A comparison between Neumann and Steklov eigenvalues

TL;DR

This paper compares Neumann and Steklov eigenvalues for planar domains, establishing bounds and behavior of their ratio, especially for convex and thin convex sets, and visualizes the relationship through Blaschke-Santaló diagrams.

Contribution

It provides the first explicit bounds for the ratio of Neumann to Steklov eigenvalues in convex domains and analyzes the ratio's behavior in thin convex sets.

Findings

The ratio can be arbitrarily small or large without restrictions.

Explicit bounds are derived for convex domains.

The relationship is visualized with Blaschke-Santaló diagrams.

Abstract

This paper is devoted to a comparison between the normalized first (non-trivial) Neumann eigenvalue $|\Omega| \mu_1(\Omega)$ for a Lipschitz open set $\Omega$ in the plane, and the normalized first (non-trivial) Steklov eigenvalue $P(\Omega) \sigma_1(\Omega)$. More precisely, we study the ratio $F(\Omega):=|\Omega| \mu_1(\Omega)/P(\Omega) \sigma_1(\Omega)$. We prove that this ratio can take arbitrarily small or large values if we do not put any restriction on the class of sets $\Omega$. Then we restrict ourselves to the class of plane convex domains for which we get explicit bounds. We also study the case of thin convex domains for which we give more precise bounds. The paper finishes with the plot of the corresponding Blaschke-Santal\'o diagrams $(x,y)=\left(|\Omega| \mu_1(\Omega), P(\Omega) \sigma_1(\Omega) \right)$.