*Accepted Paper*

**Inserted:** 21 jan 2015

**Last Updated:** 7 jan 2016

**Journal:** Math. Ann.

**Pages:** 38

**Year:** 2015

**Abstract:**

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.

**Keywords:**
regularity of minimizers, Degenerate and singular problems, uniform convexity

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