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Zbl 1124.47055
Arendt, Wolfgang; Daners, Daniel
Uniform convergence for elliptic problems on varying domains. (English)
[J] Math. Nachr. 280, No. 1-2, 28-49 (2007). ISSN 0025-584X; ISSN 1522-2616

In this paper, the Poisson equation $$\cases \lambda u-\Delta u= f&\text{in }D(\Omega)',\\ u/\partial\Omega= 0,\endcases\tag{$P_\Omega$}$$ is considered, where $\Omega$ is an open set in $\bbfR^N$ and $\lambda\ge 0$. If $\Omega$ is Dirichlet regular, then for each $f\in L^\infty(\Omega)$ there exists a unique solution $u\in C(\overline\Omega)$ solving the problem $(P_\Omega)$. For given sequence $\{\Omega_n\}$ of open sets $\Omega_n\subset\bbfR^N$, $n\in\bbfN$, the notion of regular convergence of $\Omega_n$ to $\Omega$ as $n\to\infty$ is introduced, which involves somehow Dirichlet regularity. Consider the solutions $u_n$ of $(P_{\Omega_n})$. Extending $u_n$ and $u$ by zero to $\bbfR^N$, we obtain uniformly bounded functions defined on $\bbfR^N$. The main theorem shows that regular convergence of $\Omega_n$ to $\Omega$ implies that the solutions $u_n$ of $(P_{\Omega_n})$ converge locally uniformly on $\bbfR^N$ to the solution $u$ of $(P_\Omega)$. At the end, many examples of regularly converging sequences are given, showing that the notion of regular convergence is very general. Applications to spectral theory, parabolic equations and nonlinear equations are given, too.
[A. Cichocka (Katowice)]

MSC 2000:: *47N20 Appl. of operator theory to differential and integral equations
35K05 Heat equation
35J25 Second order elliptic equations, boundary value problems
35B25 Singular perturbations (PDE)
47D06 One-parameter semigroups and linear evolution equations
47F05 Partial differential operators

Keywords: elliptic boundary value problem; domain variation; uniform convergence; nonlinear elliptic equation

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