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| Auteurs principaux: | , , |
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| Format: | Preprint |
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2024
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| Accès en ligne: | https://arxiv.org/abs/2412.20335 |
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| _version_ | 1866910766304591872 |
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| author | Du, Wenkui Wang, Ling Yang, Yang |
| author_facet | Du, Wenkui Wang, Ling Yang, Yang |
| contents | We prove a half-space Bernstein theorem for Allen-Cahn equation. More precisely, we show that every solution $u$ of the Allen-Cahn equation in the half-space $\overline{\mathbb{R}^n_+}:=\{(x_1,x_2,\cdots,x_n)\in\mathbb{R}^n:\,x_1\geq 0\}$ with $|u|\leq 1$, boundary value given by the restriction of a one-dimensional solution on $\{x_1=0\}$ and monotone condition $\partial_{x_n}u>0$ as well as limiting condition $\lim_{x_n\to\pm\infty}u(x',x_n)=\pm 1$ must itself be one-dimensional, and the parallel flat level sets and $\{x_1=0\}$ intersect at the same fixed angle in $(0, \fracπ{2}]$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_20335 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Flat level sets of Allen-Cahn equation in half-space Du, Wenkui Wang, Ling Yang, Yang Analysis of PDEs We prove a half-space Bernstein theorem for Allen-Cahn equation. More precisely, we show that every solution $u$ of the Allen-Cahn equation in the half-space $\overline{\mathbb{R}^n_+}:=\{(x_1,x_2,\cdots,x_n)\in\mathbb{R}^n:\,x_1\geq 0\}$ with $|u|\leq 1$, boundary value given by the restriction of a one-dimensional solution on $\{x_1=0\}$ and monotone condition $\partial_{x_n}u>0$ as well as limiting condition $\lim_{x_n\to\pm\infty}u(x',x_n)=\pm 1$ must itself be one-dimensional, and the parallel flat level sets and $\{x_1=0\}$ intersect at the same fixed angle in $(0, \fracπ{2}]$. |
| title | Flat level sets of Allen-Cahn equation in half-space |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2412.20335 |