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Neumann--Dirichlet eigenvalue comparison at the first Dirichlet threshold in the plane
Abstract
Let Ω⊂ℝ2 be a bounded connected Lipschitz domain, and let {μj(Ω)}j≥1 and {λj(Ω)}j≥1 denote the Neumann and Dirichlet Laplacian eigenvalues, respectively, counted with multiplicity. We prove that μ3(Ω)<λ1(Ω), thereby removing the simple-connectivity assumption from the previously known planar result at the first Dirichlet threshold. We also establish a three-spectrum counting inequality relating the Dirichlet, Neumann, and conductivity spectra.
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BibTeX
@misc{tang2026neumanndirichlet,
title = {{Neumann--Dirichlet eigenvalue comparison at the first Dirichlet threshold in the plane}},
author = {Quanyu Tang and Haiqi Zhang},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.05893},
eprint = {2609.05893},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.05893},
}
RIS
TY - UNPB
TI - Neumann--Dirichlet eigenvalue comparison at the first Dirichlet threshold in the plane
AU - Quanyu Tang
AU - Haiqi Zhang
PY - 2026
DA - 2026/09/05
PB - arXiv
DO - 10.48550/arxiv.2609.05893
UR - https://arxiv.org/abs/2609.05893
AB - Let Ω⊂ℝ^2 be a bounded connected Lipschitz domain, and let {μ_j(Ω)}_j≥1 and {λ_j(Ω)}_j≥1 denote the Neumann and Dirichlet Laplacian eigenvalues, respectively, counted with multiplicity. We prove that μ_3(Ω)<λ_1(Ω), thereby removing the simple-connectivity assumption from the previously known planar result at the first Dirichlet threshold. We also establish a three-spectrum counting inequality relating the Dirichlet, Neumann, and conductivity spectra.
ER -