Regularity of a transmission problem and periodic homogenization

TL;DR

This paper develops new regularity estimates for transmission problems in composite materials, including uniform Lipschitz bounds across interfaces with periodic coefficients oscillating at different scales.

Contribution

It provides a self-contained proof of $C^{k,eta}$ estimates and establishes uniform Lipschitz regularity for periodic coefficients with independent structures.

Findings

$C^{k,eta}$ estimates on both sides of the interface

Uniform Lipschitz estimate across $C^{1,eta}$ interfaces

Regularity results under minimal assumptions

Abstract

This paper is concerned with the regularity theory of a transmission problem arising in composite materials. We give a new self-contained proof for the $C^{k,\alpha}$ estimates on both sides of the interface under the minimal assumptions on the interface and data. Moreover, we prove the uniform Lipschitz estimate across a $C^{1,\alpha}$ interface when the coefficients on both sides of the interface are periodic with independent structures and oscillating at different microscopic scales.