Homogenization of Elliptic Problems with Neumann Boundary Conditions in Non-smooth Domains

We consider a family of second-order elliptic operators {L_ε} in divergence form with rapidly oscillating and periodic coefficients in Lipschitz and convex domains in Rⁿ. We are able to show that the uniform W^1,p estimate of second order elliptic systems holds for (2n)/(n+1)-δ < p < (2n)/(n-1)+δ where δ > 0 is independent of ε and the ranges are sharp for n=2, 3. And for elliptic equations in Lipschitz domains, the W^1,p estimate is true for (3)/2 -δ < p < 3 + δ if n ≥ 4, similar estimate was extended to convex domains for 1 < p < ∞.