Regularities for solutions to the $L_p$ dual Minkowski problem for unbounded closed sets
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866914773308801024 |
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| author | Chen, Li Tu, Qiang |
| author_facet | Chen, Li Tu, Qiang |
| contents | Recently, the $L_p$ dual Minkowski problem for unbounded closed convex sets in a pointed closed convex cone was proposed and a weak solution to this problem was provided. In smooth setting, this problem is equivalent to solving the Dirichlet problem for a class of Monge-Ampère type equations. In this paper, we show the existence, regularity and uniqueness of solutions to this Monge-Ampère type equation in the case $p\geq 1$ by studying variational properties for a family of Monge-Ampère functionals. Moreover, the existence and optimal global Hölder regularity in the case $p<1$ and $q\geq n$ is also be discussed. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2403_00651 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Regularities for solutions to the $L_p$ dual Minkowski problem for unbounded closed sets Chen, Li Tu, Qiang Analysis of PDEs Differential Geometry Recently, the $L_p$ dual Minkowski problem for unbounded closed convex sets in a pointed closed convex cone was proposed and a weak solution to this problem was provided. In smooth setting, this problem is equivalent to solving the Dirichlet problem for a class of Monge-Ampère type equations. In this paper, we show the existence, regularity and uniqueness of solutions to this Monge-Ampère type equation in the case $p\geq 1$ by studying variational properties for a family of Monge-Ampère functionals. Moreover, the existence and optimal global Hölder regularity in the case $p<1$ and $q\geq n$ is also be discussed. |
| title | Regularities for solutions to the $L_p$ dual Minkowski problem for unbounded closed sets |
| topic | Analysis of PDEs Differential Geometry |
| url | https://arxiv.org/abs/2403.00651 |