Gradient estimates for the p(x)-Laplacean system

We prove Calder\'on \& Zygmund type estimates for a class of elliptic problems whose model is the non-homogeneous $p(x)$-Laplacean system $$-div \ (|Du|^{p(x)-2}Du) =-div \ (|F|^{p(x)-2}F)\;. $$ Under optimal continuity assumptions on the function $p(x)>1$ we prove that $$ |F|^{p(x)} \in L_{loc} ^{ q} \Longrightarrow |Du|^{p(x)} \in L^{q}_{loc} \qquad \forall \ q>1\;.$$ Our estimates are motivated by recent developments in non-Newtonian fluid mechanics and elliptic problems with non-standard growth conditions, and are the natural, ``non-linear" counterpart of those obtained by Diening \& Ruzicka [12] in the linear case.