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 04 Sep 2023


Accepted Paper

Inserted: 19 jan 2023
Last Updated: 4 sep 2023

Journal: J. London Math. Soc.
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