Isoperimetric bounds for the weighted Neumann and Steklov eigenvalues in dimensions three through six
Year: 2026
Type: preprint
Abstract: For $3\le n\le6$, 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 $\overlineΩ$, \begin{equation*} \overlineλ_1^N(Ω,μ)\le\frac{n(n-1)}{n-2}ω_n\sin^2\vartheta_n\left(\frac{|Ω|}{ω_n}\right)^{(n-2)/n}, \end{equation*} where $ω_n$ is the volume of the unit ball and $\vartheta_n\in(π/2,π)$ 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 \begin{equation*} σ_1(Ω)|\partialΩ|\,|Ω|^{(2-n)/n}<\frac{n(n-1)}{n-2}ω_n^{2/n}\sin^2\vartheta_n, \end{equation*} whose constant is approached by $C^1$ perforated domains.
Source: arXiv (Cornell University)
Authors: Chen, Daguang, Liu, Hao, Yang, Chengxi
Language: English
Subfield: Mathematical Physics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
Award: G1506100397
OpenAlex ID: W7214791244