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