The $L_p$ dual Minkowski problem for unbounded closed convex sets
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arXiv
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| Auteurs principaux: | , , |
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866910410919116800 |
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| author | Ai, Wen Yang, Yunlong Ye, Deping |
| author_facet | Ai, Wen Yang, Yunlong Ye, Deping |
| contents | The central focus of this paper is the $L_p$ dual Minkowski problem for $C$-compatible sets, where $C$ is a pointed closed convex cone in $\mathbb{R}^n$ with nonempty interior. Such a problem deals with the characterization of the $(p, q)$-th dual curvature measure of a $C$-compatible set. It produces new Monge-Ampère equations for unbounded convex hypersurface, often defined over open domains and with non-positive unknown convex functions. Within the family of $C$-determined sets, the $L_p$ dual Minkowski problem is solved for $0\neq p\in \mathbb{R}$ and $q\in \mathbb{R}$; while it is solved for the range of $p\leq 0$ and $p<q$ within the newly defined family of $(C, p, q)$-close sets. When $p\leq q$, we also obtain some results regarding the uniqueness of solutions to the $L_p$ dual Minkowski problem for $C$-compatible sets. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_09804 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The $L_p$ dual Minkowski problem for unbounded closed convex sets Ai, Wen Yang, Yunlong Ye, Deping Metric Geometry Analysis of PDEs 52A20, 52A39 The central focus of this paper is the $L_p$ dual Minkowski problem for $C$-compatible sets, where $C$ is a pointed closed convex cone in $\mathbb{R}^n$ with nonempty interior. Such a problem deals with the characterization of the $(p, q)$-th dual curvature measure of a $C$-compatible set. It produces new Monge-Ampère equations for unbounded convex hypersurface, often defined over open domains and with non-positive unknown convex functions. Within the family of $C$-determined sets, the $L_p$ dual Minkowski problem is solved for $0\neq p\in \mathbb{R}$ and $q\in \mathbb{R}$; while it is solved for the range of $p\leq 0$ and $p<q$ within the newly defined family of $(C, p, q)$-close sets. When $p\leq q$, we also obtain some results regarding the uniqueness of solutions to the $L_p$ dual Minkowski problem for $C$-compatible sets. |
| title | The $L_p$ dual Minkowski problem for unbounded closed convex sets |
| topic | Metric Geometry Analysis of PDEs 52A20, 52A39 |
| url | https://arxiv.org/abs/2404.09804 |