IUMJ

Title: $W^{1,p}$ estimates for elliptic homogenization problems in nonsmooth domains

Authors: Zhongwei Shen

Issue: Volume 57 (2008), Issue 5, 2283-2298

Abstract:

Let $ \mathcal{L}_{\epsilon} = -\mathrm{Div} (A(x/\epsilon)\nabla)$, $\epsilon > 0$ be a family of second order elliptic operators with real, symmetric coefficients on a bounded Lipschitz domain $\Omega$ in $\mathbb{R}^{n}$, subject to the Dirichlet boundary condition. Assuming that $A(x)$ is periodic and belongs to $\mathrm{VMO}$, we show that there exists $\delta > 0$ independent of $\epsilon$ such that the Riesz transforms $\nabla (\mathcal{L}_{\epsilon})^{-1/2}$ are uniformly bounded on $L^{p}(\Omega)$, where $1 < p < 3 + \delta$ if $n \ge 3$, and $1 < p < 4 + \delta$ 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 $\nabla(\mathcal{L}_{\epsilon})^{-1/2}$ for $1 < p < \infty$ and $n \ge 2$. As a consequence, we obtain the uniform $W^{1,p}$ estimates for the elliptic homogenization problem $\mathcal{L}_{\epsilon}u_{\epsilon} = \mathrm{Div} f$ in $\Omega$, $u_{\epsilon} = 0$ on $\partial\Omega$.