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

Delfina Gómez, Sergei A. Nazarov, Rafael Orive‐Illera and María‐Eugenia Pérez‐Martínez

Mathematische Nachrichten, 2023, vol. 296, issue 10, 4888-4910

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)$O(\varepsilon )$ between them, where ε is a parameter that converges toward zero. Another parameter η, the Floquet‐parameter, ranges in the interval [−π,π]$[-\pi ,\pi ]$. 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 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.

Date: 2023
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https://doi.org/10.1002/mana.202100589

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