Homogenization of the planar waveguide with frequently alternating boundary conditions

TL;DR

This paper studies the homogenization of a planar waveguide with rapidly alternating boundary conditions, showing that the effective operator behaves like a Dirichlet Laplacian and analyzing the spectral properties.

Contribution

It proves the homogenized operator is the Dirichlet Laplacian under certain conditions and provides asymptotic expansions for spectral bands and the spectrum's bottom.

Findings

Homogenized operator is the Dirichlet Laplacian.

Spectrum consists of a band structure with only essential spectrum.

Asymptotic expansions for spectral bands and spectrum bottom are constructed.

Abstract

We consider Laplacian in a planar strip with Dirichlet boundary condition on the upper boundary and with frequent alternation boundary condition on the lower boundary. The alternation is introduced by the periodic partition of the boundary into small segments on which Dirichlet and Neumann conditions are imposed in turns. We show that under the certain condition the homogenized operator is the Dirichlet Laplacian and prove the uniform resolvent convergence. The spectrum of the perturbed operator consists of its essential part only and has a band structure. We construct the leading terms of the asymptotic expansions for the first band functions. We also construct the complete asymptotic expansion for the bottom of the spectrum.