Homogenization of monotone operators in divergence form with x-dependent multivalued graphs

Abstract

In a previous paper [4], we proved the existence of solutions to -div a(x, grad u) = f, together with appropriate boundary conditions, whenever a(x, e) belongs, for every fixed x, to a certain class of maximal monotone graphs in e. Here, we derive the corresponding homogenization result, letting a(x, e) depend upon a parameter ε, and imposing adequate ε-uniform boundedness and coercivity properties. The resulting homogenized graphs belong to the same class of maximal monotone graphs. Our results do not assume any kind of periodicity. © Fondazione Annali di Matematica Pura ed Applicata and Springer-Verlag 2009.