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

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