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

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\