Geometric bounds for Steklov and weighted Neumann eigenvalues on Euclidean domains

Summary · qwen2.5:32b

The study establishes sharp upper bounds for the first two nonzero Steklov eigenvalues in Euclidean spaces of dimension $d \geq 7$, normalized by volume and boundary measure, and derives strict upper bounds for dimensions $3 \leq d \leq 6$. Additionally, it extends previous results to provide strict upper bounds for all higher Steklov eigenvalues on planar simply connected domains with continuous boundaries.

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Exploring new bounds in Steklov and weighted Neumann eigenvalues, advancing real-time rendering and mathematics research.

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arXiv:2604.03418v3 Announce Type: replace Abstract: We obtain sharp upper bounds for the first two nonzero Steklov eigenvalues among bounded domains in Euclidean spaces of dimension $d \geq 7$ under a natural normalization involving volume and boundary measure. These bounds are derived from a characterization of optimal domains and weights for the first two nonzero weighted Neumann eigenvalues. In dimensions $3 \leq d \leq 6$, we obtain strict upper bounds. We further establish strict upper bounds for all higher Steklov eigenvalues on planar simply connected domains with continuous boundary, extending previous results which, beyond the second nonzero eigenvalue, were known only for smooth planar domains.