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

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(& epsilon;)$O(\varepsilon )$ between them, where & epsilon; is a parameter that converges toward zero. Another parameter & eta;, the Floquet-parameter, ranges in the interval [-& pi;,& pi;]$[-\pi ,\pi ]$. 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 & epsilon; and & eta; 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.

Automated summary

In this paper, we establish uniform bounds for the convergence rates of low frequencies in a parametric family of Laplace operator problems on a rectangular perforated domain. The perforations are periodically placed at a distance O(ε) along the ordinate axis, where ε is a parameter that tends to zero. The Floquet-parameter, η, varies within the interval [-π, π]. The boundary conditions consist of quasi-periodicity conditions on the lateral sides of the rectangle and Neumann conditions elsewhere. Precise bounds for the convergence rates, which are uniform on both ε and η parameters, are obtained and heavily depend on the height H. The analysis is particularly challenging near the nodes of the limit dispersion curves.