Global regularity for almost minimizers of nonconvex variational problems

We prove some global, up to the boundary of a domain \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega \subset {\mathbb{R}}^{n}$$\end{document} , continuity and Lipschitz regularity results for almost minimizers of 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}$${\bf {\rm u}} \mapsto \int_{\Omega} g({\bf {\rm x}}, {\bf {\rm u}}({\bf {\rm x}}), \nabla{\bf {\rm u}}({\bf {\rm x}}))\,{\rm d}{\bf x}.$$\end{document}The main assumption for g is that it be asymptotically convex with respect its third argument. For the continuity results, the integrand is allowed to have some discontinuous behavior with respect to its first and second arguments. For the global Lipschitz regularity result, we require g to be Hölder continuous with respect to its first two arguments.