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Planar Dirichlet eigenvalue bounds from a trace identity
Abstract
We prove an asymptotically sharp lower bound for the eigenvalues λk of the Dirichlet Laplacian on bounded open sets in the plane with area A>0. The proof employs a trace identity for the commutator between a certain multiplication operator and its adjoint. As a corollary, we obtain (2/3)· 4πk/A≤ λk for every k, which is two thirds of the bound conjectured by Pólya.
Related
- A priori Dirichlet energy bounds for positive Trudinger--Moser critical points on simply connected planar domains
- Neumann--Dirichlet eigenvalue comparison at the first Dirichlet threshold in the plane
- Arbitrary Dirichlet Eigenvalue Multiplicities on Smooth Star-Shaped Euclidean Domains
- Counterexamples to a higher-index Dirichlet eigenvalue-ratio conjecture
- Dirichlet eigenvalue asymptotics on conformally compact Einstein manifolds
BibTeX
@misc{speciel2026planar,
title = {{Planar Dirichlet eigenvalue bounds from a trace identity}},
author = {Romain Speciel},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.05636},
eprint = {2610.05636},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.05636},
}
RIS
TY - UNPB
TI - Planar Dirichlet eigenvalue bounds from a trace identity
AU - Romain Speciel
PY - 2026
DA - 2026/10/05
PB - arXiv
DO - 10.48550/arxiv.2610.05636
UR - https://arxiv.org/abs/2610.05636
AB - We prove an asymptotically sharp lower bound for the eigenvalues λ_k of the Dirichlet Laplacian on bounded open sets in the plane with area A>0. The proof employs a trace identity for the commutator between a certain multiplication operator and its adjoint. As a corollary, we obtain (2/3)· 4πk/A≤ λ_k for every k, which is two thirds of the bound conjectured by Pólya.
ER -