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Let $\mathcal{L}_\epsilon$ be a family of elliptic systems of linearelasticity with rapidly oscillating periodic coefficients. We obtain theuniform $W^{1,p}$ estimate in a Lipschitz domain for solutions to the Dirichletproblem, where $(2n/(n+1)) -\delta<p<(2n/(n-1))+\delta$. The ranges of $p$'sare sharp for $n=2$ or 3.

Uniform W^{1,p} Estimates for Systems of Linear Elasticity in a Periodic Medium