Calculus of Variations and Geometric Measure Theory

F. Bianchi - L. Brasco

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

created by bianchi on 19 Jan 2023
modified by brasco on 03 Feb 2023


Submitted Paper

Inserted: 19 jan 2023
Last Updated: 3 feb 2023

Year: 2023


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.

Keywords: capacity, Poincare inequality, fractional Laplacian, fractional Sobolev spaces, Inradius, eigenvalue estimates