Boundary Layers in Periodic Homogenization of Neumann Problems

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 L² in dimension three or higher. Regularity estimates are also obtained for the homogenized boundary data in both Dirichlet and Neumann problems. The results are used to establish a higher‐order convergence rate for Neumann problems with nonoscillating data. © 2018 Wiley Periodicals, Inc.