Asymptotic behaviour of nonlinear Dirichlet problems in perforated domains (Q1568756)

The authors consider the sequence of boundary value problems: \[ \begin{cases} \sum^N_{j=1} {\partial\over \partial x_j}\left( a_j\left( x,u_s(x), {\partial u_s(x) \over\partial x}\right) \right)= a_0\left( x,u_s(x), {\partial u_s(x) \over\partial x}\right) \text{ in }\Omega\\ u_s(x)= f(x)\text{ in } \partial \Omega_s,\end{cases} \] where \(s=1,2,\dots\) and \(\Omega_s\) is an arbitrary sequence of open subsets of bounded set \(\Omega\subset \mathbb{R}^N\). The goal of this paper is to study the asymptotic behaviour of \(u_s(x)\) as \(s\to\infty\) under very weak assumptions on the sets \(\Omega_s\), that is under very mild assumptions on the capacity of the holes.