کد مقاله کد نشریه سال انتشار مقاله انگلیسی نسخه تمام متن
4612423 1338684 2009 12 صفحه PDF دانلود رایگان
عنوان انگلیسی مقاله ISI
Estimates for eigenvalues on Riemannian manifolds
موضوعات مرتبط
مهندسی و علوم پایه ریاضیات آنالیز ریاضی
پیش نمایش صفحه اول مقاله
Estimates for eigenvalues on Riemannian manifolds
چکیده انگلیسی

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 RnRn, the conjecture of Pólya is well known: the k  th eigenvalue λkλk of the Dirichlet eigenvalue problem of Laplacian satisfiesλk⩾4π2(ωnvolΩ)2nk2n,for k=1,2,…. 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 proved1k∑i=1kλi⩾nn+24π2(ωnvolΩ)2nk2n,for k=1,2,…, 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)Hn(−1). From it, we prove that when the bounded domain Ω   tends to Hn(−1)Hn(−1), all eigenvalues tend to (n−1)24.

ناشر
Database: Elsevier - ScienceDirect (ساینس دایرکت)
Journal: Journal of Differential Equations - Volume 247, Issue 8, 15 October 2009, Pages 2270–2281
نویسندگان
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