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

TL;DR

This paper studies how solutions to the Laplace equation behave in a domain with a tiny hole near the boundary, showing that the solution depends analytically on the hole's size and position as it shrinks.

Contribution

It introduces a new analytical framework for understanding the dependence of Laplace equation solutions on small boundary-adjacent holes using layer potentials.

Findings

Solution map is real analytic near zero parameters.

Solution behavior is well-characterized as the hole shrinks.

Analytic continuation provides insights into boundary perturbations.

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 $\epsilon\rightarrow(0, 0)$, 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 $(0, 0)$.