Calculus of Variations and Geometric Measure Theory

V. Felli - B. Noris - R. Ognibene

Eigenvalues of the Laplacian with moving mixed boundary conditions: the case of disappearing Dirichlet region

created by ognibene on 15 Oct 2021


Published Paper

Inserted: 15 oct 2021

Journal: Calculus of Variations and Partial Differential Equations
Year: 2021

ArXiv: 2001.07989 PDF


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.