in $V_{loc}^{2,2}$ where $u_{0}\in V_{loc}^{2,2}$ is the solution of the limit problem. In addition, we have

[TABLE]

Proof.

Let $u_{0}\in V^{1,2}$ be the solution of the limit problem and let $(u_{k})_{k\in\mathbb{N}},$ $u_{k}=u_{\epsilon_{k}}\in W_{0}^{1,2}(\Omega)\cap W_{loc}^{2,2}(\Omega)$ be a sequence of solutions to (12) with $\epsilon$ replaced by $\epsilon_{k}$. Then Proposition 2 shows that the hypothesis of the Riesz-Fréchet-Kolmogorov theorem are fulfilled (For the statement of the theorem, see for instance [3]). Whence, it follows that $\left\{\nabla_{X_{2}}^{2}u_{k}\right\}_{k\in\mathbb{N}}$ is relatively compact in $L^{2}(\omega)$ for every $\omega\subset\subset\Omega$ open. Now, for $\omega\subset\subset\Omega$ fixed there exists $u_{0}^{\omega}\in L^{2}(\omega)$ and a subsequence still labeled $(\nabla_{X_{2}}^{2}u_{k})_{k\in\mathbb{N}}$ such that $\nabla_{X_{2}}^{2}u_{k}\rightarrow u_{0}^{\omega}$ in $L^{2}(\omega)$ strongly. Since $u_{k}\rightarrow u_{0}$ in $L^{2}(\omega)$ and the second order differential operators $\partial_{ij}^{2}$ are continuous on $\mathcal{D}^{\prime}(\omega)$ then $u_{0}^{\omega}=\nabla_{X_{2}}u_{0}$ on $\omega.$ Whence, since $\omega$ is arbitrary we get $\nabla_{X_{2}}^{2}u_{0}\in L_{loc}^{2}(\Omega)$, i.e. $u_{0}\in V_{loc}^{2,2}.$

Now, Let $(\omega_{n})$ be a countable covering of $\Omega$ with $\omega_{n}\subset\subset\Omega,$ $\omega_{n}\subset\omega_{n+1}$,$\forall n\in\mathbb{N}.$ Then by the diagonal process one can construct a subsequence still labeled $(u_{k})$ such that

[TABLE]

Combining this with the convergence $u_{k}\rightarrow u_{0}$ of (4) we get

[TABLE]

where $d$ is the distance of the Fréchet space $V_{loc}^{2,2}.$

To prove the convergence of the whole sequence $(u_{\epsilon})_{0<\epsilon\leq 1}$ we can reason by contradiction. Suppose that there exists $\delta>0$ and a subsequence $(u_{k})$ such that $d(u_{k},u_{0})>\delta$. It follows by the first part of this proof that there exists a subsequence still labeled $(u_{k})$ such that $d(u_{k},u_{0})\rightarrow 0$, which is a contradiction..

By using the same arguments we can show easily ( see the end of subsection 4.1) that

[TABLE]

4.

General elliptic problems

4.1. Proof of the main theorem

In this subsection we shall prove Theorem 1. Firstly, we suppose that the coefficients of $A$ are constants then we have the following

Proposition 3**.**

Suppose that the coefficients of $A$ are constants and assume (2), let $(u_{\epsilon})_{0<\epsilon\leq 1}$ be a sequence in $W^{2,2}(\mathbb{R}^{N})$ such that $-\mathop{\displaystyle\sum}\limits_{i,j}a_{ij}^{\epsilon}\partial_{ij}^{2}u_{\epsilon}=f$, with $f\in L^{2}(\mathbb{R}^{N})$ then we have for every $\epsilon\in(0,1]:$

[TABLE]

Proof.

As in proof of Lemma 1, we use the Fourier transform and we obtain

[TABLE]

From the ellipticity assumption (2) we deduce

[TABLE]

Thus, similarly we obtain the desired bounds.

