Homogenization of the Poisson equation with Dirichlet conditions in random perforated domains

摘要

We consider a sequence of open sets Oε contained in a fixed bounded open set O of RN, N≥3, which vary randomly with ε>0. The corresponding distribution function is given by an ergodic measure preserving dynamical system in such a way that O∖Oε is a union of closed sets of size εNN−2 and the distance between them of order ε. For this sequence Oε we study the asymptotic behavior of the solutions of the Poisson equation with Dirichlet conditions on ∂Oε. Similarly to the classical Cioranescu–Murat result for the deterministic problem we show the existence of a new term of zero order in the limit equation. We emphasize the fact that this new term is deterministic.