文摘
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 only outside a ball and no growth assumptions are made.