Regularity of a transmission problem and periodic homogenization

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.