Flow by Gauss curvature to the $ L_p $ dual Minkowski problem

Qiang Guang; Qi-Rui Li; Xu-Jia Wang; Qiang Guang; Qi-Rui Li; Xu-Jia Wang

doi:10.3934/mine.2023049

Mathematics in Engineering

2023, Volume 5, Issue 3: 1-19. doi: 10.3934/mine.2023049

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Flow by Gauss curvature to the $ L_p $ dual Minkowski problem

1.
Mathematical Sciences Institute, The Australian National University, Canberra, ACT 2601, Australia
2.
School of Mathematical Sciences, Zhejiang University, Hangzhou 310027, China

Received: 22 March 2022 Revised: 01 August 2022 Accepted: 01 August 2022 Published: 16 August 2022

In the paper ^[20], the authors introduced a Gauss curvature flow to study the Aleksandrov problem and the dual Minkowski problem. The paper ^[20] treated the cases when one can establish the uniform estimate for the Gauss curvature flow. In this paper, we study the $ L_p $ dual Minkowski problem, an extension of the dual Minkowski problem. We deal with some cases in which there is no uniform estimate for the Gauss curvature flow. We adopt the topological method from ^[13] to find a special initial condition such that the Gauss curvature flow converges to a solution of the $ L_p $ dual Minkowski problem.
- Gauss curvature flow,
- Monge-Ampere equation,
- Minkowski problem,
- variational method,
- a priori estimates
Citation: Qiang Guang, Qi-Rui Li, Xu-Jia Wang. Flow by Gauss curvature to the $ L_p $ dual Minkowski problem[J]. Mathematics in Engineering, 2023, 5(3): 1-19. doi: 10.3934/mine.2023049

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Abstract

In the paper ^[20], the authors introduced a Gauss curvature flow to study the Aleksandrov problem and the dual Minkowski problem. The paper ^[20] treated the cases when one can establish the uniform estimate for the Gauss curvature flow. In this paper, we study the $ L_p $ dual Minkowski problem, an extension of the dual Minkowski problem. We deal with some cases in which there is no uniform estimate for the Gauss curvature flow. We adopt the topological method from ^[13] to find a special initial condition such that the Gauss curvature flow converges to a solution of the $ L_p $ dual Minkowski problem.

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