Isoperimetric Bounds for Weighted Steklov Eigenvalues with Radial Weights

We study the following class of Steklov eigenvalue problems: \[ \nabla \cdot \bigl( w \nabla u \bigr) = 0 \quad \text{in } \Omega, \qquad \frac{\partial u}{\partial \nu} = \gamma v u \quad \text{on } \partial \Omega, \] where $w$ and $v$ are prescribed positive radial functions, $\Omega$ is a Lipschitz domain in $\mathbb{R}^N$ with $N \geq 2$ and $\nu$ denotes its outward unit normal. Extending classical results in the unweighted case due to Weinstock, the first author, and others, we establish isoperimetric inequalities for low-order eigenvalues under suitable symmetry assumptions on the domain. In the first part, we consider the case $w(x) = |x|^{\alpha}$ and $v(x) = |x|^{\beta-\alpha}$, where the parameters $\alpha, \beta \in \mathbb{R}$ satisfy appropriate constraints. Our analysis relies on an explicit computation of the spectrum in the radial case, variational principles, and a family of weighted isoperimetric inequalities with ``double density''. In the second part, we address the case $v \equiv 1$ and $w(x) = W(|x|)$, where $W$ is a non-decreasing, log-convex function. In this setting, the proof relies, among other tools, on a new weighted isoperimetric inequality, which may be of independent interest.