An obstacle problem for nonlocal equations in perforated domains (Q1745342)

Let \(\Omega \subset \mathbb{R}^N\) be an open bounded domain; \(\Omega^\epsilon \subset \Omega\) for a positive parameter \(\epsilon\) a family of open sets considered as perforated domains for which the sets \(A^\epsilon = \Omega \setminus \Omega^\epsilon\) describe the holes inside \(\Omega.\) It is assumed that there exists a function \(\mathcal{X} \in L^\infty (\mathbb{R}^N)\) strictly positive inside \(\Omega\) such that \(\chi_\epsilon \rightharpoonup \mathcal{X}\) weakly\(^\star\) in \(L^\infty (\Omega)\), where \(\chi_\epsilon\) denotes the characteristic function of \(\Omega^\epsilon\). The authors study the behavior of solutions to a nonlocal equation of the form \[ J \star u(x) - u(x) = f(x) \] in a domain \(\Omega^\epsilon\) with \(u = 0\) in \(A^\epsilon \cup \Omega^c\) and an obstacle constraint \(u \geq \psi\) in \(\Omega^\epsilon\). It is shown that there exists a weak limit \(u^\star\) in \(L^2(\Omega)\) of solutions \(u^\epsilon\) as \(\epsilon \to 0\) and the limit problem which \(u^\star\) satisfies is found.