Sharp behavior of Dirichlet--Laplacian eigenvalues for a class of singularly perturbed problems

TL;DR

This paper analyzes how Dirichlet eigenvalues in bounded domains are affected by attaching a thin tube, providing sharp asymptotic descriptions of the spectrum's behavior as the tube shrinks, with applications to Neumann boundary perturbations.

Contribution

It introduces a quantitative measure for eigenvalue perturbations and describes their asymptotic behavior in singularly perturbed boundary problems, including multiple eigenvalues.

Findings

Eigenvalues converge with a rate measured by a torsional rigidity-like quantity.

Asymptotic formulas describe eigenvalue behavior even for multiple eigenvalues.

Techniques extend to Neumann boundary perturbations on small boundary parts.

Abstract

We deepen the study of Dirichlet eigenvalues in bounded domains where a thin tube is attached to the boundary. As its section shrinks to a point, the problem is spectrally stable and we quantitatively investigate the rate of convergence of the perturbed eigenvalues. We detect the proper quantity which sharply measures the perturbation's magnitude. It is a sort of torsional rigidity of the tube's section relative to the domain. This allows us to sharply describe the asymptotic behavior of the perturbed spectrum, even when eigenvalues converge to a multiple one. The final asymptotics of eigenbranches depend on the local behavior near the junction of eigenfunctions chosen in a special way. The present techniques also apply when the perturbation of the Dirichlet eigenvalue problem consists in prescribing homogeneous Neumann boundary conditions on a small portion of the boundary of the domain.