Homogenization of nonlinear Dirichlet problems in random perforated domains

Abstract

The present paper is devoted to study the asymptotic behavior of the solutions of a Dirichlet nonlinear elliptic problem posed in a perforated domain O∖Kε, where O⊂RN is a bounded open set and Kε⊂RN a closed set. Similarly to the classical paper by D. Cioranescu and F. Murat, each set Kε is the union of disjoint closed sets Kεi, with critical size. But while there the sets Kεi were balls periodically distributed, here the main novelty is that the positions and the shapes of these sets are random, with a distribution given by a preserving measure N-dynamical system not necessarily ergodic. As in the classical result, the limit problem contains an extra term of zero order, the “strange term” which depends on the capacity of the holes relative to the nonlinear operator and also of its distribution. To prove these results we introduce an original adaptation of the two scale convergence method combined with the ergodic theory.