Uniform [formula omitted] estimates for systems of linear elasticity in a periodic medium

Abstract

Let { L ε } be a family of elliptic systems of linear elasticity with rapidly oscillating periodic coefficients. We obtain the uniform W 1 , p estimate ‖ ∇ u ε ‖ p ⩽ C ‖ f ‖ p in a Lipschitz domain Ω in R n for solutions to the Dirichlet problem: L ε ( u ε ) = div ( f ) in Ω and u ε = 0 on ∂ Ω, where | 1 p − 1 2 | < 1 2 n + δ and C, δ > 0 are constants independent of ε > 0 . The ranges are sharp for n = 2 or 3.