Isodiametric-type upper bounds for Steklov eigenvalues
Abstract
Let n≥ 2. We establish isodiametric-type upper bounds for Steklov eigenvalues of bounded Lipschitz domains Ω in Euclidean space ℝn and hyperbolic space ℍn. In Euclidean space, we prove D(Ω)σk(Ω)≤ C(n)k, with optimal dependence on k. For domains in either ℝn or ℍn, we establish D(Ω)σk(Ω)≤ C(n)k2(|Ω|1/n/D(Ω))n/(n−1). The exponent of the volume-diameter ratio is optimal in both settings, and with this exponent fixed, the quadratic dependence on k is also optimal. Interpolating the preceding volume-diameter estimate with a volume-normalized Steklov bound, we further obtain D(Ω)σk(Ω)≤ C(n)k1+1/(2n−1) in hyperbolic space. These results answer Open Question 4.37 affirmatively and resolve the hyperbolic part of Open Question 4.35, both posed in [5].
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BibTeX
@misc{gu2026isodiametrictype,
title = {{Isodiametric-type upper bounds for Steklov eigenvalues}},
author = {Pingxin Gu and Yao Wan},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2608.30807},
eprint = {2608.30807},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.30807},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Isodiametric-type upper bounds for Steklov eigenvalues
AU - Pingxin Gu
AU - Yao Wan
PY - 2026
DA - 2026/10/19
PB - arXiv
DO - 10.48550/arxiv.2608.30807
UR - https://arxiv.org/abs/2608.30807
AB - Let n≥ 2. We establish isodiametric-type upper bounds for Steklov eigenvalues of bounded Lipschitz domains Ω in Euclidean space ℝ^n and hyperbolic space ℍ^n. In Euclidean space, we prove D(Ω)σ_k(Ω)≤ C(n)k, with optimal dependence on k. For domains in either ℝ^n or ℍ^n, we establish D(Ω)σ_k(Ω)≤ C(n)k^2(|Ω|^1/n/D(Ω))^n/(n−1). The exponent of the volume-diameter ratio is optimal in both settings, and with this exponent fixed, the quadratic dependence on k is also optimal. Interpolating the preceding volume-diameter estimate with a volume-normalized Steklov bound, we further obtain D(Ω)σ_k(Ω)≤ C(n)k^1+1/(2n−1) in hyperbolic space. These results answer Open Question 4.37 affirmatively and resolve the hyperbolic part of Open Question 4.35, both posed in [5].
ER -