Extrinsic estimates for eigenvalues of the Laplace operator

Chen, Daguang; Cheng, Qing-Ming

doi:10.2969/jmsj/06020325

Cited by 68 publications

(52 citation statements)

References 13 publications

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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: 97%

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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Section: Introductionmentioning

confidence: 97%

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

2009

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“…One can find more discussions about inequalities (1.4)-(1.7) in [1,2] of Ashbaugh. In 2008, Chen and Cheng [3] obtained (1.6) for a bounded domain in an n-dimensional complete Riemannian manifold. Some remarkable results for problem (1.3) were also derived on some types of manifolds (see [4,5,10,11,17,20]).…”

Section: Introductionmentioning

confidence: 99%

Yang-type inequalities for weighted eigenvalues of a second order uniformly elliptic operator with a nonnegative potential

Sun

2010

Proc. Amer. Math. Soc.

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Abstract. In this paper, we investigate the Dirichlet weighted eigenvalue problem of a second order uniformly elliptic operator with a nonnegative potential on a bounded domain Ω ⊂ R n . First, we prove a general inequality of eigenvalues for this problem. Then, by using this general inequality, we obtain Yang-type inequalities which give universal upper bounds for eigenvalues. An explicit estimate for the gaps of any two consecutive eigenvalues is also derived. Our results contain and extend the previous results for eigenvalues of the Laplacian, the Schrödinger operator and the second order elliptic operator on a bounded domain Ω ⊂ R n .

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“…The inequalities on the higher eigenvalues of the Laplacian on a connected bounded domain in R n obtained by Payne-Pólya-Weinberger, Hile-Protter, Yang have also been extended to some other eigenvalue problems (cf. [1][2][3][4][5][8][9][10][11][12][13][14][15][16][17][19][20][21][22][23]26,27,[32][33][34][35]37], etc). Here let us consider the case of the eigenvalue problem for the Dirichlet biharmonic operator or the clamped plate problem which describes the characteristic vibrations of a clamped plate.…”

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confidence: 99%

“…For the Dirichlet eigenvalue problem on a complete Riemannian manifold isometrically immersed in a Euclidean space, Chen and Cheng [8] and El Soufi et al [17] have proved, independently,…”

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confidence: 99%

Inequalities for eigenvalues of a clamped plate problem

Wang

Xia

2010

Calc. Var.

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In this paper we study eigenvalues of a clamped plate problem on compact domains in complete manifolds. For complete manifolds admitting special functions, we prove universal inequalities for eigenvalues of clamped plate problem independent of the domains of Payne-Pólya-Weinberger-Yang type. These manifolds include Hadamard manifolds with Ricci curvature bounded below, a class of warped product manifolds, the product of Euclidean spaces with any complete manifolds and manifolds admitting eigenmaps to a sphere. In the case of warped product manifolds, our result implies a universal inequality on hyperbolic space proved by Cheng-Yang. We also strengthen an inequality for eigenvalues of clamped plate problem on submanifolds in a Euclidean space obtained recently by Cheng, Ichikawa and Mametsuka.

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Extrinsic estimates for eigenvalues of the Laplace operator

Cited by 68 publications

References 13 publications

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

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

Yang-type inequalities for weighted eigenvalues of a second order uniformly elliptic operator with a nonnegative potential

Inequalities for eigenvalues of a clamped plate problem

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