Cheng, Shiu-Yuen; Oden, Kevin Isoperimetric inequalities and the gap between the first and second eigenvalues of an Euclidean domain. (English) Zbl 0920.58054 J. Geom. Anal. 7, No. 2, 217-239 (1997). Let \(M\) be a compact connected domain in \(\mathbb{R}^n\) and let \(\Delta\) be the usual Laplace operator. Impose Dirichlet boundary conditions and let \(0<\lambda_1<\lambda_2\leq\lambda_3\dots\) be the associated eigenvalues; the ground state \(\lambda_1\) has simple multiplicity. The authors introduce a weighted Cheeger constant and estimate the gap \(\lambda_2-\lambda_1\) in terms of this constant. If the domain satisfies an interior rolling sphere condition, the authors give an estimate on the weighted Cheeger constant in terms of the rolling sphere radius, the volume, a bound on the principal curvatures of the boundary, and the dimension. Reviewer: P.Gilkey (Eugene) Cited in 11 Documents MSC: 58J50 Spectral problems; spectral geometry; scattering theory on manifolds Keywords:eigenvalue; eigenfunction; isoperimetric; Cheeger’s constant × Cite Format Result Cite Review PDF Full Text: DOI References: [1] Ashbaugh, M.; Benguria, R., A sharp bound for the ratio of the first two eigenvalues of Dirichlet Laplacian and extensions, Annals of Math., 135, 601-628 (1992) · Zbl 0757.35052 · doi:10.2307/2946578 [2] Brooks, B., The bottom of the spectrum of a Riemannian covering, Crelles J., 357, 101-114 (1985) · Zbl 0553.53027 [3] Brascamp, H.; 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, 366-389 (1976) · Zbl 0334.26009 · doi:10.1016/0022-1236(76)90004-5 [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. · Zbl 0212.44903 [5] Chern, S. S., and Lashof, R. K., On the total curvature of immersed manifolds I,American J. of Math.79, 306-318. · Zbl 0078.13901 [6] Gilbarg, G., and Trudinger, N.,Elliptic Differential Equations of Second Order, 2nd edn., Springer-Verlag, 1984. · Zbl 1042.35002 [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. · Zbl 0441.58014 [8] Payne, L.; Polya, G.; Weinberger, H., On the ratio of consecutive eigenvalues, J. Math. and Phys., 35, 289-298 (1956) · Zbl 0073.08203 [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. · Zbl 0603.35070 [10] Schoen, R.; Yau, S. T., Differential Geometry, Series in Pure & Applied Math. (1988), Peking: Science Press, Peking [11] Yau, S. T., Isoperimetric constants and the eigenvalue of a compact Riemannian manifold, Ann. Scient. Ecole Norm. Sup., 8, 487-507 (1975) · Zbl 0325.53039 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.