Isoperimetric bounds for the weighted Neumann and Steklov eigenvalues in dimensions three through six

Abstract

For $3\le n\le 6$, we determine the sharp isoperimetric bound for the mass-normalized first weighted Neumann eigenvalue on bounded Euclidean domains. For every finite nonnegative nonatomic measure $\mu$ on $\overline{\Omega }$, \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 ${\omega }_n$ is the volume of the unit ball and ${\vartheta }_n\in (\pi /2,\pi )$ 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.