Asymptotic behavior for eigenvalues and eigenfunctions associated to Stokes operator in the presence of a rapidly oscillating boundary

Abstract

In this paper, we analyze rigorously the asymptotic behaviors for perturbation of eigenvalues and eigenfunctions associated to the Stokes eigenvalue problem in the presence of a rapidly oscillating boundary. The aim is to construct asymptotic approximations, as , of the eigenvalues and corresponding eigenfunctions for the case where the eigenvalue of the reference problem is simple or multiple. Taking advantage of small oscillations, we use the method of matching of asymptotic expansions to construct the associated leading terms. We believe that our results are ambitious tools for determining shape and/or size of the small perturbed part of the domain by taking eigenvalue measurements.