Quantitative analysis of boundary layers in periodic homogenization
  We prove quantitative estimates on the rate of convergence for the oscillating Dirichlet problem in periodic homogenization of divergence-form uniformly elliptic systems. The estimates are optimal in dimensions larger than three and new in every dimension. We also prove a regularity estimate on the homogenized boundary condition. 
Scott Armstrong, Tuomo Kuusi, Jean-Christophe Mourrat and Christophe   Prange
math.AP