Spectral Problem in a Domain Perforated along the Boundary. Perturbation of a Multiple Eigenvalue

Abstract

We consider the boundary value problem for the Laplace operator in a two-dimensional domain which is partially perforated along the boundary. The homogeneous Neumann condition is imposed on the outer boundary, whereas the homogeneous Dirichlet condition is stated on the boundary of small cavities. We construct two-term asymptotics with respect to a small parameter for eigenvalues of this boundary value problem converging to a multiple eigenvalue of the homogenized (limit) problem. We obtain conditions under which a multiple eigenvalue of the homogenized problem splits into simple ones. We also obtain the limit of the corresponding eigenfunctions.