Eigenvalues and eigenfunctions of double layer potentials

Abstract:

Eigenvalues and eigenfunctions of two- and three-dimensional double layer potentials are considered. Let $\Omega$ be a $C^2$ bounded region in ${\mathbf {R}}^n$ ($n=2, 3$). The double layer potential $K: L^2(\partial \Omega ) \rightarrow L^2(\partial \Omega )$ is defined by \[ (K \psi )(x) \equiv \int _{\partial \Omega } \psi (y)\cdot \nu _{y} E(x, y) \; ds_y, \] where \[ E(x, y)= \begin {cases} \frac {1}{\pi } \log \frac {1}{|x-y|}, \quad \;\mbox {if}\; n=2, \\ \frac {1}{2\pi } \frac {1}{|x-y|}, \quad \hspace {4mm}\;\mbox {if} \; n=3, \end {cases} \] $ds_y$ is the line or surface element and $\nu _y$ is the outer normal derivative on $\partial \Omega$. It is known that $K$ is a compact operator on $L^2(\partial \Omega )$ and consists of at most a countable number of eigenvalues, with $0$ as the only possible limit point. This paper aims to establish some relationships among the eigenvalues, the eigenfunctions, and the geometry of $\partial \Omega$.