Boundary estimates in elliptic homogenization

For a family of systems of linear elasticity with rapidly oscillating periodic coef?cients, we establishsharp boundary estimates with either Dirichlet or Neumann conditions, uniform down to the microscopicscale, without smoothness assumptions on the coef?cients. 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-α domains. The Cα, W^1,p, and L ^pestimates in C¹ domains for systems with VMO coefficients are also studied. The approach is based on certain estimates on convergence rates. As a biproduct, we obtain sharp O(ε ) error estimates in L^q(Ω)for q=2d/(d-1)and a Lipschitz domain Ω, with no smoothness assumption on the coefficients.