Isoperimetric Bounds for Weighted Steklov Eigenvalues with Radial Weights
Friedemann Brock, Francesco Chiacchio
TL;DR
The paper addresses isoperimetric bounds for low-order Steklov eigenvalues in weighted settings with radial weights. It develops two regimes: (i) w=|x|^{α}, v=|x|^{β−α}, and (ii) w=W(|x|) with W log-convex and v≡1, deriving explicit radial spectra on balls and proving weighted isoperimetric inequalities under symmetry assumptions (S1) and (S2). The main results establish γ_1(Ω)≤γ_1(B_R) and, under (S2), a lower bound for the sum of reciprocals ⨍_{i=1}^N 1/γ_i(Ω), with sharp behavior in symmetric domains. Tools include separation of variables, two-density and weighted-perimeter inequalities, and Hardy-Littlewood-type rearrangement arguments, which may be of independent interest for weighted spectral problems.
Abstract
We study the following class of Steklov eigenvalue problems: \[ \nabla \cdot \bigl( w \nabla u \bigr) = 0 \quad \text{in } Ω, \qquad \frac{\partial u}{\partial ν} = γv u \quad \text{on } \partial Ω, \] where $w$ and $v$ are prescribed positive radial functions, $Ω$ is a Lipschitz domain in $\mathbb{R}^N$ with $N \geq 2$ and $ν$ 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 $w(x) = |x|^α$ and $v(x) = |x|^{β-α}$, where the parameters $α, β\in \mathbb{R}$ 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.