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| Natura: | Preprint |
| Pubblicazione: |
2024
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| Accesso online: | https://arxiv.org/abs/2406.04618 |
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| _version_ | 1866917459579109376 |
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| author | Thompson, Jack |
| author_facet | Thompson, Jack |
| contents | We prove that measurable sets $E\subset \mathbb R^n$ with locally finite perimeter and zero $s$-mean curvature satisfy the surface density estimates: \begin{align*}
\operatorname{Per} (E; B_R(x)) \geq CR^{n-1} \end{align*} for all $R>0$, $x\in \partial^\ast E$. The $C$ depends only on $n$ and $s$, and remains bounded as $s\to 1^-$. As an application, we prove that the fractional Sobolev inequality holds on the boundary of sets with zero $s$-mean curvature. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_04618 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Density estimates and the fractional Sobolev inequality for sets of zero $s$-mean curvature Thompson, Jack Analysis of PDEs 35R11 (Primary), 53A10 (Secondary) We prove that measurable sets $E\subset \mathbb R^n$ with locally finite perimeter and zero $s$-mean curvature satisfy the surface density estimates: \begin{align*} \operatorname{Per} (E; B_R(x)) \geq CR^{n-1} \end{align*} for all $R>0$, $x\in \partial^\ast E$. The $C$ depends only on $n$ and $s$, and remains bounded as $s\to 1^-$. As an application, we prove that the fractional Sobolev inequality holds on the boundary of sets with zero $s$-mean curvature. |
| title | Density estimates and the fractional Sobolev inequality for sets of zero $s$-mean curvature |
| topic | Analysis of PDEs 35R11 (Primary), 53A10 (Secondary) |
| url | https://arxiv.org/abs/2406.04618 |