Half-space Liouville-type theorems for minimal graphs with capillary boundary
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866917261742178304 |
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| author | Wang, Guofang Wei, Wei Zhang, Xuwen |
| author_facet | Wang, Guofang Wei, Wei Zhang, Xuwen |
| contents | In this paper, we prove two Liouville-type theorems for capillary minimal graph over $\mathbb{R}^n_+$. First, if $u$ has linear growth, then for $n=2,3$ and for any $θ\in(0,π)$, or $n\geq4$ and $θ\in(\fracπ6,\frac{5π}6)$, $u$ must be flat. Second, if $u$ is one-sided bounded on $\mathbb{R}^n_+$, then for any $n$ and $θ\in(0,π)$, $u$ must be flat. The proofs build upon gradient estimates for the mean curvature equation over $\mathbb{R}^n_+$ with capillary boundary condition, which are based on carefully adapting the maximum principle to the capillary setting. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_03417 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Half-space Liouville-type theorems for minimal graphs with capillary boundary Wang, Guofang Wei, Wei Zhang, Xuwen Differential Geometry Analysis of PDEs 53A10, 35J93, 35J25 In this paper, we prove two Liouville-type theorems for capillary minimal graph over $\mathbb{R}^n_+$. First, if $u$ has linear growth, then for $n=2,3$ and for any $θ\in(0,π)$, or $n\geq4$ and $θ\in(\fracπ6,\frac{5π}6)$, $u$ must be flat. Second, if $u$ is one-sided bounded on $\mathbb{R}^n_+$, then for any $n$ and $θ\in(0,π)$, $u$ must be flat. The proofs build upon gradient estimates for the mean curvature equation over $\mathbb{R}^n_+$ with capillary boundary condition, which are based on carefully adapting the maximum principle to the capillary setting. |
| title | Half-space Liouville-type theorems for minimal graphs with capillary boundary |
| topic | Differential Geometry Analysis of PDEs 53A10, 35J93, 35J25 |
| url | https://arxiv.org/abs/2506.03417 |