𝐶¹-regularity of planar ∞-harmonic functions—revisited

Zhang, Yi Ru-Ya; Zhou, Yuan

doi:10.1090/proc/14810

$C^{1}$-regularity of planar $\infty$-harmonic functions—revisited
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by Yi Ru-Ya Zhang and Yuan Zhou

Proc. Amer. Math. Soc. 148 (2020), 1187-1193

DOI: https://doi.org/10.1090/proc/14810

Published electronically: October 28, 2019

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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, $W^{1,2}_{\mathrm { loc }}$-regularity of $|Du|$, and Crandall’s flow for infinity harmonic functions.

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