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Zbl 1170.35377
Biegert, Markus; Daners, Daniel
Local and global uniform convergence for elliptic problems on varying domains. (English)
[J] J. Differ. Equations 223, No. 1, 1-32 (2006). ISSN 0022-0396

Let $u_n$ be the unique (weak) solution of $$-\Delta u+\lambda u= f\quad\text{in }\Omega_n,$$ $$u= 0\quad\text{on }\partial\Omega,$$ where $\Omega_n$ is a given sequence of open sets in $\Bbb R^N$, $\lambda> 0$ and $f_n\in L^\infty(\Bbb R^N)$. The authors extend $u_n$ by zero outside $\Omega_n$ to get a sequence of functions defined on $\Bbb R^N$. The purpose of this paper is to study necessary and sufficient conditions on $\Omega$ providing uniform convergence, that is, convergence in $L^\infty(\Bbb R^N)$ of $u_n$ to the solution of $$-\Delta u+\lambda u= f\quad\text{in }\Omega,$$ $$u= 0\quad\text{on }\partial\Omega$$ on a limit domain $\Omega$. The authors deal with two cases, namely local and global uniform convergence.
[Messoud A. Efendiev (Berlin)]

MSC 2000:: *35J25 Second order elliptic equations, boundary value problems
35B27 Homogenization, etc.

Keywords: Elliptic partial differential equations; Domain perturbation; Uniform convergence; Shape stability

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