Regularity of Homogenized Boundary Data in Periodic Homogenization of Elliptic Systems

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 $1<p<\mathrm{\infty }$. In particular, this implies that the boundary layer tails are H"older continuous of order $\alpha$ for any $\alpha \in (0,1)$.