  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.
