Singular asymptotic expansions for Dirichlet eigenvalues and eigenfunctions of the Laplacian on thin planar domains

TL;DR

This paper develops asymptotic expansions for Dirichlet eigenvalues and eigenfunctions of the Laplacian on thin planar domains, providing approximations for the spectrum as the domain is scaled in one direction.

Contribution

It introduces a method to derive singular asymptotic expansions for eigenvalues and eigenfunctions under domain scaling, applicable to a broad class of planar domains.

Findings

Derived asymptotic formulas for eigenvalues and eigenfunctions

Provided approximations for the first Dirichlet eigenvalue in scaled domains

Applicable to a wide range of planar domains with mild assumptions

Abstract

We consider the Laplace operator with Dirichlet boundary conditions on a planar domain and study the effect that performing a scaling in one direction has on the spectrum. We derive the asymptotic expansion for the eigenvalues and corresponding eigenfunctions as a function of the scaling parameter around zero. This method allows us, for instance, to obtain an approximation for the first Dirichlet eigenvalue for a large class of planar domains, under very mild assumptions.