Random Homogenization and Singular Perturbations¶in Perforated Domains

The paper considers the singularly perturbed Dirichlet problem −ɛΔuɛ+uɛ=f in a randomly perforated domain Ωɛ, which is obtained from a bounded open set Ω in RN after removing many holes of size ɛq. 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 uɛ when ɛ→ 0 in Lebesgue spaces Ln(Ω) is studied. Test functions together with the Birkhoff ergodic theorem are the main tools of analysis. The Poisson distribution of holes of size ɛp with the intensity λɛ−r is then considered. The above results apply in some cases; other cases are treated by the Wiener sausage approach.