Stable $s$-minimal cones in $\mathbb{R}^2$ are flat for $s \sim 0$
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arXiv
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866913782435938304 |
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| author | Caselli, Michele |
| author_facet | Caselli, Michele |
| contents | For $s \in (0,1)$ small, we show that the only cones in $\mathbb{R}^2$ stationary for the $s$-perimeter and stable in $\mathbb{R}^2 \setminus \{0\}$ are half-planes. This is in direct contrast with the case of the classical perimeter or the regime $s$ close to $1$, where nontrivial cones as $\{xy>0\} \subset \mathbb{R}^2$ are stable for inner variations. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_06318 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Stable $s$-minimal cones in $\mathbb{R}^2$ are flat for $s \sim 0$ Caselli, Michele Analysis of PDEs For $s \in (0,1)$ small, we show that the only cones in $\mathbb{R}^2$ stationary for the $s$-perimeter and stable in $\mathbb{R}^2 \setminus \{0\}$ are half-planes. This is in direct contrast with the case of the classical perimeter or the regime $s$ close to $1$, where nontrivial cones as $\{xy>0\} \subset \mathbb{R}^2$ are stable for inner variations. |
| title | Stable $s$-minimal cones in $\mathbb{R}^2$ are flat for $s \sim 0$ |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2412.06318 |