Multiscale homogenization of a class of nonlinear equations with applications in lubrication theory and applications

Abstract

We prove a homogenization result for monotone operators by using the method of multiscale convergence. More precisely, we study the asymptotic behavior as ε → 0 of the solutions uε of the nonlinear equation div⁡aε(x, ∇uε) = div⁡bε, where both aε and bε oscillate rapidly on several microscopic scales and aε satisfies certain continuity, monotonicity and boundedness conditions. This kind of problem has applications in hydrodynamic thin film lubrication where the bounding surfaces have roughness on several length scales. The homogenization result is obtained by extending the multiscale convergence method to the setting of Sobolev spaces , where 1 < p < ∞. In particular we give new proofs of some fundamental theorems concerning this convergence that were first obtained by Allaire and Briane for the case p = 2.