Existence and symmetry of minimizers for nonconvex radially symmetric variational problems

We study functionals 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}$$E(u):=\int_{B_R(0)} W(\nabla u)+G(u)\,dx,$$\end{document}where u is a real valued function over the ball \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$B_R(0)\subset {\mathbb{R}}^N$$\end{document} which vanishes on the boundary and W is nonconvex. The functional is assumed to be radially symmetric in the sense that W only depends on \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$|{\nabla u}|$$\end{document} . Existence of one and radial symmetry of all global minimizers is shown with an approach based on convex relaxation. Our assumptions on G do not include convexity, thus extending a result of A. Cellina and S. Perrotta.