Approximation of relaxed Dirichlet problems by boundary value problems in perforated domains

Abstract

Given an elliptic operator L on a bounded domain Ω ⊆ 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 ⊇ Ω with the following property: for every f ∈ H −1 (Ω) the sequence u h of the solutions of the Dirichlet problems Lu h = f in Ω h , u h = 0 on ∂Ω h , extended to 0 in Ω\Ω h , converges to the solution of the “relaxed Dirichlet problem” Lu + μu = f in Ω, u = 0 on ∂Ω.

Locations

• Proceedings of the Royal Society of Edinburgh Section A Mathematics - View
• arXiv (Cornell University) - View - PDF