Eigenvalues Estimate for the Neumann Problem ofaBounded Domain
Colbois, Bruno ; Maerten, Daniel
In: Journal of Geometric Analysis, 2008, vol. 18, no. 4, p. 1022-1032
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- 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