Isoperimetric inequalities and the gap between the first and second eigenvalues of an euclidean domain

Journal of Geometric Analysis

Volumn 7, Issue 2, 1997, Pages 217-239

  Cheng, Shiu Yuen ^a,b   Oden, Kevin ^c  

^a UNIVERSITY OF CALIFORNIA   (United States)

^b CHINESE UNIVERSITY OF HONG KONG   (Hong Kong)

^c HARVARD UNIVERSITY   (United States)

Author keywords

Cheeger's constant; Eigenfunction; Eigenvalue; Isoperimetric

Indexed keywords

EID: 14544289346     PISSN: 10506926     EISSN: None     Source Type: Journal    
DOI: 10.1007/bf02921721     Document Type: Article

References (11)

1. Ashbaugh, M., and Benguria, R. A sharp bound for the ratio of the first two eigenvalues of Dirichlet Laplacian and extensions, Annals of Math. 135 (1992), 601-628.
2. Brooks, B. The bottom of the spectrum of a Riemannian covering, Crelles J. 357 (1985), 101-114.
3. Brascamp, H., and Lieb, E. On extensions of the Brunn-Minkowski and Prekopa-Leindler Theorems, including inequalities for log concave functions and with an application to diffusion equation, Journal of Functional Analysis 22 (1976), 366-389.
4. Cheeger, J. A lower bound for the smallest eigenvalue of the Laplacian, Problems in Analysis, a Symposium in honor of S. Bochner, Princeton University Press, 1970.
5. Chern, S. S., and Lashof, R. K., On the total curvature of immersed manifolds I, American J. of Math. 79, 306-318.
6. Gilbarg, G., and Trudinger, N., Elliptic Differential Equations of Second Order, 2nd edn., Springer-Verlag, 1984.
7. Li, P., and Yau, S. T., Estimates of eigenvalues of a compact Riemannian manifold, Proceedings of Symposia in Pure Math., vol. 36, American Mathematical Society, 1980, pp. 205-230.
8. Payne, L., Polya, G., and Weinberger, H., On the ratio of consecutive eigenvalues, J. Math. and Phys. 35 (1956), 289-298.
9. Singer, I., Wong, B., Yau, S. T., and Yau, S. T. T., An estimate of the gap of the first two eigenvalues of the Schrödinger operator, Ann. Scul. Norm. Sup. Pisa, Series IV, vol. XXII, no. 2, pp. 319-333.
10. Schoen, R., and Yau, S. T., Differential Geometry, Series in Pure & Applied Math., no. 18, Science Press, Peking, 1988. (Chinese edn.)
11. Yau, S. T., Isoperimetric constants and the eigenvalue of a compact Riemannian manifold, Ann. Scient. Ecole Norm. Sup. 8 (1975), 487-507.