Asymptotics as s → 0⁺ of the fractional perimeter on Riemannian manifolds

2023/06/20 by Caselli, Michele, Gennaioli, Luca
#Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics

paper · doi:10.48550/arxiv.2306.11590

Abstract

In this work we study the asymptotics of the fractional Laplacian as s→ 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 ℝⁿ 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.

Related

• Asymptotics of the s-fractional Gaussian perimeter as s→ 0⁺
• Fractional Sobolev spaces on Riemannian manifolds
• Second-order asymptotics of the fractional perimeter as s→ 1
• Asymptotics of the Poisson kernel and Green's functions of the fractional conformal Laplacian
• Gamma-convergence of fractional Gaussian perimeter