W^1,p estimates for elliptic homogenization problems in nonsmooth domains

Let L_ε = -div(A(x/ε)∇), ε > 0 be a family of second order elliptic operators with real, symmetric coefficients on a bounded Lipschitz domain Ω in ℝⁿ, subject to the Dirichlet boundary condition. Assuming that A(x) is periodic and belongs to VMO, we show that there exists δ > 0 independent of ε such that Riesz transforms ∇(L_ε)^-1/2 are uniformly bounded on L^P(Ω), where 1 < p < 3 + δ if n ≥ 3, and 1 < p < 4 + δ if n = 2. The ranges of p's are sharp. In the case of C ¹ domains, we establish the uniform L^P boundedness of ∇(L_ε)^-1/2 for 1 < p < ∞ and n ≥ 2. As a consequence, we obtain the uniform W^1,p estimates for the elliptic homogenization problem L_εu_ε = div f in Ω, u_ε = 0 on ∂Ω.