Isoperimetric bounds for the weighted Neumann and Steklov eigenvalues in dimensions three through six
Abstract
For 3≤ n≤6, we determine the sharp isoperimetric bound for the mass-normalized first weighted Neumann eigenvalue on bounded Euclidean domains. For every finite nonnegative nonatomic measure μ on Ω̅, λ̅1N(Ω,μ)≤(n(n−1))/(n−2)ωnsin2ϑn(|Ω|/ωn)(n−2)/n, where ωn is the volume of the unit ball and ϑn∈(π/2,π) is the angle at the first stationary radius of the regular rotational harmonic-map profile. The bound is attained on balls by a smooth positive radial density. In fact, we affirmatively resolve Question~1.12 posed by Vinokurov \cite{Vinokurov2026}. Furthermore, for admissible domains with finite boundary measure, we also prove the sharp strict Steklov bound σ1(Ω)|∂Ω| |Ω|(2−n)/n<(n(n−1))/(n−2)ωn2/nsin2ϑn, whose constant is approached by C1 perforated domains.
Related
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BibTeX
@misc{chen2026isoperimetric,
title = {{Isoperimetric bounds for the weighted Neumann and Steklov eigenvalues in dimensions three through six}},
author = {Daguang Chen and Hao Liu and Chengxi Yang},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.33555},
eprint = {2609.33555},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.33555},
note = {CC BY-NC-SA 4.0},
}
RIS
TY - UNPB
TI - Isoperimetric bounds for the weighted Neumann and Steklov eigenvalues in dimensions three through six
AU - Daguang Chen
AU - Hao Liu
AU - Chengxi Yang
PY - 2026
DA - 2026/09/27
PB - arXiv
DO - 10.48550/arxiv.2609.33555
UR - https://arxiv.org/abs/2609.33555
AB - For 3≤ n≤6, we determine the sharp isoperimetric bound for the mass-normalized first weighted Neumann eigenvalue on bounded Euclidean domains. For every finite nonnegative nonatomic measure μ on Ω̅, λ̅_1^N(Ω,μ)≤(n(n−1))/(n−2)ω_nsin^2ϑ_n(|Ω|/ω_n)^(n−2)/n, where ω_n is the volume of the unit ball and ϑ_n∈(π/2,π) is the angle at the first stationary radius of the regular rotational harmonic-map profile. The bound is attained on balls by a smooth positive radial density. In fact, we affirmatively resolve Question~1.12 posed by Vinokurov \cite{Vinokurov2026}. Furthermore, for admissible domains with finite boundary measure, we also prove the sharp strict Steklov bound σ_1(Ω)|∂Ω| |Ω|^(2−n)/n<(n(n−1))/(n−2)ω_n^2/nsin^2ϑ_n, whose constant is approached by C^1 perforated domains.
ER -