# Explicit solutions of Jensen's auxiliary equations via extremal Lipschitz extensions

## TL;DR

This paper demonstrates that McShane and Whitney's Lipschitz extensions are viscosity solutions of Jensen's auxiliary equations, linking classical extension methods to modern PDE theory, and providing new insights into the uniqueness of infinity harmonic functions.

## Contribution

It establishes a novel connection between Lipschitz extensions and viscosity solutions of Jensen's equations, a result not previously documented.

## Key findings

- Lipschitz extensions are viscosity solutions of Jensen's auxiliary equations
- This connection aids in understanding the uniqueness of infinity harmonic functions
- Provides a new perspective on Absolutely Minimizing Lipschitz Extensions

## Abstract

In this note we prove that McShane and Whitney's Lipschitz extensions are viscosity solutions of Jensen's auxiliary equations, known to have a key role in Jensen's celebrated proof of uniqueness of infinity harmonic functions, and therefore of Absolutely Minimizing Lipschitz Extensions. To the best of the author's knowledge, this result does not appear to be known in the literature in spite of the vast amount of work around the topic.