Global Lipschitz regularity for almost minimizers of asymptotically convex variational problems

We prove 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}$${{\it \Omega}\subset\mathbb {R}^n}$$\end{document}, 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}$$u \mapsto \int \limits_{\Omega} g(x, u(x), \nabla u(x))\,dx.$$\end{document}The main assumption for g is that it is asymptotically convex with respect its third argument. We also require g to satisfy some growth conditions and to be Hölder continuous with respect to its first two arguments.