Calculus of Variations and Geometric Measure Theory

F. Bianchi - L. Brasco

The fractional Makai-Hayman inequality

created by bianchi on 14 Oct 2021
modified by brasco on 28 Mar 2022


Accepted Paper

Inserted: 14 oct 2021
Last Updated: 28 mar 2022

Journal: Ann. Mat. Pura Appl.
Pages: 30
Year: 2022


We prove that the first eigenvalue of the fractional Dirichlet-Laplacian of order $s$ on a simply connected set of the plane can be bounded from below in terms of its inradius only. This is valid for $1/2<s<1$ and we show that this condition is sharp, i.\,e. for $0<s\le 1/2$ such a lower bound is not possible. The constant appearing in the estimate has the correct asymptotic behaviour with respect to $s$, as it permits to recover a classical result by Makai and Hayman in the limit $s\nearrow 1$. The paper is as self-contained as possible.

Keywords: Poincare inequality, fractional Laplacian, Inradius, simply connected sets, Cheeger inequality