An optimal lower bound in fractional spectral geometry for planar sets with topological constraints

TL;DR

This paper establishes an optimal lower bound for the first eigenvalue of the fractional Dirichlet-Laplacian on planar sets with topological constraints, connecting spectral properties with geometry and topology.

Contribution

It introduces a new optimal lower bound for fractional spectral geometry that generalizes classical results and analyzes the limits as the fractional order approaches 1 and 1/2.

Findings

The lower bound is proven to be optimal in many cases.

The classical result is recovered as the fractional order approaches 1.

The behavior as the fractional order approaches 1/2 is thoroughly analyzed.

Abstract

We prove a lower bound on the first eigenvalue of the fractional Dirichlet-Laplacian of order $s$ on planar open sets, in terms of their inradius and topology. The result is optimal, in many respects. In particular, we recover a classical result proved independently by Croke, Osserman and Taylor, in the limit as $s$ goes to $1$. The limit as $s$ goes to $1/2$ is carefully analyzed, as well.