Estimates for eigenvalues on Riemannian manifolds
Estimates for eigenvalues on Riemannian manifolds
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DOI：
10.1016/j.jde.2009.07.015
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发表时间：
2009-10
影响因子：
2.4
通讯作者：
Q. Cheng;Hongcang Yang
中科院分区：
文献类型：
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作者：
Q. Cheng;Hongcang Yang
本文研究了n维完备黎曼流形M中有界区域Ω上拉普拉斯算子的Dirichlet特征值问题的特征值。当M是n维欧氏空间Rn时，Pólya猜想是众所周知的：Laplacian的Dirichlet特征值问题的第k个特征值λ k满足Li和Yau [P. Li，S.T. Yau，On the Schrödinger equation and the eigenvalue problem，Comm. Math. Phys. 88（1983）309-318]（cf. Lieb [E. Lieb，单体薛定谔算子的束缚态数和Weyl问题，在：Proc. Sympos。纯数学，第36卷，1980年，第36页。241-252]）给出了一个部分解决方案的猜想波利亚，也就是说，他们已经证明，这是尖锐的意义上的平均。本文考虑完备黎曼流形的一个一般设定。本文建立了完备黎曼流形中有界区域上拉普拉斯算子Dirichlet特征值问题的特征值的李和丘不等式的一个类似结果。进一步，我们得到了双曲空间Hn（−1）中有界区域上Laplacian算子Dirichlet特征值问题的特征值的一个普适不等式.由此，我们证明了当有界区域Ω趋于Hn（−1）时，所有特征值趋于（n−1）24。
In this paper, we investigate eigenvalues of the Dirichlet eigenvalue problem of Laplacian on a bounded domain Ω in an n-dimensional complete Riemannian manifold M. When M is an n-dimensional Euclidean space Rn, the conjecture of Pólya is well known: the kth eigenvalue λkof the Dirichlet eigenvalue problem of Laplacian satisfies Li and Yau [P. Li, S.T. Yau, On the Schrödinger equation and the eigenvalue problem, Comm. Math. Phys. 88 (1983) 309–318] (cf. Lieb [E. Lieb, The number of bound states of one-body Schrödinger operators and the Weyl problem, in: Proc. Sympos. Pure Math., vol. 36, 1980, pp. 241–252]) have given a partial solution for the conjecture of Pólya, that is, they have proved which is sharp in the sense of average. In this paper, we consider a general setting for complete Riemannian manifolds. We establish an analog of the Li and Yau's inequality for eigenvalues of the Dirichlet eigenvalue problem of Laplacian on a bounded domain in a complete Riemannian manifold. Furthermore, we obtain a universal inequality for eigenvalues of the Dirichlet eigenvalue problem of Laplacian on a bounded domain in a hyperbolic space Hn(−1). From it, we prove that when the bounded domain Ω tends to Hn(−1), all eigenvalues tend to (n−1)24.