Sharp inequalities for Neumann eigenvalues on the sphere

TL;DR

This paper establishes sharp inequalities for Neumann eigenvalues on the sphere, showing the union of two equal geodesic balls maximizes the second eigenvalue, and explores implications for the first eigenvalue and densities.

Contribution

It proves that the union of two equal geodesic balls maximizes the second Neumann eigenvalue on the sphere, extending to densities and weaker conditions than previous results.

Findings

Union of two equal geodesic balls maximizes second eigenvalue

Geodesic ball is maximal for first eigenvalue under relaxed conditions

Numerical evidence suggests densities with higher first eigenvalue than geodesic balls

Abstract

We prove that the second nontrivial Neumann eigenvalue of the Laplace-Beltrami operator on the unit sphere $\mathbb{S}^n \subseteq \mathbb{R}^{n+1}$ is maximized by the union of two disjoint, equal, geodesic balls among all subsets of $\mathbb{S}^n$ of prescribed volume. In fact, the result holds in a stronger version, involving the harmonic mean of the eigenvalues of order $2$ to $n$, and extends to densities. A (surprising) consequence occurs on the maximality of a geodesic ball for the first nontrivial eigenvalue under the volume constraint: the hemisphere inclusion condition of the Ashbaugh-Benguria result can be relaxed to a weaker one, namely empty intersection with a geodesic ball of the prescribed volume. Although we do not prove that this last inclusion result is sharp, for a mass less than the half of the sphere, we numerically identify a density with higher first eigenvalue than the corresponding geodesic ball and with support equal to the full sphere $\mathbb{S}^2$.