Nonlinear Analysis 133 (2016) 250–274 Contents lists available at ScienceDirect Nonlinear Analysis www.elsevier.com/locate/na Homogenization of nonlinear Dirichlet problems in random perforated domains Carmen Calvo-Jurado a , Juan Casado-D´ ıaz b , Manuel Luna-Laynez b,* a Dpto. de Matem´ aticas, Escuela Polit´ ecnica, Avenida de la Universidad s/n, 10003 C´aceres, Spain b Dpto. de Ecuaciones Diferenciales y An´ alisis Num´ erico, Facultad de Matem´ aticas, Calle Tarfia s/n, 41012 Sevilla, Spain article info Article history: Received 14 April 2015 Accepted 10 December 2015 Communicated by Enzo Mitidieri MSC: 35R60 35B27 Keywords: Homogenization Monotone operators Random perforated domains Dirichlet conditions 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 ⊂ R N is a bounded open set and Kε ⊂ R N a closed set. Similarly to the classical paper by D. Cioranescu and F. Murat, each set Kε is the union of disjoint closed sets K i ε , with critical size. But while there the sets K i ε 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. © 2015 Elsevier Ltd. All rights reserved. 1. Introduction Probably, the most well known result [15] in the homogenization of an elliptic problem with Dirichlet conditions posed in a sequence of perforated domains is the following one: If O is a bounded open set of R N ,N ≥ 3, and K ε is the union of closed balls (the holes) of critical size ε N N−2 , periodically distributed with period ε, i.e. K ε =  k∈Z N B  εk, Cε N N−2  , then the solution of the Poisson equation in O ε = O \ K ε with right-hand side f ∈ H −1 (O) and homogeneous Dirichlet boundary condition, converges weakly in H 1 0 (O) to the solution of ∗ Corresponding author. E-mail addresses: ccalvo@unex.es (C. Calvo-Jurado), jcasadod@us.es (J. Casado-D´ ıaz), mllaynez@us.es (M. Luna-Laynez). http://dx.doi.org/10.1016/j.na.2015.12.008 0362-546X/© 2015 Elsevier Ltd. All rights reserved.