Uniform Estimates in Periodic Homogenization of Fully Nonlinear Elliptic Equations

TL;DR

This paper establishes uniform regularity estimates in periodic homogenization for fully nonlinear elliptic equations using novel approximation techniques that handle the nonlinearity and oscillation of the operators.

Contribution

It introduces new approximation methods to control the Hessian of correctors, enabling uniform estimates for a broader class of nonlinear elliptic equations.

Findings

Achieved $C^{1,eta}$ and $C^{1,1}$ estimates in homogenization.

Developed new techniques for decomposing regular and irregular parts of the process.

Extended methods to non-concave operators.

Abstract

This article is concerned with uniform $C^{1,\alpha}$ and $C^{1,1}$ estimates in periodic homogenization of fully nonlinear elliptic equations. The analysis is based on the compactness method, which involves linearization of the operator at each approximation step. Due to the nonlinearity of the equations, the linearized operators involve the Hessian of correctors, which appear in the previous step. The involvement of the Hessian of the correctors deteriorates the regularity of the linearized operator, and sometimes even changes its oscillating pattern. These issues are resolved with new approximation techniques, which yield a precise decomposition of the regular part and the irregular part of the homogenization process, along with a uniform control of the Hessian of the correctors in an intermediate level. The approximation techniques are even new in the context of linear equations. Our argument can be applied not only to concave operators, but also to certain class of non-concave operators.