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 $s$ 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.
- Publication:
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arXiv e-prints
- Pub Date:
- January 2023
- DOI:
- 10.48550/arXiv.2301.08017
- arXiv:
- arXiv:2301.08017
- Bibcode:
- 2023arXiv230108017B
- Keywords:
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- Mathematics - Analysis of PDEs;
- Mathematics - Spectral Theory;
- 47A75;
- 39B72;
- 35R11
- E-Print:
- 38 pages, 6 figures