Capillary John ellipsoid theorem with applications to capillary curvature problems
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866910104636358656 |
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| author | Hu, Jinrong Yang, Bo |
| author_facet | Hu, Jinrong Yang, Bo |
| contents | In this paper, we apply a capillary John ellipsoid theorem for capillary convex bodies in the Euclidean half-space $\overline{\mathbb{R}^{n+1}_{+}}$. This theorem yields a non-collapsing estimate for capillary hypersurfaces, which provides a new approach to obtaining $C^{0}$ estimates for solutions to some capillary curvature problems (including the capillary $L_{p}$ Christoffel-Minkowski problem and the capillary $L_{p}$ curvature problem), based on the corresponding gradient estimates. As an application, we study the capillary $L_{p}$ dual Minkowski problem. By deriving a gradient estimate, refining a $C^{2}$ estimate, and combining these with the non-collapsing estimate, we establish existence in the case $1<p\leq q\leq 3$ and improve upon the existing existence result for the case $p > q$ in $\overline{\mathbb{R}^3_{+}}$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2603_27252 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Capillary John ellipsoid theorem with applications to capillary curvature problems Hu, Jinrong Yang, Bo Analysis of PDEs Differential Geometry In this paper, we apply a capillary John ellipsoid theorem for capillary convex bodies in the Euclidean half-space $\overline{\mathbb{R}^{n+1}_{+}}$. This theorem yields a non-collapsing estimate for capillary hypersurfaces, which provides a new approach to obtaining $C^{0}$ estimates for solutions to some capillary curvature problems (including the capillary $L_{p}$ Christoffel-Minkowski problem and the capillary $L_{p}$ curvature problem), based on the corresponding gradient estimates. As an application, we study the capillary $L_{p}$ dual Minkowski problem. By deriving a gradient estimate, refining a $C^{2}$ estimate, and combining these with the non-collapsing estimate, we establish existence in the case $1<p\leq q\leq 3$ and improve upon the existing existence result for the case $p > q$ in $\overline{\mathbb{R}^3_{+}}$. |
| title | Capillary John ellipsoid theorem with applications to capillary curvature problems |
| topic | Analysis of PDEs Differential Geometry |
| url | https://arxiv.org/abs/2603.27252 |