Now, suppose that $A\in L^{\infty}(\Omega)\cap C^{1}(\Omega)$ and assume (2), and let $u_{\epsilon}\in W_{0}^{1,2}(\Omega)$ be the unique weak solution to (1), then it follows by the elliptic regularity that $u_{\epsilon}\in W_{loc}^{2,2}(\Omega)$. We denote $u_{k}=u_{\epsilon_{k}}$the solution to (1) where $(\epsilon_{k})$ is a sequence in $(0,1]$ such that, $\epsilon_{k}\rightarrow 0$ as $k\rightarrow\infty.$

Under the above assumption we can prove the following

Proposition 4**.**

Let $z_{0}\in\Omega$ fixed then there exists $\omega_{0}\subset\subset\Omega$ open with $z_{0}\in\omega_{0}$ such that the sequences $\left(\nabla_{X_{2}}^{2}u_{k}\right)$, $\left(\nabla_{X_{1}}^{2}u_{k}\right)$ and $\left(\nabla_{X_{1}X_{2}}^{2}u_{k}\right)$ are bounded in $L^{2}(\omega_{0}).$

Proof.

Since $u_{k}\in W_{0}^{1,2}(\Omega)\cap W_{loc}^{2,2}(\Omega)$ and $A\in C^{1}(\Omega)$ then $u_{k}$ satisfies

[TABLE]

where we have set $a_{ij}^{k}=a_{ij}^{\epsilon_{k}}$.

Let $z_{0}\in\Omega$ fixed, and let $\theta>0$ such that

[TABLE]

By using the continuity of the $a_{ij}$ one can choose $\omega_{1}\subset\subset\Omega$, $z_{0}\in\omega_{1}$ such that

[TABLE]

Let $\omega_{0}\subset\subset\omega_{1}$ open with $z_{0}\in\omega_{0}$ and let $\rho\in\mathcal{D}(\mathbb{R}^{N})$ such $\rho=1$ on $\omega_{0}$, $0\leq\rho\leq 1$ and $Supp(\rho)\subset\omega_{1}$. We set $U_{k}=\rho u_{k}$, and we extend it by [math] on the outside of $\omega_{1}$ then $U_{k}\in W^{2,2}(\mathbb{R}^{N})$. Therefore from (13) we obtain

[TABLE]

where $g_{k}$ is given by

[TABLE]

and we have extended $g_{k}$ by [math] outside of $\omega_{1}.$

Now, applying Proposition 3 to the above differential equality we get

[TABLE]

Whence, by using (15) we get

[TABLE]

and thus by the discrete Cauchy-Schwarz inequality we deduce

[TABLE]

and thus

[TABLE]

Hence, by (14) we get

[TABLE]

To complete the proof, we will show the boundedness of $(g_{k})$ in $L^{2}(\omega_{1})$. Indeed, $\rho$ and its derivatives, $a_{ij}$ and their first derivatives are bounded on $\omega_{1}$, moreover (4) shows that the sequences $\left(\epsilon_{k}\nabla_{X_{1}}u_{k}\right)$, $\left(\nabla_{X_{2}}u_{k}\right)$ and $\left(u_{k}\right)$ are bounded in $L^{2}(\Omega),$ and therefore from (4.1) the boundedness of $(g_{k})$ in $L^{2}(\omega_{1})$ follows.

Corollary 1**.**

The sequences $\left(\nabla_{X_{2}}^{2}u_{k}\right)$, $\left(\epsilon_{k}^{2}\nabla_{X_{1}}^{2}u_{k}\right)$, $\left(\epsilon_{k}\nabla_{X_{1}X_{2}}^{2}u_{k}\right)$ are bounded in $L_{loc}^{2}(\Omega).$

Proof.

Let $\omega\subset\subset\Omega$ open, for every $z\in\bar{\omega}$ there exists $\omega_{z}\subset\subset\Omega$, $z\in\omega_{z}$ which satisfies the affirmations of **Proposition 4 **in $L^{2}(\omega_{z})$. By using the compacity of $\bar{\omega}$, one can extract a finite cover $(\omega_{z_{i}})$, and hence the sequences $\left(\nabla_{X_{2}}^{2}u_{k}\right)$, $\left(\epsilon_{k}^{2}\nabla_{X_{1}}^{2}u_{k}\right)$, $\left(\epsilon_{k}\nabla_{X_{1}X_{2}}^{2}u_{k}\right)$ are bounded in $L^{2}(\omega)$.

Proposition 5**.**

Let $z_{0}\in\Omega$ then there exists $\omega_{0}\subset\subset\Omega$, $z_{0}\in\omega_{0}$ such that

[TABLE]

Proof.

