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This paper is concerned with periodic homogenization of second-order elliptic systems in divergence form with oscillating Dirichlet data or Neumann data of first order. We prove that the homogenized boundary data belong to $W^{1, p}$ for any $1<p<\infty$. In particular, this implies that the boundary layer tails are H\"older continuous of order $\alpha$ for any $\alpha \in (0,1)$.

Regularity of homogenized boundary data in periodic homogenization of elliptic systems