Compactness of the $L_p$ dual Minkowski problem in $\mathbb{R}^3$
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| Format: | Preprint |
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2025
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| _version_ | 1866910964096434176 |
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| author | Boroczky, Karoly J. Chen, Shibing Liu, Weiru Saroglou, Christos |
| author_facet | Boroczky, Karoly J. Chen, Shibing Liu, Weiru Saroglou, Christos |
| contents | We prove the $C^0$ estimate for the $L_p$ $q$th dual Minkowski problem on $S^2$ under fairly general conditions; namely, when $p$ lies in [0,1) and $q>2+p$, and the $L_p$ $q$th dual curvarture is bounded and bounded away from zero. We note that it is known that the analogous $C^0$ estimate does not hold if $p<-1$ and $q=3$. As a corollary of our $C^0$ estimate, we deduce the uniqueness of the solution of the near isotropic $q$th $L_p$ dual Minkowski problem on $S^2$ if $q$ is close to 3 and the $q$th $L_p$ dual curvature is Holder close to be the constant one function. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2505_17219 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Compactness of the $L_p$ dual Minkowski problem in $\mathbb{R}^3$ Boroczky, Karoly J. Chen, Shibing Liu, Weiru Saroglou, Christos Analysis of PDEs We prove the $C^0$ estimate for the $L_p$ $q$th dual Minkowski problem on $S^2$ under fairly general conditions; namely, when $p$ lies in [0,1) and $q>2+p$, and the $L_p$ $q$th dual curvarture is bounded and bounded away from zero. We note that it is known that the analogous $C^0$ estimate does not hold if $p<-1$ and $q=3$. As a corollary of our $C^0$ estimate, we deduce the uniqueness of the solution of the near isotropic $q$th $L_p$ dual Minkowski problem on $S^2$ if $q$ is close to 3 and the $q$th $L_p$ dual curvature is Holder close to be the constant one function. |
| title | Compactness of the $L_p$ dual Minkowski problem in $\mathbb{R}^3$ |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2505.17219 |