Global Lipschitz continuity for minima of degenerate problems

TL;DR

This paper proves boundary Lipschitz regularity for solutions to a degenerate minimization problem involving a convex Lagrangian, under minimal growth and convexity assumptions, extending regularity theory in calculus of variations.

Contribution

It establishes boundary Lipschitz regularity for minimizers with convex Lagrangians that are only uniformly convex outside a ball, without growth restrictions.

Findings

Lipschitz regularity up to the boundary for solutions

Regularity holds under convexity and bounded slope conditions

No growth assumptions needed for the convex function

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.