Higher differentiability for solutions of elliptic systems with Sobolev coefficients

Let n \ge 2 and let \Omega be a bounded open set in R^n. We provide higher differentiability results for local solutions of elliptic systems of the type div A(x, Du)=0 in \Omega, with coefficients in Sobolev spaces. We find that differentiability assumptions on A transfer to Du with no losses in the order of differentiation.