Uniform W1,p estimates for systems of linear elasticity in a periodic medium

Let {Lε} be a family of elliptic systems of linear elasticity with rapidly oscillating periodic coefficients. We obtain the uniform W1,p estimate ∇uε p C f p in a Lipschitz domain Ω in Rn for solutions to the Dirichlet problem: Lε(uε) = div(f ) in Ω and uε = 0 on ∂Ω, where | 1 p − 12 | < 1 2n +δ and C, δ >0 are constants independent of ε >0. The ranges are sharp for n = 2 or 3. © 2011 Elsevier Inc. All rights reserved.