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This paper is concerned with a family of second-order elliptic systems indivergence form with rapidly oscillating periodic coefficients. We initiate thestudy of homogenization and boundary layers for Neumann problems withfirst-order oscillating boundary data. We identify the homogenized system andestablish the sharp rate of convergence in $L^2$ in dimension three or higher.Sharp regularity estimates are also obtained for the homogenized boundary datain both Dirichlet and Neumann problems. The results are used to obtain ahigher-order convergence rate for Neumann problems with non-oscillating data.

Boundary Layers in Periodic Homogenization of Neumann Problems