A monotone boundary condition for a domain with many tiny spherical holds (Q1066008)

This work considers the behavior, as \(\epsilon \to 0^+\), of the solutions \(u_{\epsilon}\) of the problems \[ -\Delta u_{\epsilon}=f\quad a.e.\quad in\quad \Omega_{\epsilon},\quad \partial u_{\epsilon}/\partial \nu +\beta_{\epsilon}(u_{\epsilon})=0\quad a.e.\quad on\quad \partial \Omega_{\epsilon} \] where \(\nu\) is the outer normal to \(\partial \Omega_{\epsilon}\), \(\beta_{\epsilon}\) is a suitable monotone function; \(\Omega_{\epsilon}=\Omega \setminus F_{\epsilon}\), \(\Omega\) is a bounded domain in \(R^ N\) with smooth boundary \(\partial \Omega\), \(F_{\epsilon}\) is the union of all balls \(B^ i(r_{\epsilon})\) (contained in \(\Omega)\) such that \(dist(B^ i(r_{\epsilon}),\partial \Omega)\leq \epsilon\), \(B^ i(r_{\epsilon})\) has radius \(r_{\epsilon}<\epsilon\) and the same center as \(C^ i_{\epsilon}\), and finally \(C^ i_{\epsilon}\) are cubes, with volume \((2\epsilon)^ N\), which divide \(R^ N\).