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January/February 2011 Singular behavior of the solution of the Helmholtz equation in weighted $L^p$-Sobolev spaces
Colette De Coster, Serge Nicaise
Adv. Differential Equations 16(1/2): 165-198 (January/February 2011).

Abstract

We study the Helmholtz equation \[ (1)\quad -\Delta u+zu=g \hbox{ in } \Omega, \] with Dirichlet boundary conditions in a polygonal domain $\Omega$, where $z$ is a complex number. Here $g$ belongs to $L^p_\mu(\Omega)=\{v \in L^p_{loc}(\Omega): r^\mu v\in L^p(\Omega)\},$ with a real parameter $\mu$ and $r(x)$ the distance from $x$ to the set of corners of $\Omega$. We give sufficient conditions on $\mu,$ $ p$, and $\Omega$ that guarantee that problem (1) has a unique solution $u\in H^1_0(\Omega)$ that admits a decomposition into a regular part in weighted $L^p$-Sobolev spaces and an explicit singular part. We further obtain some estimates where the explicit dependence on $|z|$ is given.

Citation

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Colette De Coster. Serge Nicaise. "Singular behavior of the solution of the Helmholtz equation in weighted $L^p$-Sobolev spaces." Adv. Differential Equations 16 (1/2) 165 - 198, January/February 2011.

Information

Published: January/February 2011
First available in Project Euclid: 18 December 2012

zbMATH: 1219.35049
MathSciNet: MR2766898

Subjects:
Primary: 35B65, 35J05

Rights: Copyright © 2011 Khayyam Publishing, Inc.

JOURNAL ARTICLE
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Vol.16 • No. 1/2 • January/February 2011
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