Uniform W^{1,p} Estimates for Systems of Linear Elasticity in a Periodic   Medium
  Let $\mathcal{L}_\epsilon$ be a family of elliptic systems of linear elasticity with rapidly oscillating periodic coefficients. We obtain the uniform $W^{1,p}$ estimate in a Lipschitz domain for solutions to the Dirichlet problem, where $(2n/(n+1)) -\delta<p<(2n/(n-1))+\delta$. The ranges of $p$'s are sharp for $n=2$ or 3. 
Jun Geng, Zhongwei Shen, Liang Song
math.AP