Asymptotics as $s \to 0^+$ of the fractional perimeter on Riemannian manifolds

Michele Caselli; Luca Gennaioli

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arXiv · 2306.11590

Asymptotics as $s \to 0^+$ of the fractional perimeter on Riemannian manifolds

Michele Caselli·Luca Gennaioli

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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, 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.

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Michele Caselli, Luca Gennaioli. 2023-06-20. Asymptotics as $s \to 0^+$ of the fractional perimeter on Riemannian manifolds. https://arxiv.org/abs/2306.11590

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