Convergence of Dirichlet eigenvalues for elliptic systems on perturbed domains (Q361719)

Summary: We consider the eigenvalues of an elliptic operator \[ (Lu)^{\beta}=-\frac{\partial}{\partial x_j}\left(a^{\alpha\beta}_{ij}\frac{\partial u^{\alpha}}{\partial x_i}\right),\quad \beta=1,\ldots,m, \] where \(u=(u^1\ldots,u^m)^t\) is a vector valued function and \(a^{\alpha\beta}(x)\) are \((n \times n)\) matrices whose elements \(a^{\alpha \beta}_{ij}(x)\) are at least uniformly bounded measurable real-valued functions such that \[ a^{\alpha \beta}_{ij}(x)=a^{\beta \alpha}_{ji}(x) \] 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 \(T_{\varepsilon}\) of small measure to form the domain \(\Omega_{\varepsilon}\). Then we show that as \(\varepsilon \rightarrow 0^+\), the Dirichlet eigenvalues corresponding to the family of domains \(\{\Omega_{\varepsilon}\}_{\varepsilon>0}\) 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.