Geometric bounds for Steklov and weighted Neumann eigenvalues on Euclidean domains

TL;DR

This paper derives sharp upper bounds for the first two nonzero Steklov eigenvalues in high-dimensional Euclidean domains and extends bounds for higher eigenvalues in planar domains, advancing spectral geometry understanding.

Contribution

It provides new sharp bounds for Steklov eigenvalues in high dimensions and extends bounds for higher eigenvalues in planar domains, including non-smooth boundaries.

Findings

Sharp upper bounds for first two nonzero Steklov eigenvalues in dimensions ≥7.

Non-sharp bounds for dimensions 3 to 6.

Strict upper bounds for all higher Steklov eigenvalues in planar simply connected domains.

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 upper bounds that are not sharp. 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.