Eigenvalues Estimate for the Neumann Problem of a Bounded Domain

Abstract

In this note, we investigate upper bounds of the Neumann eigenvalue problem for the Laplacian of a domain Ω in a given complete (not compact a priori) Riemannian manifold (M,g). 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 applications, we prove that if the Ricci curvature of (M,g) is bounded below Ric ^g≥−(n−1)a ², a≥0, then there exist constants A _n >0,B _n >0 only depending on the dimension, such that

$$\lambda _{k}(\Omega)\le A_{n}a^{2}+B_{n}\left(\frac{k}{V}\right)^{2/n},$$

where λ _k (Ω) (k∈ℕ^*) denotes the k-th eigenvalue of the Neumann problem on any bounded domain Ω⊂M of volume V=Vol (Ω,g). Furthermore, this upper bound is clearly in agreement with the Weyl law. As a corollary, we get also an estimate which is analogous to Buser’s upper bounds of the spectrum of a compact Riemannian manifold with lower bound on the Ricci curvature.