Homogenization of maximal monotone vector fields via selfdual variational calculus

TL;DR

This paper introduces a variational approach using selfdual Lagrangians to homogenize divergence-form equations driven by periodic maximal monotone vector fields, leveraging $3$-convergence methods.

Contribution

It develops a novel variational framework for homogenization of monotone vector fields using selfdual Lagrangians, extending classical convex potential techniques.

Findings

Provides a new variational formulation for homogenized equations.

Utilizes $3$-convergence methods instead of graph convergence.

Applies to equations driven by periodic maximal monotone vector fields.

Abstract

We use the theory of selfdual Lagrangians to give a variational approach to the homogenization of equations in divergence form, that are driven by a periodic family of maximal monotone vector fields. The approach has the advantage of using $\Gamma$-convergence methods for corresponding functionals just as in the classical case of convex potentials, as opposed to the graph convergence methods used in the absence of potentials. A new variational formulation for the homogenized equation is also given.