Convex capillary hypersurfaces of prescribed curvature problem
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866910915195043840 |
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| author | Mei, Xinqun Wang, Guofang Weng, Liangjun |
| author_facet | Mei, Xinqun Wang, Guofang Weng, Liangjun |
| contents | In this paper, we study the prescribed $k$-th Weingarten curvature problem for convex capillary hypersurfaces in $\overline{\mathbb{R}^{n+1}_+}$. This problem naturally extends the prescribed $k$-th Weingarten curvature problem for closed convex hypersurfaces, previously investigated by Guan-Guan in [19], to the capillary setting. We reformulate the problem as the solvability of a Hessian quotient equation with a Robin boundary condition on a spherical cap. Under a natural sufficient condition, we establish the existence of a strictly convex capillary hypersurface with the prescribed $k$-th Weingarten curvature. This also extends our recent work on the capillary Minkowski problem in [40]. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2504_14392 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Convex capillary hypersurfaces of prescribed curvature problem Mei, Xinqun Wang, Guofang Weng, Liangjun Differential Geometry Analysis of PDEs Primary: 53C21, 35B65 Secondary: 35J60, 53C42 In this paper, we study the prescribed $k$-th Weingarten curvature problem for convex capillary hypersurfaces in $\overline{\mathbb{R}^{n+1}_+}$. This problem naturally extends the prescribed $k$-th Weingarten curvature problem for closed convex hypersurfaces, previously investigated by Guan-Guan in [19], to the capillary setting. We reformulate the problem as the solvability of a Hessian quotient equation with a Robin boundary condition on a spherical cap. Under a natural sufficient condition, we establish the existence of a strictly convex capillary hypersurface with the prescribed $k$-th Weingarten curvature. This also extends our recent work on the capillary Minkowski problem in [40]. |
| title | Convex capillary hypersurfaces of prescribed curvature problem |
| topic | Differential Geometry Analysis of PDEs Primary: 53C21, 35B65 Secondary: 35J60, 53C42 |
| url | https://arxiv.org/abs/2504.14392 |