Regularity of boundary data in periodic homogenization of elliptic systems in layered media

TL;DR

This paper investigates the regularity of boundary data in the periodic homogenization of elliptic systems with oscillating coefficients and boundary conditions, addressing a key challenge in understanding the limiting behavior.

Contribution

It initiates the study of boundary data regularity in homogenization, providing new regularity results for the fixed boundary data in layered media.

Findings

Established regularity results for boundary data in homogenized elliptic systems.

Clarified the relationship between oscillating boundary conditions and their homogenized limits.

Abstract

In this note we study periodic homogenization of Dirichlet problem for divergence type elliptic systems when both the coefficients and the boundary data are oscillating. One of the key difficulties here is the determination of the fixed boundary data corresponding to the limiting (homogenized) problem. This issue has been addressed in recent papers by D. G\'{e}rard-Varet and N. Masmoudi, and by C. Prange, however, not much is known about the regularity of this fixed data. The main objective of this note is to initiate a study of this problem, and to prove several regularity results in this connection.