Geometric bounds for Steklov and weighted Neumann eigenvalues on Euclidean domains

Abstract

We obtain sharp upper bounds for the first two nonzero Steklov eigenvalues among bounded domains in Euclidean spaces of dimension $d\ge 7$ 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 $3\le d\le 6$, 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.

16 pages; minor revision: correction of the exact values in Theorems 1.5 and 1.6 and Corollary 1.7; added more details on the proof Theorem 1.8