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 and C, δ>0 are constants independent of ε>0. The ranges are sharp for n=2 or 3.