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~
L
u
h
=f
in~
Ω
h
,
u
h
=0
on~
∂
Ω
h
, extended to 0 in~
Ω∖
Ω
h
, converges to the solution of the \lq\lq relaxed Dirichlet problem\rq\rq\
Lu+μu=f
in~
Ω
,
u=0
on~
∂Ω
.