Infinity-Harmonic Potentials and Their Streamlines

TL;DR

This paper investigates solutions to the Infinity-Laplace Equation in convex rings, revealing unique ascending streamlines, potential bifurcations in descending streamlines, and implications for the regularity of solutions' gradients.

Contribution

It demonstrates that bifurcations in streamlines are generic and shows solutions generally lack Lipschitz continuous gradients.

Findings

Ascending streamlines are unique

Descending streamlines can bifurcate

Solutions typically do not have Lipschitz continuous gradients

Abstract

We consider certain solutions of the Infinity-Laplace Equation in planar convex rings. Their ascending streamlines are unique while the descending ones may bifurcate. We prove that bifurcation occurs in the generic situation and as a consequence, the solutions cannot have Lipschitz continuous gradients.