Regularity of Relaxed Minimizers of Quasiconvex Variational Integrals with (p, q)-growth

We consider autonomous integrals \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$F[u]:=\int_\Omega f(Du) \, {\rm d}x \quad \rm{for} \, u:\mathbb R^n\supset\Omega\to \mathbb{R}^N$$\end{document}in the multidimensional calculus of variations, where the integrand f is a strictly quasiconvex C2-function satisfying the (p, q)-growth conditions \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma|A|^p \leqq f(A)\leqq\Gamma (1+|A|^q)\quad \rm{for\, every} \, A\in\mathbb{R} ^{nN}$$\end{document}with exponents \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${1 < p\leqq q < \infty}$$\end{document}.