Isoperimetric bounds for Wentzel-Laplace eigenvalues on Riemannian manifolds

TL;DR

This paper establishes isoperimetric bounds for Wentzel-Laplace eigenvalues on Riemannian manifolds, providing asymptotically optimal estimates that relate the spectrum to the domain's isoperimetric ratio.

Contribution

It introduces new bounds linking the spectrum of the Wentzel-Laplace operator to isoperimetric ratios, extending spectral estimates to Riemannian manifolds.

Findings

Bounds are asymptotically optimal according to Weyl's law.

Isoperimetric ratio controls the entire spectrum of the operator.

Results apply to various ambient Riemannian spaces.

Abstract

In this paper, we investigate eigenvalues of the Wentzel-Laplace operator on a bounded domain in some Riemannian manifold. We prove asymptotically optimal estimates, according to the Weyl's law through bounds that are given in terms of the isoperimetric ratio of the domain. Our results show that the isoperimetric ratio allows to control the entire spectrum of the Wentzel-Laplace operator in various ambient spaces.