Estimates on Eigenvalues of Laplacian

Cheng, Qing-Ming; Yang, Hang

doi:10.1007/s00208-004-0589-z

Cited by 95 publications

(63 citation statements)

References 15 publications

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“…Recently, for eigenvalue problems of Dirichlet Laplacian on either a bounded domain in an n-dimensional unit sphere, or an n-dimensional compact minimal submanifold in a unit sphere, or a bounded domain in an n-dimensional complex projective space, or an n-dimensional compact homogeneous Riemannian manifold, or a compact complex submanifold in an m-dimensional complex projective space, we also obtained the universal inequalities on higher eigenvalues in [6] and [8], which are sharper than the old results in corresponding cases (cf. 6-12, 14, 17, 20, 23).…”

Section: Introductionmentioning

confidence: 99%

Bounds on eigenvalues of Dirichlet Laplacian

Cheng

Yang

2006

Math. Ann.

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In this paper, we investigate an eigenvalue problem of Dirichlet Laplacian on a bounded domain in an n-dimensional Euclidean space R n . If λ k+1 is the (k+1)th eigenvalue of Dirichlet Laplacian on , then, we prove that, for n ≥ 41 and k ≥ 41, λ k+1 ≤ k 2 n λ 1 and, for any n and k, λ k+1 ≤ C 0 (n, k)k 2 n λ 1 with C 0 (n, k) ≤ j 2 n/2,1 /j 2 n/2−1,1 , where j p,k denotes the k-th positive zero of the standard Bessel function J p (x) of the first kind of order p. From the asymptotic formula of Weyl and the partial solution of the conjecture of Pólya, we know that our estimates are optimal in the sense of order of k. Mathematics Subject Classification (2001) 35P15 · 58G25Q.-M.

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Section: Introductionmentioning

confidence: 99%

Bounds on eigenvalues of Dirichlet Laplacian

Cheng

Yang

2006

Math. Ann.

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“…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]). The spectrum of problem (1.1) has been discussed from some angles, but fewer universal inequalities of eigenvalues have been derived.…”

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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“…Recently, there are many mathematicians who have investigated properties of the eigenvalues of p-Laplacian on Finsler manifolds and Riemannian manifolds to estimate the spectrum in terms of the other geometric quantities of the manifold. (see [3,4,9,11,18,20]). …”

Section: Introductionmentioning

confidence: 99%

Eigenvalues Variation of the P-Laplacian Under the Ricci Flow on Sm

Azami¹

2017

JIMS

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Abstract. Let (M, F ) be a compact Finsler manifold. Studying the eigenvalues and eigenfunctions for the linear and nonlinear geometric operators is a known problem. In this paper we will consider the eigenvalue problem for the p-laplace operator for Sasakian metric acting on the space of functions on SM . We find the first variation formula for the eigenvalues of p-Laplacian on SM evolving by the Ricci flow on M and give some examples. Kami memperoleh rumus variasi pertama dan memberikan beberapa contoh untuk nilai eigen dari p-Laplacian pada SM yang melibatkan aliran Ricci pada M .Kata kunci: aliran Ricci, manifold Finsler, operator p-Laplace

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Estimates on Eigenvalues of Laplacian

Cited by 95 publications

References 15 publications

Bounds on eigenvalues of Dirichlet Laplacian

Bounds on eigenvalues of Dirichlet Laplacian

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

Eigenvalues Variation of the P-Laplacian Under the Ricci Flow on Sm

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