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

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 $\mathrm{\Omega }$ 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 $\mathrm{\Omega }$, we compute an asymptotic expansion of the $u$-capacity ${\mathrm{C}\mathrm{a}\mathrm{p}}_{\mathrm{\Omega }}(\epsilon \omega ,u)$ 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 $\Omega \setminus (\varepsilon \overline{\omega})$for$\varepsilon$closeto$0$. 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.