Convergence of Dirichlet Eigenvalues for Elliptic Systems on Perturbed Domains

TL;DR

This paper proves that Dirichlet eigenvalues of elliptic systems on domains with small perturbations converge to those of the original domain, with a rate independent of the eigenvalues, covering various ellipticity conditions.

Contribution

It establishes convergence of eigenvalues for elliptic systems on perturbed domains, including the Lamé system and systems with different ellipticity conditions, with a rate independent of eigenvalue index.

Findings

Eigenvalues converge as domain perturbation diminishes

Convergence rate is independent of eigenvalue index

Applicable to Lamé and other elliptic systems

Abstract

We consider the eigenvalues of an elliptic operator for systems with bounded, measurable, and symmetric coefficients. We assume we have two non-empty, open, disjoint, and bounded sets and add a set of small measure to form the perturbed domain. Then we show that the Dirichlet eigenvalues corresponding to the family of perturbed domains converge to the Dirichlet eigenvalues corresponding to the unperturbed domain. Moreover, our rate of convergence is independent of the eigenvalues. In this paper, we consider the Lam\'{e} system, systems which satisfy a strong ellipticity condition, and systems which satisfy a Legendre-Hadamard ellipticity condition.