Sharp bounds for the first two eigenvalues of an exterior Steklov eigenvalue problem

TL;DR

This paper establishes sharp bounds for the first two eigenvalues of the Steklov problem on exterior domains, linking geometric properties of the boundary to eigenvalue estimates, with implications for capacity and spectral geometry.

Contribution

It provides the first sharp bounds for the first and second Steklov eigenvalues on exterior domains, connecting geometric conditions to spectral estimates.

Findings

Sharp lower bound for the first eigenvalue in terms of support and distance functions.

Sharp upper bounds for the first eigenvalue under various geometric conditions.

Upper bound for the capacity of the boundary in three dimensions.

Abstract

Let $U\subset \mathbb{R}^n$ ($n\geq 3$) be an exterior Euclidean domain with smooth boundary $\partial U$. We consider the Steklov eigenvalue problem on $U$. First we derive a sharp lower bound for the first eigenvalue in terms of the support function and the distance function to the origin of $\partial U$. Second under various geometric conditions on $\partial U$ we obtain sharp upper bounds for the first eigenvalue. Along the proof, we get a sharp upper bound for the capacity of $\partial U$ when $n=3$ and $\partial U$ is connected. Last we also discuss an upper bound for the second eigenvalue.