A Dirichlet problem for the Laplace operator in a domain with a small hole close to the boundary

Abstract

We take an open regular domain $\Omega$ in ${\mathbb{R}}^n$ with $n\ge 3$. We introduce a pair of positive parameters ${\epsilon }_1$ and ${\epsilon }_2$ and we set $\epsilon \equiv ({\epsilon }_1,{\epsilon }_2)$. Then we define the perforated domain ${\Omega }_{\epsilon }$ by making in $\Omega$ a small hole of size ${\epsilon }_1{\epsilon }_2$ at distance ${\epsilon }_1$ from the boundary. When $\epsilon \to (0,0)$, the hole approaches the boundary while its size shrinks at a faster rate. In ${\Omega }_{\epsilon }$ we consider a Dirichlet problem for the Laplace equation and we denote its solution by $u_{\epsilon }$. By an approach based on functional analysis and on the introduction of special layer potentials we show that the map which takes $\epsilon$ to (a restriction of) $u_{\epsilon }$ has a real analytic continuation in a neighbourhood of $(0,0)$.

This work has been combined into arXiv:1612.05115