Shape analyticity and singular perturbations for layer potential operators

Abstract

We study the effect of regular and singular domain perturbations on layer potential operators for the Laplace equation. First, we consider layer potentials supported on a diffeomorphic image $\varphi (\partial \Omega )$ of a reference set $\partial \Omega$ and we present some real analyticity results for the dependence upon the map $\varphi$. Then we introduce a perforated domain $\Omega (\epsilon )$ with a small hole of size $\epsilon$ and we compute power series expansions that describe the layer potentials on $\partial \Omega (\epsilon )$ when the parameter $\epsilon$ approximates the degenerate value $\epsilon =0$.