# C^1-Regularity of planar \infty-harmonic functions - REVISIT

## TL;DR

This paper offers a new perspective on the $C^1$-regularity of planar infinity-harmonic functions by simplifying previous proofs and utilizing capacity, topology, and flow techniques.

## Contribution

It provides a simplified proof of $C^1$-regularity for planar infinity-harmonic functions using capacity and topological methods, removing complex technical arguments.

## Key findings

- Established $C^1$-regularity of planar infinity-harmonic functions.
- Connected regularity to capacity and topological properties.
- Utilized flow methods and flat estimates for proof.

## Abstract

In the seminal paper [Arch. Ration. Mech. Anal. 176 (2005), 351--361], Savin proved the $C^1$-regularity of planar $\infty$-harmonic functions $u$. Here we give a new understanding of it from a capacity viewpoint and drop several high technique arguments therein. Our argument is essentially based on a topological lemma of Savin, a flat estimate by Evans and Smart, % \cite{es11a}, $W^{1,2}_{loc}$-regularity of $|Du|$ and Crandall's flow for infinity harmonic functions.