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

Abstract

We consider a family of second-order elliptic operators {Le} in divergence form with rapidly oscillating and periodic coefficients in Lipschitz and convex domains in ℝn. We are able to show that the uniform W1,p estimate of second order elliptic systems holds for \(\frac{{2n}}{{n + 1}} - \delta 0 is independent of e and the ranges are sharp for n = 2, 3. And for elliptic equations in Lipschitz domains, the W1,p estimate is true for \(\frac{3}{2} - \delta < p < 3 + \delta \) if n ≥ 4, similar estimate was extended to convex domains for 1 < p < ∞.