W 1,p ESTIMATES FOR ELLIPTIC HOMOGENIZATION PROBLEMS IN NONSMOOTH DOMAINS

Let Lε = −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 ∇(Lε)−1/2 are uniformly bounded on Lp(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 C1 domains, we establish the uniform Lp boundedness of ∇(Lε)−1/2 for 1 < p < ∞ and n ≥ 2. As a consequence, we obtain the uniform W 1,p estimates for the elliptic homogenization problem Lεuε = divf in Ω, uε = 0 on ∂Ω.