PreprintCC BY 4.0arXiv
Inequalities between Dirichlet and Neumann eigenvalues in large dimensions
Abstract
Let Ω be a bounded domain in Rd. Denote by λk (resp. μk) the eigenvalues of the Laplace operator in Ω with Dirichlet (resp. Neumann) boundary conditions. Denote by Ψ= Ψ(d,k,Ω) the shift of indices in the inequality μk+Ψ ≤ λk. We are interested to describe the behaviour of Ψ for large d. We prove that a) Ψ(d,1,Ω) ≥ C (e/2)d for all domains Ω; and b) Ψ(d,k,Ω) ≥ C (e/2)d for all k and all convex domains Ω.
Related
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BibTeX
@misc{filonov2026inequalities,
title = {{Inequalities between Dirichlet and Neumann eigenvalues in large dimensions}},
author = {N. Filonov},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2606.30120},
eprint = {2606.30120},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2606.30120},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Inequalities between Dirichlet and Neumann eigenvalues in large dimensions
AU - N. Filonov
PY - 2026
DA - 2026/10/31
PB - arXiv
DO - 10.48550/arxiv.2606.30120
UR - https://arxiv.org/abs/2606.30120
AB - Let Ω be a bounded domain in R^d. Denote by λ_k (resp. μ_k) the eigenvalues of the Laplace operator in Ω with Dirichlet (resp. Neumann) boundary conditions. Denote by Ψ= Ψ(d,k,Ω) the shift of indices in the inequality μ_k+Ψ ≤ λ_k. We are interested to describe the behaviour of Ψ for large d. We prove that a) Ψ(d,1,Ω) ≥ C (e/2)^d for all domains Ω; and b) Ψ(d,k,Ω) ≥ C (e/2)^d for all k and all convex domains Ω.
ER -