Asymptotic property for eigenelements of the Laplace operator in a domain with an oscillating boundary

Abstract

AbstractThis work deals with the variations of eigenvalues and eigenfunctions for the Laplace operator in a bounded domain of with Dirichlet boundary conditions, a part of which boundary, depending on a small parameter ε, is rapidly oscillating. By using surface potentials we show that the eigenvalues are exactly built with the characteristic values of meromorphic operator‐valued functions that are of Fredholm type with index 0. Then, we proceed from the generalized Rouché's theorem to study the splitting problem.