In this paper, we investigate non-zero positive eigenvalues of the Laplacian with Dirichlet boundary condition in an n-dimentional Euclidean space Rn, then we obtain an new upper bound of the (k + 1)-th eigenvalue λk+1, which improve the previous estimate which was obtained by Cheng and Yang, see (1.8).
Zhengchao JI
. NEW BOUNDS ON EIGENVUALUES OF LAPLACIAN[J]. Acta mathematica scientia, Series B, 2019
, 39(2)
: 545
-550
.
DOI: 10.1007/s10473-019-0217-3
[1] Shao Z Q, Hong J X. The eigenvalue problem for the Laplacian equations. Acta Math Sci, 2007, 27B(2):329-337
[2] Huang G Y, Chen W Y. Universal bounds for eigenvalues of Laplacian operator of any order. Acta Math Sci, 2010, 30B(3):939-948
[3] Hu Y X, Xu H W. An eigenvalue pinching theorem for compact hypersurfaces in a sphere. J Geom Anal, 2017, 27(3):2472-2489
[4] Hu Y X, Xu H W, Zhao E T. First eigenvalue pinching for Euclidean hypersurfaces via k-th mean curvatures. Ann Global Anal Geom, 2015, 48(1):23-35
[5] Li P, Yau S T. On the Schödinger equation and the eigenvalue problem. Comm Math Phys, 1983, 88(3):309-318
[6] Stewarson K, Waechter R T. On hearing the shape of a drum:further results. Proc Camb Philol Soc, 1971, 69(2):353-369
[7] Payne L E, Pölya G, Weinberger H F. Sur le quotient de deux fréquence propers consécutives. C R Acad Sci Paris, 1955, 241:917-919
[8] Hile G N, Potter M H. Inequalities for eigenvalues of the Laplacian. Indiana Univ Math J, 1980, 29(4):255-306
[9] Yang H C. An estimate of the difference between consecutive eigenvalues. preprint. 1991, IC/91/60 of the Intl Centre for Theoretical Physics
[10] Cheng Q M, Yang H C. Bounds on eigenvaluesof Dirichlet Laplacian. Math Ann, 2007, 337(1):159-175
