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

TL;DR

This paper investigates how solutions to the Laplace equation behave in domains with small holes near the boundary, revealing different asymptotic behaviors and analyticity properties depending on the dimension and the approach to the limit.

Contribution

It provides a detailed analysis of the asymptotic behavior and analyticity of solutions in perforated domains with small holes near the boundary, including dimension-specific regimes.

Findings

For n≥3, the solution map is real analytic near the limit.

For n=2, solutions exhibit logarithmic behavior as parameters tend to zero.

Different regimes in n=2 lead to distinct limiting values of energy and flux.

Abstract

We study the Dirichlet problem in a domain with a small hole close to the boundary. To do so, for each pair $\boldsymbol\varepsilon = (\varepsilon_1, \varepsilon_2 )$ of positive parameters, we consider a perforated domain $\Omega_{\boldsymbol\varepsilon}$ obtained by making a small hole of size $\varepsilon_1 \varepsilon_2 $ in an open regular subset $\Omega$ of $\mathbb{R}^n$ at distance $\varepsilon_1$ from the boundary $\partial\Omega$. As $\varepsilon_1 \to 0$, the perforation shrinks to a point and, at the same time, approaches the boundary. When $\boldsymbol\varepsilon \to (0,0)$, the size of the hole shrinks at a faster rate than its approach to the boundary. We denote by $u_{\boldsymbol\varepsilon}$ the solution of a Dirichlet problem for the Laplace equation in $\Omega_{\boldsymbol\varepsilon}$. For a space dimension $n\geq 3$, we show that the function mapping $\boldsymbol\varepsilon$ to $u_{\boldsymbol\varepsilon}$ has a real analytic continuation in a neighborhood of $(0,0)$. By contrast, for $n=2$ we consider two different regimes: $\boldsymbol\varepsilon$ tends to $(0,0)$, and $\varepsilon_1$ tends to $0$ with $\varepsilon_2$ fixed. When $\boldsymbol\varepsilon\to(0,0)$, the solution $u_{\boldsymbol\varepsilon}$ has a logarithmic behavior; when only $\varepsilon_1\to0$ and $\varepsilon_2$ is fixed, the asymptotic behavior of the solution can be described in terms of real analytic functions of $\varepsilon_1$. We also show that for $n=2$, the energy integral and the total flux on the exterior boundary have different limiting values in the two regimes. We prove these results by using functional analysis methods in conjunction with certain special layer potentials.