Asymptotic behavior of $u$-capacities and singular perturbations for the Dirichlet-Laplacian

TL;DR

This paper analyzes the asymptotic behavior of $u$-capacities of small sets and applies these results to understand how small holes in a domain affect the eigenvalues of the Dirichlet-Laplacian, providing explicit formulas.

Contribution

It provides new asymptotic expansions for $u$-capacities and eigenvalues in perforated domains, revealing their dependence on the shape and eigenfunction behavior.

Findings

Asymptotic expansion of $u$-capacity as $ o 0$

Explicit formula for eigenvalue shifts due to small holes

Dependence of eigenvalue asymptotics on hole shape and eigenfunction

Abstract

In this paper we study the asymptotic behavior of $u$-capacities of small sets and its application to the analysis of the eigenvalues of the Dirichlet-Laplacian on a bounded planar domain with a small hole. More precisely, we consider two (sufficiently regular) bounded open connected sets $\Omega$ and $\omega$ of $\mathbb{R}^2$, containing the origin. First, if $\varepsilon$ is positive and small enough and if $u$ is a function defined on $\Omega$, we compute an asymptotic expansion of the $u$-capacity $\mathrm{Cap}_\Omega(\varepsilon \omega, u)$ as $\varepsilon \to 0$. As a byproduct, we compute an asymptotic expansion for the $N$-th eigenvalues of the Dirichlet-Laplacian in the perforated set $\Omega \setminus (\varepsilon \overline{\omega})$ for $\varepsilon$ close to $0$. Such formula shows explicitly the dependence of the asymptotic expansion on the behavior of the corresponding eigenfunction near $0$ and on the shape $\omega$ of the hole.