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

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.