Homogenization and reduction of dimension for nonlinear parametric variational inequalities

Abstract

In this paper we study the limit behavior of the solution u ϵ of a parametric variational inequality, governed by a nonlinear differential operator, the gradient operator ∇ being replaced by another operator ∇ ϵ , with the positive parameter ϵ ( ϵ → 0 ). Generalizing earlier results of Courilleau and Mossino [P. Courilleau, J. Mossino, Compensated compactness for nonlinear homogenization and reduction of dimension, Calculus of Variations 20 (2004) 65–91], we show that, up to a subsequence, u ϵ weakly converges to a solution of a limit problem. For quasilinear operators, we show that the limit problem can be formulated on a lower dimensional domain.