New regularity theorems for non-autonomous variational integrals with (p, q)-growth

One prominent problem in the calculus of variations is minimizing anisotropic integrals with a (p, q)-elliptic density F depending on the gradient of a function omega : Omega -> R-n. The best known sufficient bound for regularity of solutions is q <p (n + 2)/n. On the other hand, if we allow an additional x-dependence of the density we have the much weaker result q <p (n + 1)/n. If one additionally imposes the local boundedness of the minimizer, then these bounds can be improved to q <p + 2 and q <p + 1. In this paper we give natural assumptions for F closing the gap between the autonomous and non-autonomous situation.