Geometric versus spectral convergence for the Neumann Laplacian under exterior perturbations of the domain

TL;DR

This paper investigates how the eigenvalues and eigenfunctions of the Neumann Laplacian change under exterior domain perturbations, revealing that spectral convergence implies measure convergence but not Hausdorff distance convergence.

Contribution

It establishes a link between spectral convergence and measure of domain perturbations, providing examples where spectra converge despite unbounded Hausdorff distance.

Findings

Spectral convergence implies measure of perturbation tends to zero.

Spectral convergence does not necessarily imply Hausdorff distance tends to zero.

Constructed example shows spectra can converge with unbounded Hausdorff distance.

Abstract

We analyze the behavior of the eigenvalues and eigenfunctions of the Laplace operator with homogeneous Neumann boundary conditions when the domain is perturbed. We focus on exterior perturbations of the domain, that is, the limit domain is contained in each of the perturbed domains. Moreover, the family of perturbed domains is made up of bounded domains although not necessarily uniformly bounded. We show that if we know that the eigenvalues and eigenfunctions of the perturbed domains converge to the ones of the limit domain, then necessarily we have that the measure of the perturbation tends to zero, while it is not necessarily true for the Hausdorff distance of the perturbation. As a matter of fact we will construct an example of a perturbation where the spectra behave continuously but the distance tends to infinity.