Approximation of Relaxed Dirichlet Problems by Boundary Value problems in perforated domains

Abstract

Given an elliptic operator~$L$ on a bounded domain~$\Omega \subseteq {\mathbf{R}}^n$, and a positive Radon measure~$\mu$ on~$\Omega$, not charging polar sets, we discuss an explicit approximation procedure which leads to a sequence of domains~${\Omega }_h\subseteq \Omega$ with the following property: for every~$f\in H^{-1}(\Omega )$ the sequence~$u_h$ of the solutions of the Dirichlet problems~$L{u}_h=f$ in~${\Omega }_h$, $u_h=0$ on~$\partial {\Omega }_h$, extended to 0 in~$\Omega \setminus {\Omega }_h$, converges to the solution of the \lq\lq relaxed Dirichlet problem\rq\rq\ $Lu+\mu u=f$ in~$\Omega$, $u=0$ on~$\partial \Omega$.

19 pages