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 $f\in L^2(\Omega) \bigcap L^s(\Omega)$ $$|\Delta u|_{2} \leq \begin{cases} c_1|f|_2 + c_2 |\nabla \mathbf{A}|_q^2|f|_s, & \text{if } 1 < s < d/2, \frac{1}{2}=\frac{2}{q}+ \frac{1}{s} - \frac{2}{d}\ c_1|f|_2 + c_2 |\nabla \mathbf{A}|_4^2|f|_s, & \text{if } s > d/2 \end{cases}.$$