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Anisotropic Gauss curvature flows and their associated Dual Orlicz-Minkowski problems

Part of: Elliptic equations and systems Global differential geometry

Published online by Cambridge University Press:  01 November 2021

Li Chen ,
Qiang Tu ,
Di Wu  and
Ni Xiang
Li Chen
Affiliation:
Faculty of Mathematics and Statistics, Hubei Key Laboratory of Applied Mathematics, Hubei University, Wuhan430062, People's Republic of China (nixiang@hubu.edu.cn)
Qiang Tu
Affiliation:
Faculty of Mathematics and Statistics, Hubei Key Laboratory of Applied Mathematics, Hubei University, Wuhan430062, People's Republic of China (nixiang@hubu.edu.cn)
Di Wu
Affiliation:
Faculty of Mathematics and Statistics, Hubei Key Laboratory of Applied Mathematics, Hubei University, Wuhan430062, People's Republic of China (nixiang@hubu.edu.cn)
Ni Xiang
Affiliation:
Faculty of Mathematics and Statistics, Hubei Key Laboratory of Applied Mathematics, Hubei University, Wuhan430062, People's Republic of China (nixiang@hubu.edu.cn)
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Abstract

In this paper we study a normalized anisotropic Gauss curvature flow of strictly convex, closed hypersurfaces in the Euclidean space. We prove that the flow exists for all time and converges smoothly to the unique, strictly convex solution of a Monge-Ampère type equation and we obtain a new existence result of solutions to the Dual Orlicz-Minkowski problem for smooth measures, especially for even smooth measures.

Keywords

Gauss curvature flowconvex hypersurfaceMonge-Ampère type equation

MSC classification

Primary: 35J96: Elliptic Monge-Ampère equations
Secondary: 53C45: Global surface theory (convex surfaces à la A. D. Aleksandrov)
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 152 , Issue 1 , February 2022 , pp. 148 - 162
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Copyright
Copyright © The Author(s), 2021. Published by Cambridge University Press on behalf of The Royal Society of Edinburgh

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