On a waveguide with frequently alternating boundary conditions: homogenized Neumann condition

TL;DR

This paper analyzes a waveguide with periodically alternating boundary conditions, demonstrating homogenization to a Neumann condition and providing detailed spectral and resolvent convergence results with improved estimates.

Contribution

It establishes uniform resolvent convergence for the waveguide with alternating boundary conditions and derives precise asymptotics for spectral properties, including an improved convergence rate.

Findings

Uniform resolvent convergence with explicit rate estimates

Two-term asymptotics for band functions

Complete asymptotic expansion for the spectrum's bottom

Abstract

We consider a waveguide modeled by the Laplacian in a straight planar strip. The Dirichlet boundary condition is taken on the upper boundary, while on the lower boundary we impose periodically alternating Dirichlet and Neumann condition assuming the period of alternation to be small. We study the case when the homogenization gives the Neumann condition instead of the alternating ones. We establish the uniform resolvent convergence and the estimates for the rate of convergence. It is shown that the rate of the convergence can be improved by employing a special boundary corrector. Other results are the uniform resolvent convergence for the operator on the cell of periodicity obtained by the Floquet-Bloch decomposition, the two-terms asymptotics for the band functions, and the complete asymptotic expansion for the bottom of the spectrum with an exponentially small error term.