Dissertation Title:          Boundary layers in periodic homogenization

Abstract Title:  This dissertation is devoted to the homogenization theory of the second-order linear elliptic system in divergence form with periodically oscillating coefficients, subject to periodically oscillating Neumann (or Dirichlet) boundary condition. Assuming the domain is uniformly convex, we identify the homogenized system and establish the almost sharp rate of convergence in $L^2$. For either Neumann or Dirichlet problem, we prove that the homogenized boundary data is in the Sobolev space $W^{1,p}$ for any $p\in (1,\infty)$, which implies the $C^\alpha$-H\"{o}lder continuity for any $\alpha\in (0,1)$. Further applications and generalizations of these results will also be discussed.