W$^{1,p}$ estimates for elliptic homogenization problems in nonsmooth domains
Zhongwei Shen · Indiana University Mathematics Journal · 2008
Let Le = -div(A(x/e)∇), e > 0 be a family of second order elliptic operators with real, symmetric coefficients on a bounded Lipschitz domain Q in R n , 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 e such that Riesz transforms ∇ (Le) -1/2 are uniformly bounded on L p (Ω), where 1 2. As a consequence, we obtain the uniform W 1,p estimates for the elliptic homogenization problem L e u e = divf in Ω, u e = 0 on δΩ.