Let $z_{0}\in\Omega$ fixed and let $\theta>0$ then using the continuity of the $a_{ij}$ one can choose $\omega_{1}\subset\subset\Omega$, $z_{0}\in\omega_{1}$ such that we have (15) with $\theta$ is chosen as in (14). Let $\omega_{0}\subset\subset\omega_{1},$ with $z_{0}\in\omega_{0},$ and let $\rho\in\mathcal{D}(\mathbb{R}^{N})$ with $\rho=1$ on $\omega_{0}$, $0\leq\rho\leq 1,$ and $Supp(\rho)\subset\omega_{1}$. Let $0<h<dist(\omega_{1},\partial\Omega),$ we set $\mathcal{W}_{k}^{h}=\rho U_{h}^{k}$, with $U_{k}^{h}=(\tau_{h}u_{k}-u_{k})$ and extend it by [math] on the outside of $\omega_{1}$ then $\mathcal{W}_{k}^{h}\in W^{2,2}(\mathbb{R}^{N})$, therefore using (13) we have$:$

[TABLE]

where

[TABLE]

and $G_{k}^{h}$ is extended by [math] outside of $\omega_{1}.$

Then, as in proof of Proposition 4, we obtain

[TABLE]

To complete the proof, we have to show that $\underset{h\rightarrow 0}{\lim}\underset{k\in\mathbb{N}}{\text{ }\sup}\left\|G_{k}^{h}\right\|_{L^{2}(\omega_{1})}=0$.

Using the boundedness of the $a_{ij}$ and the boundedness of $\rho$ and its derivatives on $\omega_{1}$ we get from (4.1)

[TABLE]

where $M\geq 0$ is independent of $h$ and $k.$ Now, estimating the fifth term of the right hand side of the above inequality

[TABLE]

where $C_{q,N}>0$ is only depends in $q$ and $N$.

Let $\delta>0$ small enough such that for every $\left|h\right|\leq\delta$ we have $\omega_{1}+h$ $\subset\subset\Omega$. Then it follows by **Corollary 1, **applied on $\omega_{1}+h$, that the quantity

[TABLE]

is uniformly bounded in $k$ and $h$ (for $\left|h\right|\leq\delta$). Since the $a_{ij}$ are uniformly continuous on every $\omega\subset\subset\Omega$ open then

[TABLE]

and hence

[TABLE]

Now, estimating the last term of (4.1). By the triangular inequality we obtain

[TABLE]

and thus, by using the boundedness of the first derivatives of the $a_{ij}$ on $\omega_{1}$ we get

[TABLE]

where $M^{\prime}\geq 0$ and $C_{q,N}^{\prime}>0$ are independent of $h$ and $k$. Now, since the $\partial_{i}a_{ij}$ are uniformly continuous (recall that $A\in C^{1}(\Omega)$) on every $\omega\subset\subset\Omega$ then

[TABLE]

and therefore, from the above inequality we get

[TABLE]

where we have used (4) and Lemma 2.

Passing to the limit in (4.1) by using (19), (20) and (4) with Lemma 2 we deduce

[TABLE]

and the proposition follows.

Corollary 2**.**

For every $\omega\subset\subset\Omega$ open we have

[TABLE]

Proof.

Similar to proof of Corollary 1,where we use the compacity of $\bar{\omega}$ and Proposition 5.

Now, we are able to give the proof of the main theorem. Indeed it is similar to proof of **Theorem 2, **where we will use **Corollary 1 **and **Corollary 2. **Let us prove the convergence

[TABLE]

Fix $\omega\subset\subset\Omega$ open, and let $u_{k}\in W_{0}^{1,2}(\Omega)\cap W_{loc}^{2,2}(\Omega)$ be a sequence of solutions of (1), then it follows from** Corollary 1** and **2 **that the subset $\left\{\epsilon_{k}^{2}\nabla_{X_{1}}^{2}u_{k}\right\}_{k\in\mathbb{N}}$ is relatively compact in $L^{2}(\omega)$ then there exists $v^{\omega}\in L^{2}(\omega)$ and a subsequence still labeled $(\epsilon_{k}^{2}\nabla_{X_{1}}^{2}u_{k})$ such that

[TABLE]

and since $\epsilon_{k}^{2}u_{k}\rightarrow 0$ in $L^{2}(\omega)$ then $v^{\omega}=0$ (we used the continuity of $\nabla_{X_{1}}^{2}$on $\mathcal{D}^{{}^{\prime}}(\omega)$). Hence by the diagonal process one can construct a sequence still labeled $(\epsilon_{k}^{2}\nabla_{X_{1}}^{2}u_{k})$ such that

