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 ω and ω of R 2, containing the origin. First, if ϵ is close to 0 and if u is a function defined on ω, we compute an asymptotic expansion of the u-capacity Capω(ϵω¯,u) as ϵ → 0. As a byproduct, we compute an asymptotic expansion for the Nth eigenvalues of the Dirichlet-Laplacian in the perforated set ω(ϵω¯) for ϵ 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 ω of the hole.