Higher differentiability of minimizers of variational integrals with Sobolev coefficients

In this paper we consider integral functionals of the form 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.