Global Lipschitz continuity for minima of degenerate problems

We consider the problem of minimizing the Lagrangian $\int [F(\nabla u)+f\,u]$ among functions on $\Omega\subset\mathbb{R}^N$ with given boundary datum $\varphi$. We prove Lipschitz regularity up to the boundary for solutions of this problem, provided $\Omega$ is convex and $\varphi$ satisfies the bounded slope condition. The convex function $F$ is required to satisfy a qualified form of uniform convexity {\it only outside a ball} and no growth assumptions are made.