Asymptotic expansions of eigenvalues and eigenfunctions of an elliptic operator in a domain with many “light” concentrated masses situated on the boundary. Two-dimensional case

We consider vibrations of a membrane which contains many “light” concentrated masses on the boundary. We study the asymptotic behaviour of the frequencies of eigenvibrations of the membrane as the small parameter (which characterizes the diameter and density of the concentrated masses) tends to zero. We construct asymptotic expansions of eigenelements of the corresponding problems and carefully justify these expansions.