Convergence of Dirichlet eigenvalues for elliptic systems on perturbed domains

Abstract

We consider the eigenvalues of an elliptic operator

where $u=(u^1,\dots ,u^m{)}^t$ is a vector valued function and $a^{\alpha \beta }(x)$ are $(n\times n)$ matrices whose elements $a_{ij}^{\alpha \beta }(x)$ are at least uniformly bounded measurable real-valued functions such that

for any combination of $\alpha ,\beta ,i,$ and $j$. We assume we have two non-empty, open, disjoint, and bounded sets, $\Omega$ and $\tilde{\Omega }$, in ${\mathbb{R}}^n$, and add a set $T_{\epsilon }$ of small measure to form the domain ${\Omega }_{\epsilon }$. Then we show that as $\epsilon \to 0^{+}$, the Dirichlet eigenvalues corresponding to the family of domains $\{{\Omega }_{\epsilon }{\}}_{\epsilon >0}$ converge to the Dirichlet eigenvalues corresponding to ${\Omega }_0=\Omega \cup \tilde{\Omega }$. 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.