ASYMPTOTICS OF A SPECTRAL PROBLEM ASSOCIATED WITH THE NEUMANN SIEVE

Abstract

In this paper, we analyze the asymptotic behavior of a Stekloff spectral problem associated with the Neumann Sieve model, i.e. a three-dimensional set Ω, cut by a hyperplane Σ where each of the two-dimensional holes, ∊-periodically distributed on Σ, have diameter r∊. Depending on the asymptotic behavior of the ratios [Formula: see text] we find the limit problem of the ∊ spectral problem and prove that the sequences [Formula: see text], formed by the nth eigenvalue of the ∊ problem, converge to λn, the nth eigenvalue of the limit problem, for any n ∈ N. We also prove the weak convergence, on a subsequence, of the associated sequence of eigenvectors [Formula: see text], to an eigenvector associated with λn. When λn is a simple eigenvalue, we show that the entire sequence of the eigenvectors converges. As a consequence, similar results hold for the spectrum of the DtN map associated to this model.