Advances in Differential Equations

Singular behavior of the solution of the Helmholtz equation in weighted $L^p$-Sobolev spaces

Colette De Coster and Serge Nicaise

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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.

Article information

Adv. Differential Equations, Volume 16, Number 1/2 (2011), 165-198.

First available in Project Euclid: 18 December 2012

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 35J05: Laplacian operator, reduced wave equation (Helmholtz equation), Poisson equation [See also 31Axx, 31Bxx] 35B65: Smoothness and regularity of solutions


De Coster, Colette; Nicaise, Serge. Singular behavior of the solution of the Helmholtz equation in weighted $L^p$-Sobolev spaces. Adv. Differential Equations 16 (2011), no. 1/2, 165--198.

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