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 similar to and ! of R2, containing the origin. First, if " is close to 0 and if u is a function defined on similar to, we compute an asymptotic expansion of the u-capacity Cap similar to("!; u) as " ! 0. As a byproduct, we compute an asymptotic expansion for the Nth eigenvalues of the Dirichlet-Laplacian in the perforated set similar to n ("!) for " 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 ! of the hole.