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

TL;DR

This paper presents an explicit method to approximate relaxed Dirichlet problems involving elliptic operators and measures by boundary value problems on perforated domains, ensuring convergence of solutions.

Contribution

It introduces a novel approximation procedure for relaxed Dirichlet problems using perforated domains, bridging boundary value problems and measure-driven elliptic equations.

Findings

Solutions of boundary problems on perforated domains converge to the relaxed problem solution.

The method explicitly constructs domain sequences for approximation.

The approach handles measures not charging polar sets effectively.

Abstract

Given an elliptic operator~$L$ on a bounded domain~$\Omega \subseteq {\bf R}^n$, 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\ $L\,u+\mu u=f$ in~$\Omega$, $u=0$ on~$\partial \Omega$.