Asymptotic stability of the spectrum of a parametric family of homogenization problems associated with a perforated waveguide

Abstract

In this paper, we provide uniform bounds for convergence rates of the low frequencies of a parametric family of problems for the Laplace operator posed on a rectangular perforated domain of the plane of height 𝐻. The perforations are periodically placed along the ordinate axis at a distance 𝑂(𝜀) between them, where 𝜀 is a parameter that converges toward zero. Another parameter 𝜂, the Floquet-parameter, ranges in the interval [−𝜋, 𝜋]. The boundary conditions are quasi-periodicity conditions on the lateral sides of the rectangle and Neumann over the rest. We obtain precise bounds for convergence rates which are uniform on both parameters 𝜀 and 𝜂 and strongly depend on 𝐻. As a model problem associated with a waveguide, one of the main difficulties in our analysis comes near the nodes of the limit dispersion curves.