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

Let ${\mathcal{L}}_{ϵ}$ 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 $(2n\mathrm{/}(n+1))-\delta <p<(2n\mathrm{/}(n-1))+\delta$. The ranges of $p$’s are sharp for $n=2$ or 3.