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.

在这项工作中，我们考虑有界Lipschitz域上的Laplacian的齐次Neumann特征值问题及其奇异摄动，这包括在边界的一小部分上规定零Dirichlet边界条件。我们首先描述一个扰动特征值的尖锐渐近行为，在这种情况下，它正在收敛到极限Neumann问题的一个简单特征值。渐近展开的第一项取决于受干扰本征函数消失的子集的Sobolev容量。然后，我们将重点放在Dirichlet边界条件施加到一个按比例缩放到一个点的子集中的情况；通过对容量潜力的爆破分析，我们检测到了这种收缩的Dirichlet边界部分的Sobolev容量的消失阶。