Boundary Estimates in Elliptic Homogenization

For a family of systems of linear elasticity with rapidly oscillating periodic coefficients, we establish sharp boundary estimates with either Dirichlet or Neumann conditions, uniform down to the microscopic scale, without smoothness assumptions on the coefficients. Under additional smoothness conditions, these estimates, combined with the corresponding local estimates, lead to the full Rellich type estimates in Lipschitz domains and Lipschitz estimates in $C^{1,\alpha }$ domains. The $C^{\alpha }$, $W^{1,p}$, and $L^p$ estimates in $C^1$ domains for systems with VMO coefficients are also studied. The approach is based on certain estimates on convergence rates. As a bi-product, we obtain sharp $O(\epsilon )$ error estimates in $L^q(\mathrm{\Omega })$ for $q=\frac{2d}{d-1}$ and a Lipschitz domain $\mathrm{\Omega }$, with no smoothness assumption on the coefficients.