  We provide a full series expansion of a generalization of the so-called
$u$-capacity related to the Dirichlet-Laplacian in dimension three and higher,
extending previous results of the authors, and of the authors together with
Virginie Bonnaillie-No\"el, dealing with the planar case. We apply the result
in order to study the asymptotic behavior of perturbed eigenvalues when
Dirichlet conditions are imposed on a small regular subset of the domain of the
eigenvalue problem.
