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

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Submitted Paper

Inserted: 19 jan 2023
Last Updated: 3 feb 2023

Year: 2023

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.

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


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