The fractional Makai-Hayman inequality

TL;DR

This paper establishes a lower bound for the first eigenvalue of the fractional Dirichlet-Laplacian on simply connected planar sets based solely on inradius, valid for 1/2<s<1, with sharpness and asymptotic analysis.

Contribution

It introduces a fractional Makai-Hayman inequality, providing the first eigenvalue bound in terms of inradius for fractional Laplacians, and analyzes the sharpness and asymptotic behavior of the constant.

Findings

Lower bound valid for 1/2<s<1

Bound is sharp at s=1/2

Constant matches classical results as s approaches 1

Abstract

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.