Partial regularity for non autonomous functionals with non standard growth conditions

We prove a C1,μ partial regularity result for minimizers of a non autonomous integral funcitional of the form \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal{F}(u; \Omega):=\int_{\Omega}f(x, Du)\ dx$$\end{document}under the so-called non standard growth conditions. More precisely we assume that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$c |z|^{p}\leq f(x ,z) \leq L (1+|z|^{q}),$$\end{document}for 2 ≤ p < q and that Dzf(x, z) is α-Hölder continuous with respect to the x-variable. The regularity is obtained imposing that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\frac{p}{q} < \frac{n+\alpha}{n}}$$\end{document} but without any assumption on the growth of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${D^{2}_{z}f}$$\end{document}.