A spectral isoperimetric inequality on the n-sphere for the Robin-Laplacian with negative boundary parameter

TL;DR

This paper investigates the maximization of the first Robin eigenvalue of the Laplacian on convex sets on the n-sphere with negative boundary parameter, establishing that geodesic balls are optimal for fixed perimeter and providing a stability estimate.

Contribution

It proves that geodesic balls maximize the Robin eigenvalue among convex sets with fixed perimeter on the sphere for negative boundary parameters, including a quantitative stability result.

Findings

Geodesic balls maximize the Robin eigenvalue for fixed perimeter when the boundary parameter is negative.

A stability estimate quantifies how close a set is to a geodesic ball based on volume difference.

The result holds for convex, not necessarily smooth, sets on the n-sphere.

Abstract

For every given $\beta<0$, we study the problem of maximizing the first Robin eigenvalue of the Laplacian $\lambda_\beta(\Omega)$ among convex (not necessarily smooth) sets $\Omega\subset\mathbb{S}^{n}$ with fixed perimeter. In particular, denoting by $\sigma_n$ the perimeter of the $n$-dimensional hemisphere, we show that for fixed perimeters $P<\sigma_n$, geodesic balls maximize the eigenvalue. Moreover, we prove a quantitative stability result for this isoperimetric inequality in terms of volume difference between $\Omega$ and the ball $D$ of the same perimeter.