A note on the first Steklov eigenvalue on planar domains

TL;DR

This paper investigates properties of the first Steklov eigenvalue on planar domains, providing examples, bounds, and simplicity results that extend previous mathematical understanding of these eigenvalues.

Contribution

It offers new examples of eigenfunctions with closed nodal lines, establishes lower bounds for symmetric domains, and proves eigenvalue simplicity for all ellipses.

Findings

Existence of a planar domain with a closed nodal line for the first eigenfunction

Lower bounds for the first eigenvalue on symmetric domains

Simplicity of the first eigenvalue for all ellipses

Abstract

We consider the first positive Steklov eigenvalue on planar domains. First, we provide an example of a planar domain for which a first eigenfunction has a closed nodal line. Second, we establish a lower bound for the first positive eigenvalue on certain symmetric domains and show that this eigenvalue is simple for all ellipses. These results complement two statements contained in a work by Kuttler and Sigillito (Proc. Amer. Math. Soc. 20, 1969).