preprint
Inserted: 12 jun 2024
Year: 2019
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.