Eigenvalues estimate for the Neumann problem on bounded domains

TL;DR

This paper derives upper bounds for Neumann eigenvalues of the Laplacian on bounded domains within Riemannian manifolds, extending classical spectral bounds to more general geometric contexts.

Contribution

It introduces a new method using metric-based test functions to estimate Neumann eigenvalues on domains in non-compact Riemannian manifolds with Ricci curvature bounds.

Findings

Upper bounds align with Weyl law predictions

Bounds are analogous to Buser's bounds for closed manifolds

Method applicable to non-compact manifolds with smooth boundaries

Abstract

In this note, we investigate upper bounds of the Neumann eigenvalue problem for the Laplacian of a bounded domain (with smooth boundary) in a given complete (not compact a priori) Riemannian manifold with Ricci bounded below . For this, we use test functions for the Rayleigh quotient subordinated to a family of open sets constructed in a general metric way, interesting for itself. As application, we get upper bounds for the Neumann spectrum which is clearly in agreement with the Weyl law and which is analogous to Buser's upper bounds of the spectrum of a closed Riemannian manifold with lower bound on the Ricci curvature.