Universal inequalities for the eigenvalues of Laplace and Schrödinger operators on submanifolds

Soufi, Ahmad El; Harrell, Evans M.; Ilias, Saïd

doi:10.1090/s0002-9947-08-04780-6

Cited by 68 publications

(74 citation statements)

References 31 publications

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“…(See also [39], [6], [7], [17], [22].) In the present article we put those notions together with some transform techniques in order to connect together several inequalities for spectra, which have been derived by independent methods in the past.…”

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confidence: 99%

On Riesz Means of Eigenvalues

Harrell

Hermi

2011

Communications in Partial Differential Equations

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Abstract. In this article we prove the equivalence of certain inequalities for Riesz means of eigenvalues of the Dirichlet Laplacian with a classical inequality of Kac. Connections are made via integral transforms including those of Laplace, Legendre, Weyl, and Mellin, and the Riemann-Liouville fractional transform. We also prove new universal eigenvalue inequalities and monotonicity principles for Dirichlet Laplacians as well as certain Schrödinger operators. At the heart of these inequalities are calculations of commutators of operators, sum rules, and monotonic properties of Riesz means. In the course of developing these inequalities we prove new bounds for the partition function and the spectral zeta function (cf. Corollaries 3.5-3.7) and conjecture about additional bounds.Manuscript received December 25, 2007; revised XXX XX, XXXX.

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confidence: 99%

On Riesz Means of Eigenvalues

Harrell

Hermi

2011

Communications in Partial Differential Equations

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“…which is optimal since when tends to S n (1), the above inequality becomes equality for any k. Recently, Chen and Cheng [3] and El Soufi et al [9] have generalized this result to any complete Riemannian manifold, independently. When l = 2, Wang and Xia [16] have obtained…”

Section: Introductionmentioning

confidence: 94%

Estimates for eigenvalues of the poly-Laplacian with any order in a unit sphere

2009

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In this paper we study eigenvalues of the poly-Laplacian with any order on a domain in an n-dimensional unit sphere and obtain estimates for eigenvalues. In particular, the optimal result of Cheng and Yang (Math Ann 331: [445][446][447][448][449][450][451][452][453][454][455][456][457][458][459][460] 2005) is included in our ones. In order to prove our results, we introduce 2(l + 1) functions a i and b i , for i = 0, 1, . . . , l and two operators µ and η. First of all, we study properties of functions a i and b i and the operators µ and η. By making use of these properties and introducing k free constants, we obtain estimates for eigenvalues. Mathematics Subject Classification (2000) 35P15 · 58G25( 1.1) where is the Laplacian in M and ν denotes the outward unit normal. It is well known that the spectrum of this eigenvalue problem is real and discrete. 0 < λ 1 ≤ λ 2 ≤ · · · ≤ λ k ≤ · · · −→ ∞.

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“…More recently, for the Dirichlet eigenvalue problems of the Laplacian on a bounded domain in the n-dimensional unit sphere, complex projective space, and compact homogeneous Riemannian manifolds, Cheng and Yang [2005;2006b;2007] obtained the Yangtype inequalities for eigenvalues. For a bounded domain in a complete Riemannian manifold M, the first author and Cheng [Chen and Cheng 2008] proved a Yang-type inequality by using the Nash embedding theorem (compare [El Soufi et al 2009;Harrell 2007]). …”

Section: Introductionmentioning

confidence: 99%

Eigenvalue estimates on domains in complete noncompact Riemannian manifolds

Chen¹,

Zheng²,

Min³

2012

Pacific J. Math.

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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.

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Universal inequalities for the eigenvalues of Laplace and Schrödinger operators on submanifolds

Cited by 68 publications

References 31 publications

On Riesz Means of Eigenvalues

On Riesz Means of Eigenvalues

Estimates for eigenvalues of the poly-Laplacian with any order in a unit sphere

Eigenvalue estimates on domains in complete noncompact Riemannian manifolds

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