M. Caselli - L. Gennaioli
Asymptotics as $s \to 0^+$ of the fractional perimeter on Riemannian manifolds
Preprint
Inserted: 21 jun 2023
Last Updated: 27 aug 2023
Pages: 39
Year: 2023
Abstract:
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, generalizing both the existing results for $\mathbb{R}^n$ and for the Gaussian space. In doing so, from many sets $E ⊂ M$ we are able to produce a bounded harmonic function associated to $E$, which in general can be non-constant.
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