The $\infty$-harmonic potential is not always an $\infty$-eigenfunction

TL;DR

This paper demonstrates that in certain convex domains, the $ abla$-harmonic potential does not coincide with the first $ abla$-eigenfunction, challenging assumptions about their equivalence.

Contribution

It provides a counterexample showing the $ abla$-harmonic potential is not always an $ abla$-eigenfunction in convex domains.

Findings

Counterexample in convex domain

$ abla$-harmonic potential differs from $ abla$-eigenfunction

Challenges previous assumptions

Abstract

In this note we prove that there is a convex domain for which the $\infty$-harmonic potential is not a first $\infty$-eigenfunction.