Asymptotics as $s \to 0^+$ of the fractional perimeter on Riemannian manifolds
Fuente:
arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2023
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| _version_ | 1866913358818574336 |
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| author | Caselli, Michele Gennaioli, Luca |
| author_facet | Caselli, Michele Gennaioli, Luca |
| contents | In this work we study the asymptotics of the fractional Laplacian as $s\to 0^+$ on any complete Riemannian manifold $(M,g)$, both of finite and infinite volume. Surprisingly enough, when $M$ is not stochastically complete this asymptotics is related to the existence of bounded harmonic functions on $M$.
As a corollary, we can find the asymptotics of the fractional $s$-perimeter on (essentially) every complete manifold, generalising both the existing results for $\mathbb{R}^n$ and for the Gaussian space. In doing so, from many sets $E\subset M$ we are able to produce a bounded harmonic function associated to $E$, which in general can be non-constant. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2306_11590 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Asymptotics as $s \to 0^+$ of the fractional perimeter on Riemannian manifolds Caselli, Michele Gennaioli, Luca Differential Geometry Analysis of PDEs In this work we study the asymptotics of the fractional Laplacian as $s\to 0^+$ on any complete Riemannian manifold $(M,g)$, both of finite and infinite volume. Surprisingly enough, when $M$ is not stochastically complete this asymptotics is related to the existence of bounded harmonic functions on $M$. As a corollary, we can find the asymptotics of the fractional $s$-perimeter on (essentially) every complete manifold, generalising both the existing results for $\mathbb{R}^n$ and for the Gaussian space. In doing so, from many sets $E\subset M$ we are able to produce a bounded harmonic function associated to $E$, which in general can be non-constant. |
| title | Asymptotics as $s \to 0^+$ of the fractional perimeter on Riemannian manifolds |
| topic | Differential Geometry Analysis of PDEs |
| url | https://arxiv.org/abs/2306.11590 |