[TABLE]

To prove the convergence for the whole sequence $(\epsilon^{2}\nabla_{X_{1}}^{2}u_{\epsilon})_{0<\epsilon\leq 1}$, we can reason by contradiction (recall that $L_{loc}^{2}(\Omega)$ equipped with the family of semi norms $(\left\|\cdot\right\|_{L^{2}(\omega)})_{\omega\subset\subset\Omega}$ is a Fréchet space), and the proof of the main theorem is finished.

4.2. A convergence result for some class of semilinear problem

In this section we deal with the following semilinear elliptic problem

[TABLE]

where $a:\mathbb{R}\rightarrow\mathbb{R}$ a continuous nonincreasing real valued function which satisfies the growth condition

[TABLE]

for some $c\geq 0.$ This problem has been treated in [6] for $f\in L^{p}(\Omega)$, $1<p\leq 2$, and the author have proved the convergences

[TABLE]

where $u_{0}$ is the solution of the limit problem.

Let $f\in L^{2}(\Omega)$ and assume $A$ as in Theorem 1 then the unique $W_{0}^{1,2}(\Omega)$ weak solution $u_{\epsilon}$ to (21) belongs to $W_{loc}^{2,2}(\Omega)$. Following the same arguments exposed in the above subsection one can prove the theorem

Theorem 3**.**

Under the above assumptions we have $u_{\epsilon}\rightarrow u_{0}$ in $V_{loc}^{2,2}$, $\epsilon^{2}\nabla_{X_{1}}^{2}u_{\epsilon}\rightarrow 0$ and $\epsilon\nabla_{X_{1}X_{2}}^{2}u_{\epsilon}\rightarrow 0$ strongly in $L_{loc}^{2}(\Omega).$

Proof.

The arguments are similar, we only give the proof for the Laplacian case, so assume that $A=Id$.

Let $\omega\subset\subset\Omega$ open, then one can choose $\omega^{\prime}$ open such that $\omega\subset\subset\omega^{\prime}\subset\subset\Omega,$ let $\rho\in\mathcal{D}(\mathbb{R}^{N})$ with $\rho=1$ on $\omega$, $0\leq\rho\leq 1$ and $Supp(\rho)\subset\omega^{\prime}$. Let $0<h<dist(\partial\omega^{\prime},$ $\Omega),$ we use the same notations of the above subsection, we set $U_{k}^{h}=\tau_{h}u_{k}-u_{k}$, then $U_{k}^{h}\in W^{1,2}(\omega^{\prime})$ and we have

[TABLE]

with $F^{h}=\tau_{h}f-f$. We set $\mathcal{W}_{k}^{h}=\rho U_{k}^{h}$ then we get as in Proposition 2

[TABLE]

We can prove easily, by using the continuity of the function $a$ and (22), that the Nemytskii operator $a$ maps continuously $L^{2}(\Omega)$ to $L^{2}(\Omega)$. Therefore, the convergence $u_{k}\rightarrow u_{0}$ in $L^{2}(\Omega)$ gives $a(u_{k})\rightarrow a(u_{0})$ in $L^{2}(\Omega)$, and hence **Lemma 2 **gives

[TABLE]

and finally the convergences (23) give

[TABLE]

Similarly, using boundedness of the sequences $(u_{k})$, $(\epsilon_{k}\nabla_{X_{1}}u_{k})$, $(\nabla_{X_{2}}u_{k})$and $a(u_{k})$ in $L^{2}(\Omega),$ and boundedness of $\rho$ and its derivatives we get

[TABLE]

and we conclude as in proof of Theorem 2.

We complete this paper by giving an open question

Problem 1**.**

Let $f\in L^{p}(\Omega)$ with $1<p<2$, and consider problem (1). In [6] the author have proved the convergence $u_{\epsilon}\rightarrow u_{0}$ in the Banach space $V^{1,p}$ defined by

[TABLE]

equipped with the norm

[TABLE]

Similarly we introduce the Fréchet space

[TABLE]

equipped with family of norms

[TABLE]

Can one prove that $u_{\epsilon}\rightarrow u_{0}$ in $V_{loc}^{2,p}$?