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

Abstract

We prove a lower bound on the first eigenvalue of the fractional Dirichlet–Laplacian of order 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 goes to 1. The limit as goes to is carefully analyzed, as well.