# Convergence of Dirichlet eigenvalues for elliptic systems on perturbed domains

### Justin L. Taylor

Murray State University, USA

## Abstract

We consider the eigenvalues of an elliptic operator

where $u=(u_{1},…,u_{m})_{t}$ is a vector valued function and $a_{αβ}(x)$ are $(n×n)$ matrices whose elements $a_{ij}(x)$ are at least uniformly bounded measurable real-valued functions such that

for any combination of $α,β,i,$ and $j$. We assume we have two non-empty, open, disjoint, and bounded sets, $Ω$ and $Ω~$, in $R_{n}$, and add a set $T_{ε}$ of small measure to form the domain $Ω_{ε}$. Then we show that as $ε→0_{+}$, the Dirichlet eigenvalues corresponding to the family of domains ${Ω_{ε}}_{ε>0}$ converge to the Dirichlet eigenvalues corresponding to $Ω_{0}=Ω∪Ω~$. Moreover, our rate of convergence is independent of the eigenvalues. In this paper, we consider the Lamé system, systems which satisfy a strong ellipticity condition, and systems which satisfy a Legendre–Hadamard ellipticity condition.

## Cite this article

Justin L. Taylor, Convergence of Dirichlet eigenvalues for elliptic systems on perturbed domains. J. Spectr. Theory 3 (2013), no. 3, pp. 293–316

DOI 10.4171/JST/46