Random homogenization and singular perturbations in perforated domains

The paper considers the singularly perturbed Dirichlet problem -epsilon Deltau(epsilon) + u(epsilon) = f in a randomly perforated domain Omega (epsilon), which is obtained from a bounded open set Omega in R-N after removing many holes of size epsilon (q). The perforated domain is described in terms of an ergodic dynamical system acting on a probability space. imposing certain conditions on the domain, the behaviour of u(epsilon) when epsilon --> 0 in Lebesgue spaces L-n(Omega) is studied. Test functions together with the Birkhoff ergodic theorem are the main tools of analysis. The Poisson distribution of holes of size FP with the intensity lambda epsilon (-r) is then considered. The above results apply in some cases; other cases are treated by the Wiener sausage approach.