Calculus of Variations and Geometric Measure Theory

P. Bousquet - L. Brasco

Global Lipschitz continuity for minima of degenerate problems

created by brasco on 21 Jan 2015
modified on 07 Jan 2016


Accepted Paper

Inserted: 21 jan 2015
Last Updated: 7 jan 2016

Journal: Math. Ann.
Pages: 38
Year: 2015


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