Higher differentiability of minimizers of variational integrals with Sobolev coefficients

In this paper we consider integral functionals of the form F(nu, Omega) = integral(Omega) F(x, D nu(x)) dx with convex integrand satisfying p growth conditions with respect to the gradient variable. As a novel feature, the dependence of the integrand on the x-variable is allowed to be through a Sobolev function. We prove local higher differentiability results for local minimizers of the functional F, establishing uniform higher differentiability estimates for solutions to a class of auxiliary problems, constructed adding singular higher order perturbations to the integrand. Furthermore, we prove a dimension free higher integrability result for the gradient of local minimizers, by the use of a weighted version of the Gagliardo-Nirenberg interpolation inequality.