Regularity of minimizers of W1,p-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}$$F[u]:=\int_\Omega f(Du)dx \quad{\rm for}\,\,u:{\mathbb{R}}^{n}\supset\Omega\to{\mathbb{R}}^{N} $$\end{document}in the multidimensional calculus of variations, where the integrand f is a strictly W1,p-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}$$ \gamma |A|^p\,\le\,f(A) \le \Gamma(1+|A|^q)\quad {\rm for \quad every}\,A \in \mathbb{R}^{nN}$$\end{document}with exponents 1 < p ≤  q < ∞. Under these assumptions we establish an existence result for minimizers of F in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$W^{1,p}(\Omega;{\mathbb{R}}^N)$$\end{document} provided \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q\quad < \quad\frac{np}{n-1}$$\end{document} . We prove a corresponding partial C1,α-regularity theorem for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q < p +\frac{{\rm min}\{2,p\}}{2n}$$\end{document} . This is the first regularity result for autonomous quasiconvex integrals with (p,q)-growth.