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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 $Ω\subset\mathbb{R}^N$ with given boundary datum $φ$. We prove Lipschitz regularity up to the boundary for solutions of this problem, provided $Ω$ is convex and $φ$ 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.