Random Homogenization and Singular Perturbations¶in Perforated Domains

Abstract:

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