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$ C^{1}$-regularity of planar $ \infty$-harmonic functions--revisited


Authors: Yi Ru-Ya Zhang and Yuan Zhou
Journal: Proc. Amer. Math. Soc. 148 (2020), 1187-1193
MSC (2010): Primary 35J60, 35J70
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}_{\mathop {\mathrm {\,loc\,}}}$-regularity of $ \vert Du\vert$, and Crandall's flow for infinity harmonic functions.


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Additional Information

Yi Ru-Ya Zhang
Affiliation: Hausdorff Center for Mathematics, Endenicher Allee 62, Bonn 53115, Germany
Email: yizhang@math.uni-bonn.de

Yuan Zhou
Affiliation: Department of Mathematics, Beihang University, Beijing 100191, People’s Republic of China
Email: yuanzhou@buaa.edu.cn

DOI: https://doi.org/10.1090/proc/14810
Received by editor(s): May 16, 2019
Received by editor(s) in revised form: July 20, 2019
Published electronically: October 28, 2019
Additional Notes: The second author is the corresponding author
Communicated by: Jeremy Tyson
Article copyright: © Copyright 2019 American Mathematical Society