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

2024/05/09 by Peter S. Morfe, Morfe, Peter S.
Physics and Astronomy · #Advanced Differential Geometry Research #Analysis of PDEs (math.AP) #FOS: Mathematics

paper · pdf · doi:10.48550/arxiv.2405.05684

openalex publication_date 2024/05/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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^∞-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 φ in ℝ^d, a function u is a φ-absolutely minimizing Lipschitz extension if and only if it is a viscosity solution of the φ-infinity Laplace equation, settling a longstanding question in the L^∞-calculus of variations. The proofs combine new geometric constructions with classical notions from convex analysis.

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