Boundary Layers in Periodic Homogenization of Neumann Problems

TL;DR

This paper investigates the homogenization process for elliptic systems with oscillating coefficients, focusing on boundary layers in Neumann problems, and establishes convergence rates and regularity estimates.

Contribution

It introduces the study of boundary layers in periodic homogenization of Neumann problems and provides sharp convergence rates and regularity results.

Findings

Sharp $L^2$ convergence rates in dimensions three and higher.

Regularity estimates for homogenized boundary data.

Higher-order convergence rates for non-oscillating Neumann problems.

Abstract

This paper is concerned with a family of second-order elliptic systems in divergence form with rapidly oscillating periodic coefficients. We initiate the study of homogenization and boundary layers for Neumann problems with first-order oscillating boundary data. We identify the homogenized system and establish the sharp rate of convergence in $L^2$ in dimension three or higher. Sharp regularity estimates are also obtained for the homogenized boundary data in both Dirichlet and Neumann problems. The results are used to obtain a higher-order convergence rate for Neumann problems with non-oscillating data.