Where to place a spherical obstacle so as to maximize the second Dirichlet eigenvalue

TL;DR

This paper proves that for doubly connected domains bounded by two spheres, the second Dirichlet eigenvalue of the Laplacian is maximized when the spheres are concentric, extending known results to various geometries.

Contribution

It establishes the maximality of the second eigenvalue for concentric spherical shells across Euclidean, spherical, and hyperbolic spaces, generalizing previous results.

Findings

Maximum second eigenvalue occurs for concentric spheres

Results extend to spherical and hyperbolic geometries

Generalizes prior work on the first eigenvalue

Abstract

We prove that among all doubly connected domains of $\mathbb{R}^n$ bounded by two spheres of given radii, the second eigenvalue of the Dirichlet Laplacian achieves its maximum when the spheres are concentric (spherical shell). The corresponding result for the first eigenvalue has been established by Hersch in dimension 2, and by Harrell, Kr\"oger and Kurata and Kesavan in any dimension. We also prove that the same result remains valid when the ambient space $\mathbb{R}^n$ is replaced by the standard sphere $\mathbb{S}^n$ or the hyperbolic space $\mathbb{H}^n$ .