Isoperimetric Bounds for Weighted Steklov Eigenvalues with Radial Weights

We study the following class of Steklov eigenvalue problems: [Formula: see text] where [Formula: see text] and [Formula: see text] are prescribed positive radial functions, [Formula: see text] is a Lipschitz domain in [Formula: see text] with [Formula: see text] and [Formula: see text] 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 [Formula: see text] and [Formula: see text], where the parameters [Formula: see text] 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 [Formula: see text] and [Formula: see text], where [Formula: see text] 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.