Higher Order Convergence Rates in Theory of Homogenization I: Equations of Non-divergence Form

TL;DR

This paper develops higher order convergence rates for periodic homogenization of linear and nonlinear elliptic equations in non-divergence form, using advanced correctors and a viscosity-based approach.

Contribution

It introduces a novel method involving higher order correctors and a new regularity theory to achieve improved convergence rates in homogenization.

Findings

Higher order correctors improve error estimates

The approach applies to both linear and nonlinear equations

Enhanced stability results for correctors are established

Abstract

We establish higher order convergence rates in the theory of periodic homogenization of both linear and fully nonlinear uniformly elliptic equations of non-divergence form. The rates are achieved by involving higher order correctors which fix the errors occurring both in the interior and on the boundary layer of our physical domain. The proof is based on a viscosity method and a new regularity theory which captures the stability of the correctors with respect to the shape of our limit profile.