Shape analyticity and singular perturbations for layer potential operators

TL;DR

This paper investigates how small and regular domain changes affect layer potential operators for the Laplace equation, providing analyticity results and power series expansions for perforated domains.

Contribution

It introduces new analyticity results for layer potentials under domain perturbations and develops power series expansions for perforated domains as the hole size approaches zero.

Findings

Layer potentials depend real analytically on domain deformations.

Power series expansions describe layer potentials on perforated domains.

Results facilitate understanding of boundary perturbation effects.

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 $\phi(\partial\Omega)$ of a reference set $\partial\Omega$ and we present some real analyticity results for the dependence upon the map $\phi$. 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 $\epsilon=0$.