Flatness of anisotropic minimal graphs in $\mathbb{R}^{n+1}$
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arXiv
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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2023
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| _version_ | 1866929303169531904 |
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| author | Du, Wenkui Yang, Yang |
| author_facet | Du, Wenkui Yang, Yang |
| contents | We prove a Bernstein theorem for $Φ$-anisotropic minimal hypersurfaces in all dimensional Euclidean spaces that the only entire smooth solutions $u: \mathbb{R}^{n}\rightarrow \mathbb{R}$ of $Φ$-anisotropic minimal hypersurfaces equation are linear functions provided the anisotropic area functional integrand $Φ$ is sufficiently $C^{3}$-close to classical area functional integrand and $|\nabla u(x)|=o(|x|^{\varepsilon})$ for $\varepsilon\leq \varepsilon_{0}(n, Φ)$ with the constant $\varepsilon_{0}(n, Φ)>0$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2311_00166 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Flatness of anisotropic minimal graphs in $\mathbb{R}^{n+1}$ Du, Wenkui Yang, Yang Analysis of PDEs Differential Geometry 35B08, 35J93, 49Q05, 53A10 We prove a Bernstein theorem for $Φ$-anisotropic minimal hypersurfaces in all dimensional Euclidean spaces that the only entire smooth solutions $u: \mathbb{R}^{n}\rightarrow \mathbb{R}$ of $Φ$-anisotropic minimal hypersurfaces equation are linear functions provided the anisotropic area functional integrand $Φ$ is sufficiently $C^{3}$-close to classical area functional integrand and $|\nabla u(x)|=o(|x|^{\varepsilon})$ for $\varepsilon\leq \varepsilon_{0}(n, Φ)$ with the constant $\varepsilon_{0}(n, Φ)>0$. |
| title | Flatness of anisotropic minimal graphs in $\mathbb{R}^{n+1}$ |
| topic | Analysis of PDEs Differential Geometry 35B08, 35J93, 49Q05, 53A10 |
| url | https://arxiv.org/abs/2311.00166 |