Variational Approach to Homogenization of Doubly-Nonlinear Flow in a Periodic Structure

Abstract

This work deals with the homogenization of an initial- and boundary-value problem for the doubly-nonlinear system $$D_{t}w-\nabla \cdot \vec{z}=\nabla \cdot \vec{h}(x,t,x/\epsilon ),w\in \alpha (u,x/\epsilon ),\vec{z}\in \vec{\gamma }(\nabla u,x/\epsilon ).$$ Here $\epsilon$ is a positive parameter, and the prescribed mappings $\alpha$ and $\vec{\gamma }$ are maximal monotone with respect to the first variable and periodic with respect to the second one. The two inclusions are here formulated as null-minimization principles, via the theory of Fitzpatrick [MR 1009594]. As $\epsilon \to 0$, a two-scale formulation is derived via Nguetseng's notion of two-scale convergence, and a (single-scale) homogenized problem is then retrieved.

21 pages, original article