Higher differentiability of solutions to variational problems

We prove local higher differentiability of solutions to the problem of minimizing \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\int_\Omega f\left( x, v(x), \nabla v(x)\right) dx$$\end{document}under broad assumptions on f.