Bounds on eigenvalues of Dirichlet Laplacian

Cheng, Qing-Ming; Yang, Hang

doi:10.1007/s00208-006-0030-x

Cited by 98 publications

(84 citation statements)

References 19 publications

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“…For the upper bounds of λ k , one can refer to [15,27]. For more results on the estimates of Dirichlet eigenvalues for the non-degenerate elliptic operators, we can see [13,14,31].…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Lower bounds of Dirichlet eigenvalues for some degenerate elliptic operators

Chen

Luo

2015

Calc. Var.

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Let be a bounded open domain in R n with smooth boundary and X = (X 1 , X 2 , . . . , X m ) be a system of real smooth vector fields defined on with the boundary ∂ which is non-characteristic for X . If X satisfies the Hörmander's condition, then the vector fields is finite degenerate and the sum of square operator X = m j=1 X 2 j is a finitely degenerate elliptic operator, otherwise the operator − X is called infinitely degenerate. If λ j is the jth Dirichlet eigenvalue for − X on , then this paper shall study the lower bound estimates for λ j . Firstly, by using the sub-elliptic estimate directly, we shall give a simple lower bound estimates of λ j for general finitely degenerate X which is polynomial increasing in j. Secondly, if X is so-called Grushin type degenerate elliptic operator, then we can give a precise lower bound estimates for λ j . Finally, by using logarithmic regularity estimate, for infinitely degenerate elliptic operator X we prove that the lower bound estimates of λ j will be logarithmic increasing in j. Mathematics Subject Classification 35J70 · 35P15

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“…For the upper bounds of λ k , one can refer to [15,27]. For more results on the estimates of Dirichlet eigenvalues for the non-degenerate elliptic operators, we can see [13,14,31].…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Lower bounds of Dirichlet eigenvalues for some degenerate elliptic operators

Chen

Luo

2015

Calc. Var.

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“…By making use of the same proof as in Cheng and Yang [9], we can complete our proof of Theorem 2.1. 2 Remark 2.1.…”

Section: A Recursion Formula Of Cheng and Yangmentioning

confidence: 98%

Estimates for the Eigenvalues of the Drifting Laplacian on Some Complete Ricci Solitons

Zeng

2018

Kyushu J. Math.

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Abstract. Ricci solitons are the self-similar solutions to the Ricci flow, which play an important role in understanding the singularity dilations of the Ricci flow. In this paper, we investigate eigenvalues of the Dirichlet problem of a drifting Laplacian on some important complete Ricci solitons: the product shrinking Ricci soliton, cigar soliton, and so on. Since eigenvalues are invariant of isometries, we can give the estimates for the eigenvalues of a drifting Laplacian on the rotationally invariant shrinking solitons. In addition, we also obtain a sharp upper bound of the kth eigenvalue of the a drifting Laplacian on the product Ricci soliton in the sense of order k.

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“…Recursive Formula (Cheng and Yang [7]). Let μ 1 ≤ μ 2 ≤ · · · ≤ μ k+1 be any non-negative real numbers satisfying…”

Section: Qing-ming Cheng and Hongcang Yangmentioning

confidence: 99%

Universal inequalities for eigenvalues of a clamped plate problem on a hyperbolic space

Cheng

Yang

2011

Proc. Amer. Math. Soc.

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Abstract. In this paper, we investigate universal inequalities for eigenvalues of a clamped plate problem on a bounded domain in an n-dimensional hyperbolic space. It is well known that, for a bounded domain in the n-dimensional Euclidean space, Payne, Pólya and Weinberger (1955), Hook (1990) and Chen and Qian (1990) studied universal inequalities for eigenvalues of the clamped plate problem. Recently, Cheng and Yang (2006) have derived the Yangtype universal inequality for eigenvalues of the clamped plate problem on a bounded domain in the n-dimensional Euclidean space, which is sharper than the other ones. For a domain in a unit sphere, Wang and Xia (2007) have also given a universal inequality for eigenvalues. For a bounded domain in the ndimensional hyperbolic space, although many mathematicians want to obtain a universal inequality for eigenvalues of the clamped plate problem, there are no results on universal inequalities for eigenvalues. The main reason that one could not derive a universal inequality is that one cannot find appropriate trial functions. In this paper, by constructing "nice" trial functions, we obtain a universal inequality for eigenvalues of the clamped plate problem on a bounded domain in the hyperbolic space. Furthermore, we can prove that if the first eigenvalue of the clamped plate problem tends to 16.

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Bounds on eigenvalues of Dirichlet Laplacian

Cited by 98 publications

References 19 publications

Lower bounds of Dirichlet eigenvalues for some degenerate elliptic operators

Lower bounds of Dirichlet eigenvalues for some degenerate elliptic operators

Estimates for the Eigenvalues of the Drifting Laplacian on Some Complete Ricci Solitons

Universal inequalities for eigenvalues of a clamped plate problem on a hyperbolic space

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