Bounds for higher Steklov and mixed Steklov Neumann eigenvalues on domains with holes

TL;DR

This paper derives bounds for higher Steklov and mixed Steklov Neumann eigenvalues on domains with holes, emphasizing symmetry effects and providing explicit formulas and numerical insights.

Contribution

It introduces explicit formulas for eigenvalues on annular domains and sharp bounds for symmetric domains with holes, highlighting the importance of symmetry in eigenvalue estimates.

Findings

Explicit second eigenvalue formula for concentric annuli

Sharp upper bounds for eigenvalues on symmetric domains with holes

Numerical observations and conjectures on eigenvalues

Abstract

In this article, we study Steklov eigenvalues and mixed Steklov Neumann eigenvalues on a smooth bounded domain in $\mathbb{R}^{n}$, $n \geq 2$, having a spherical hole. We focus on two main results related to Steklov eigenvalues. First, we obtain explicit expression for the second nonzero Steklov eigenvalue on concentric annular domain. Secondly, we derive a sharp upper bound of the first $n$ nonzero Steklov eigenvalues on a domain $\Omega \subset \mathbb{R}^{n}$ having symmetry of order $4$ and a ball removed from its center. This bound is given in terms of the corresponding Steklov eigenvalues on a concentric annular domain of the same volume as $\Omega$. Next, we consider the mixed Steklov Neumann eigenvalue problem on $4^{\text{th}}$ order symmetric domains in $\mathbb{R}^{n}$ having a spherical hole and obtain upper bound of the first $n$ nonzero eigenvalues. We also provide some examples to illustrate that symmetry assumption in our results is crucial. Finally, We make some numerical observations about these eigenvalues using FreeFEM++ and state them as conjectures.