Second order regularity of solutions of elliptic equations in divergence form with Sobolev coefficients

Abstract

We give $L^p$ estimates for the second derivatives of weak solutions to the Dirichlet problem for equation $\Div(\mathbf{A}\nabla u) = f$ in $\Omega \subset {\mathbb{R}}^d$ with Sobolev coefficients. In particular, for $f\in L^2(\Omega )\bigcap L^s(\Omega )$ $$\Vert \Delta u{\Vert }_2\le \{\begin{array}{ll}{c}_1\Vert f{\Vert }_2+c_2\Vert \nabla \mathbf{A}{\Vert }_q^2\Vert f{\Vert }_s,& \text{if }1<s<d/2,\frac{1}{2}=\frac{2}{q}+\frac{1}{s}-\frac{2}{d}\\ c_1\Vert f{\Vert }_2+c_2\Vert \nabla \mathbf{A}{\Vert }_4^2\Vert f{\Vert }_s,& \text{if }s>d/2\end{array}.$$

9 pages