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

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