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

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.

References in corpus (1)

• Estimates for Robin $p$-Laplacian eigenvalues of convex sets with prescribed perimeter