The streamlines of $\infty$-harmonic functions obey the inverse mean curvature flow

TL;DR

This paper demonstrates that the streamlines of infinity-harmonic functions follow the inverse mean curvature flow, even under weak regularity conditions, by approximating with p-harmonic functions and using conjugate functions.

Contribution

It establishes a weak solution framework linking infinity-harmonic functions to inverse mean curvature flow via approximation methods.

Findings

Streamlines of infinity-harmonic functions are level sets of a related function.

The function w solves the inverse mean curvature flow in a weak sense.

Regularity properties of the gradient magnitude are derived.

Abstract

Given an $\infty$-harmonic function $u_\infty$ on a domain $\Omega \subseteq \mathbb{R}^2$, consider the function $w = -\log |\nabla u_\infty|$. If $u_\infty \in C^2(\Omega)$ with $\nabla u_\infty \neq 0$ and $\nabla |\nabla u_\infty| \neq 0$, then it is easy to check that (1) the streamlines of $u_\infty$ are the level sets of $w$ and (2) $w$ solves the level set formulation of the inverse mean curvature flow. For less regular solutions, neither statement is true in general, but even so, $w$ is still a weak solution of the inverse mean curvature flow under far weaker assumptions. This is proved through an approximation of $u_\infty$ by $p$-harmonic functions, the use of conjugate $p'$-harmonic functions, and the known connection of the latter with the inverse mean curvature flow. A statement about the regularity of $|\nabla u_\infty|$ arises as a by-product.