Summary
In this note, we adopt a probabilistic method for estimating the first Dirichlet eigenvalue. The results improve or contain some known ones, especially for large dimension. Moreover, our estimates are sharp for some typical cases.
References
Barbosa, J.L., doCarmo, M.: Stability of minimal surfaces and eigenvalues of Laplacian. Math. Z.173, 13–28 (1980)
Chavel, I.: Eigenvalues in Riemannian geometry. New York London: Academic Press 1984
Cheng, S.Y.: Eigenvalue comparison theorems and its geometric applications. Math. Z.143, 289–297 (1975)
Gage, M.: Upper bounds for the first eigenvalue of the Laplace-Beltrami operator. Indiana Univ. Math. J.29, 897–912 (1980)
Ikeda, N., Watanabe, S.: Stochastic differential equations and diffusion processes. Amsterdam: North-Holland 1981
Matsuzawa, T., Tanno, S.: Estimates of the first eigenvalue of a big cup domain of 2-sphere. Compos. Math.47, 95–100 (1982)
McKean, H.P.: An upper bound for the spectrum of Δ on a manifold of negative curvature. J. Differ. Geom.4, 359–366 (1970)
Pinsky, M.A.: The first eigenvalue of a spherical cap. Appl. Math. Opt.7, 137–139 (1981)
Sato, S.: Barta's inequalities and the first eigenvalue of a cap domain of a 2-sphere. Math. Z.181, 313–318 (1982)
Wang, F.Y.: Application of coupling methods to the Neumann eigenvalue problem. Probab. Theory Related Fields98, 299–306 (1994)
Wang, F.Y.: A probabilistic approach to the first Dirichlet eigenvalue on non-compact Riemannian manifold. (Preprint)
Author information
Authors and Affiliations
Additional information
Research supported in part by NSFC and the State Education Commission of China
Rights and permissions
About this article
Cite this article
Wang, FY. Estimates of the first Dirichlet eigenvalue by using diffusion processes. Probab. Th. Rel. Fields 101, 363–369 (1995). https://doi.org/10.1007/BF01200501
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01200501
Mathematics Subject Classification (1991)
- 35P15
- 58C50