Approximate homogenization of convex nonlinear elliptic PDEs

TL;DR

This paper develops a variational approach to approximate the homogenization of convex nonlinear elliptic PDEs, providing new formulas, numerical methods, and convergence insights for periodic and random settings.

Contribution

It introduces a variational formula for the effective operator using invariant measures, generalizing linear homogenization and enabling new analytic and numerical techniques.

Findings

Derived new formulas for the effective operator $ar H$

Compared numerical simulations with theoretical convergence rates

Analyzed homogenization in periodic and random environments

Abstract

We approximate the homogenization of fully nonlinear, convex, uniformly elliptic Partial Differential Equations in the periodic setting, using a variational formula for the optimal invariant measure, which may be derived via Legendre-Fenchel duality. The variational formula expresses $\bar H$ as an average of the operator against the optimal invariant measure, generalizing the linear case. Several nontrivial analytic formulas for $\bar H$ are obtained. These formulas are compared to numerical simulations, using both PDE and variational methods. We also perform a numerical study of convergence rates for homogenization in the periodic and random setting and compare these to theoretical results.