Eigenvalue estimates on domains in complete noncompact Riemannian manifolds

Abstract

In this paper, we obtain universal inequalities for eigenvalues of the Dirichlet eigenvalue problem of the Laplacian and the clamped plate problem on a bounded domain in an n-dimensional (n ≥ 3) noncompact simply connected complete Riemannian manifold with sectional curvature Sec satisfying −K 2 ≤ Sec ≤ −k 2 , where K ≥ k ≥ 0 are constants. When M is ‫ވ‬ n (−1) (n ≥ 3), these inequalities become ones previously found by Cheng and Yang.

Fundamental tones of clamped plates in nonpositively curved spaces

“…One of the most central problems is to establish Payne-Pólya-Weinberger-Yang-type inequalities for the eigenvalues of the problem ∆ 2 g u = Γu in Ω, u = ∂u ∂n = 0 on ∂Ω, (1.4) where Ω is a bounded domain in an n-dimensional Riemannian manifold (M, g), ∆ 2 g stands for the biharmonic Laplace-Beltrami operator on (M, g) and ∂ ∂n is the outward normal derivative on ∂Ω, respectively; see e.g. Chen, Zheng and Lu [7], Cheng, Ichikawa and Mametsuka [8], Cheng and Yang [9,10,11], Wang and Xia [39]. Instead of (1.2), one naturally considers the fundamental tone of Ω ⊂ M by Γ g (Ω) := Γ g,n (Ω) = inf u∈W 2,2 0 (Ω)\{0} Ω (∆ g u) 2 dv g Ω u 2 dv g , (1.5) where dv g denotes the canonical measure on (M, g), and W 2,2 0 (Ω) is the usual Sobolev space on (M, g), see Hebey [19]; in fact, it turns out that Γ g (Ω) is the first eigenvalue to (1.4).…”

“…where Ω is a bounded domain in an n-dimensional Riemannian manifold (M, g), ∆ 2 g stands for the biharmonic Laplace-Beltrami operator on (M, g) and ∂ ∂n is the outward normal derivative on ∂Ω, respectively; see e.g. Chen, Zheng and Lu [7], Cheng, Ichikawa and Mametsuka [8], Cheng and Yang [9,10,11], Wang and Xia [39]. Instead of (1.2), one naturally considers the fundamental tone of Ω ⊂ M by Γ g (Ω) := Γ g,n (Ω) = inf…”

Estimates of eigenvalues of a clamped problem

“…Remark 1.4. The inequality (1.22) is better than the one in [4]. If a ¼ b ¼ 1, the inequality (1.22) is the one of Cheng and Yang [11] (see (1.8)).…”

Untitled

“…Remark 1.1. When is an n-dimensional compact homogeneous Riemannian manifold, a compact minimal submanifold without boundary and a connected bounded domain in the standard unit sphere ‫ޓ‬ N (1), and a connected bounded domain and a compact complex hypersurface without boundary of the complex projective space ‫ސރ‬ n (4) with holomorphic sectional curvature 4, many mathematicians have studied the universal inequalities for eigenvalues and the difference of the consecutive eigenvalues (see [Cheng and Yang 2005;Harrell 1993;Harrell and Michel 1994;Harrell and Stubbe 1997;Li 1980;Yang and Yau 1980;Leung 1991;Sun et al 2008;Chen et al 2012]).…”