Isoperimetric Bounds for Weighted Steklov Eigenvalues with Radial Weights Article Swipe

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Friedemann Brock , Francesco Chiacchio · YOU? · · 2025 · Open Access · · DOI: https://doi.org/10.48550/arxiv.2510.12631

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.

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preprint

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en

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http://arxiv.org/abs/2510.12631

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https://arxiv.org/pdf/2510.12631

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https://doi.org/10.48550/arxiv.2510.12631

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Title

Isoperimetric Bounds for Weighted Steklov Eigenvalues with Radial Weights

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preprint

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en

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2025

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2025-10-14

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Friedemann Brock, Francesco Chiacchio

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https://arxiv.org/abs/2510.12631

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https://arxiv.org/pdf/2510.12631

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green

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