Asymptotic behavior of u-capacities and singular perturbations for the Dirichlet-Laplacian
Résumé
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 positive and small enough and if u is a function defined on Ω, we compute an asymptotic expansion of the u-capacity Cap Ω (εω, u) as ε → 0. As a byproduct, we compute an asymptotic expansion for the N -th 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.
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