Convergence of Dirichlet eigenvalues for elliptic systems on perturbed domains
Justin L. Taylor · Journal of Spectral Theory · 2013
We consider the eigenvalues of an elliptic operator (Lu)^{\beta}=-\frac{\partial}{\partial x_j} (a^{\alpha \beta}_{ij}\frac{\partial u^{\alpha}}{\partial x_i}),\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.