Flow by Gauss Curvature to the orlicz Chord Minkowski Problem
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866910427828453376 |
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| author | Zhao, Xia Zhao, Peibiao |
| author_facet | Zhao, Xia Zhao, Peibiao |
| contents | The $L_p$ chord Minkowski problem based on Chord measures and $L_p$ chord measures introduced firstly by Lutwak, Xi, Yang and Zhang [38] is a very important and meaningful geometric measure problem in the $L_p$ Brunn-Minkowski theory. Xi, Yang, Zhang and Zhao [45] using variational methods gave a measure solution when $p > 1$ and $0<p<1$ in the symmetric case. Recently, Guo, Xi and Zhao [18] also obtained a measure solution for $0\leq p<1$ by similar methods without the symmetric assumption.
In the present paper, we investigate and confirm the orlicz chord Minkowski problem, which generalizes the $L_p$ chord Minkowski problem by replacing $p$ with a fixed continuous function $φ:(0,\infty)\rightarrow(0,\infty)$, and achieve the existence of smooth solutions to the orlicz chord Minkowski problem by using methods of Gauss curvature flows. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2308_05332 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Flow by Gauss Curvature to the orlicz Chord Minkowski Problem Zhao, Xia Zhao, Peibiao Differential Geometry Analysis of PDEs The $L_p$ chord Minkowski problem based on Chord measures and $L_p$ chord measures introduced firstly by Lutwak, Xi, Yang and Zhang [38] is a very important and meaningful geometric measure problem in the $L_p$ Brunn-Minkowski theory. Xi, Yang, Zhang and Zhao [45] using variational methods gave a measure solution when $p > 1$ and $0<p<1$ in the symmetric case. Recently, Guo, Xi and Zhao [18] also obtained a measure solution for $0\leq p<1$ by similar methods without the symmetric assumption. In the present paper, we investigate and confirm the orlicz chord Minkowski problem, which generalizes the $L_p$ chord Minkowski problem by replacing $p$ with a fixed continuous function $φ:(0,\infty)\rightarrow(0,\infty)$, and achieve the existence of smooth solutions to the orlicz chord Minkowski problem by using methods of Gauss curvature flows. |
| title | Flow by Gauss Curvature to the orlicz Chord Minkowski Problem |
| topic | Differential Geometry Analysis of PDEs |
| url | https://arxiv.org/abs/2308.05332 |