On the periodic homogenization of elliptic equations in non-divergence form with large drifts

TL;DR

This paper develops a method to analyze the homogenization of elliptic equations with large drifts in non-divergence form, transforming them into divergence form to apply existing homogenization results and obtain quantitative estimates.

Contribution

It introduces a novel approach using invariant measures to convert non-divergence form equations with large drifts into divergence form, enabling systematic quantitative homogenization analysis.

Findings

Established convergence rates for homogenization.

Achieved uniform Lipschitz regularity results.

Extended homogenization techniques to equations with large drifts.

Abstract

We study the quantitative homogenization of linear second order elliptic equations in non-divergence form with highly oscillating periodic diffusion coefficients and with large drifts, in the so-called ``centered'' setting where homogenization occurs and the large drifts contribute to the effective diffusivity. Using the centering condition and the invariant measures associated to the underlying diffusion process, we transform the equation into divergence form with modified diffusion coefficients but without drift. The latter is in the standard setting for which quantitative homogenization results have been developed systematically. An application of those results then yields quantitative estimates, such as the convergence rates and uniform Lipschitz regularity, for equations in non-divergence form with large drifts.