Sharp upper bounds for Steklov eigenvalues of a hypersurface of revolution with two boundary components in Euclidean space

TL;DR

This paper establishes sharp upper bounds for Steklov eigenvalues of revolution hypersurfaces with two boundary components in Euclidean space, including explicit bounds for the first non-zero eigenvalue and stability properties.

Contribution

It provides explicit sharp upper bounds for Steklov eigenvalues of hypersurfaces of revolution with two boundary components, including the first non-zero eigenvalue, and analyzes their stability.

Findings

Explicit sharp upper bounds for Steklov eigenvalues depending on dimension and meridian length.

Identification of a degenerated metric achieving the bounds.

Proof of stability properties of the bounds.

Abstract

We investigate the question of sharp upper bounds for the Steklov eigenvalues of a hypersurface of revolution of the Euclidean space with two boundary components isometric to two copies of $\mathbb{S}^{n-1}$. For the case of the first non zero Steklov eigenvalue, we give a sharp upper bound $B_n(L)$ (that depends only on the dimension $n \ge 3$ and the meridian length $L>0$) which is reached by a degenerated metric $g^*$, that we compute explicitly. We also give a sharp upper bound $B_n$ which depends only on $n$. Our method also permits us to prove some stability properties of these upper bounds.