Regularity of the correctors and local gradient estimate of the homogenization for the elliptic equation: linear periodic case

TL;DR

This paper investigates the regularity of correctors and provides local gradient estimates in homogenization of elliptic equations with periodic coefficients, achieving precise error bounds under smoothness assumptions.

Contribution

It establishes new $C^eta$ and $W^{1, abla}$ estimates for correctors and quantifies the local error of first-order expansions in homogenization.

Findings

Error of first-order expansion is $O(\epsilon)$ in Hölder norm for smooth data.

Error is $O(\epsilon)$ in $W^{1, abla}$ norm for Lipschitz continuous coefficients.

Results partly extend to nonlinear parabolic equations.

Abstract

$C^\alpha$ and $W^{1,\infty}$ estimates for the first-order and second-order correctors in the homogenization are presented based on the translation invariant and Li-Vogelius's gradient estimate for the second order linear elliptic equation with piecewise smooth coefficients. If the data are smooth enough, the error of the first-order expansion for piecewise smooth coefficients is locally $O(\epsilon)$ in the H\"older norm; it is locally $O(\epsilon)$ in $W^{1,\infty}$ when coefficients are Lipschitz continuous. It can be partly extended to the nonlinear parabolic equation.