Calculus of Variations and Geometric Measure Theory

M. Caselli - L. Gennaioli

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

created by caselli on 21 Jun 2023
modified on 31 May 2024



Inserted: 21 jun 2023
Last Updated: 31 may 2024

Pages: 39
Year: 2023

ArXiv: 2306.11590 PDF


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.