Comparison Principles for the Finsler Infinity Laplacian with Applications to Minimal Lipschitz Extensions

TL;DR

This paper establishes comparison principles for the Finsler infinity Laplacian, linking absolutely minimizing Lipschitz extensions to viscosity solutions, and advances the understanding of $L^{ abla}$-variational problems.

Contribution

It proves generalized cone comparison principles for the Finsler infinity Laplacian, resolving a longstanding question in the $L^{ abla}$-calculus of variations.

Findings

Functions are $ ext{phi}$-absolutely minimizing iff they solve the $ ext{phi}$-infinity Laplace equation.

New geometric and convex analysis techniques underpin the proofs.

Results unify Lipschitz extension theory with viscosity solutions.

Abstract

This paper proves comparison principles for elliptic PDE involving the Finsler infinity Laplacian, a second-order differential operator with discontinuities in the gradient variable arising in $L^{\infty}$-variational problems and tug-of-war games. The core of the paper consists in proving generalized cone comparison principles. Among other consequences, these results imply that, for any Finsler norm $\varphi$ in $\mathbb{R}^{d}$, a function $u$ is a $\varphi$-absolutely minimizing Lipschitz extension if and only if it is a viscosity solution of the $\varphi$-infinity Laplace equation, settling a longstanding question in the $L^{\infty}$-calculus of variations. The proofs combine new geometric constructions with classical notions from convex analysis.