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

Abstract

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