The Robin problems on domains with many tiny holes (Q1071999)

Let \(\Omega\) be a bounded domain in \(R^ N\) and let \(R^{\epsilon}\) be \(\Omega\) with \(n^{\epsilon}\) balls of radius \(r^{\epsilon}\) removed from \(\Omega\) in a certain manner. Consider the Robin problem \[ (*)\quad -\Delta u^{\epsilon}=f\quad a.e.\quad in\quad R^{\epsilon},\quad \partial u^{\epsilon}/\partial \nu +\alpha u^{\epsilon}=0\quad a.e.\quad or\quad \partial R^{\epsilon}, \] where \(f\in L^ 2(\Omega)\), \(\alpha\) is a positive constant, and \(\partial /\partial \nu\) is the outward normal derivative operator. Using an abstract scheme, the author shows that a suitably extended solution of (*), \(\tilde u^{\epsilon}\in H^ 1(\Omega)\), converges weakly in \(H^ 1(\Omega)\) to the solution of \(-\Delta u+\alpha S_ N\eta u/| \Omega | =f\) a.e. in \(\Omega\), \(\partial u/\partial \nu +\alpha u=0\) a.e. on \(\partial \Omega\), where \(| \Omega |\) is the volume of \(\Omega\), \(S_ N\) is the surface area of the unit sphere in \(R^ N\), and \(\eta\) is the parameter \(\eta =\lim n^{\epsilon}(r^{\epsilon})^{N-1}\) which is positive and finite.