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

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 $\epsilon$ is positive and small enough and if $u$ is a function defined on $\Omega$, we compute an asymptotic expansion of the $u$-capacity ${\mathrm{Cap}}_{\Omega }(\epsilon \omega ,u)$ as $\epsilon \to 0$. As a byproduct, we compute an asymptotic expansion for the $N$-th eigenvalues of the Dirichlet-Laplacian in the perforated set $\Omega \setminus (\epsilon \overline{\omega })$ for $\epsilon$ 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.

46 pages, 13 figures

References in corpus (1)

• On the sharpness of a certain spectral stability estimate for the Dirichlet Laplacian