  In this work we consider the homogeneous Neumann eigenvalue problem for the
Laplacian on a bounded Lipschitz domain and a singular perturbation of it,
which consists in prescribing zero Dirichlet boundary conditions on a small
subset of the boundary. We first describe the sharp asymptotic behaviour of a
perturbed eigenvalue, in the case in which it is converging to a simple
eigenvalue of the limit Neumann problem. The first term in the asymptotic
expansion turns out to depend on the Sobolev capacity of the subset where the
perturbed eigenfunction is vanishing. Then we focus on the case of Dirichlet
boundary conditions imposed on a subset which is scaling to a point; by a
blow-up analysis for the capacitary potentials, we detect the vanishing order
of the Sobolev capacity of such shrinking Dirichlet boundary portion.
