$L^p$ Neumann Problems in Homogenization of General Elliptic Systems

TL;DR

This paper extends boundary estimate techniques to nonhomogeneous elliptic systems with oscillating coefficients, improving understanding of boundary behavior in homogenization and involving a bootstrap process and Neumann boundary correctors.

Contribution

It introduces new boundary estimates for nonhomogeneous elliptic systems with oscillating coefficients, building upon previous work and addressing the complexities introduced by lower order terms.

Findings

Extended nontangential maximal function estimates to nonhomogeneous operators

Developed optimal boundary estimates based on convergence rates

Utilized Neumann boundary correctors for lower order terms

Abstract

In this paper, we extend the nontangential maximal function estimate obtained by C. Kenig, F. Lin and Z. Shen in \cite{KFS1} to the nonhomogeneous elliptic operators with rapidly oscillating periodic coefficients. The result relies on the previous work \cite{X2}, and optimal boundary estimates which is based upon certain estimates on convergence rates. Compared to the homogeneous case, the additional bootstrap process seems inevitable, and the Neumann boundary corrector caused by the lower order term are still useful here.