[1] Ashbaugh M S, Benguria R D.Bounds for ratios of the first, second, and third membrane eigenvalues// Angell T, Pamela Cook L, Kleinman R, Olmstead W. Nonlinear Problems in Applied Mathematics. Philadelphia: Soc Indu Appl Math, 1996: 30-42
[2] Ashbaugh M S, Benguria R D.The range of values of $\frac{{{\lambda _2}}}{{{\lambda _1}}}$ and $\frac{{{\lambda _3}}}{{{\lambda _1}}}$ for the fixed membrane problem. Reviews in Mathematical Physics, 1994, 6(5a): 999-1009
[3] Faber G.Beweis, dass unter allen homogenen Membranen von gleicher Fläche und gleicher Spannung die kreisförmige den tiefsten Grundton gibt// München Z. Sitzungsber Bayer Akad Wiss München Math-Phys KI, 1923: 169-172
[4] Brands J.Bounds for the ratios of the first three membrane eigenvalues. Archive for Rational Mechanics and Analysis, 1964, 16: 265-268
[5] Chavel I.Eigenvalues in Riemannian Geometry. Orlando: Academic Press, 1984
[6] Chen D G, Zheng T.Bounds for ratios of the membrane eigenvalues. Journal of Differential Equations, 2011, 250(3): 1575-1590
[7] 陈维桓, 李兴校. 黎曼几何引论. 上册. 北京: 北京大学出版社, 2002: 123-126
Chen W H, Li X X.Introduction to Riemannian Geometry. Part I. Beijing: Peking University Press, 2002: 123-126
[8] Cheng Q M.Universal estimates for eigenvalues and applications// Proceedings of the 6th International Congress of Chinese Mathematicians. Advanced Lectures in Mathematics, 2016: 37-52
[9] Cheng Q M, Qi X. Inequalities for eigenvalues of the Laplacian. arXiv:1104.5298v1
[10] Cheng Q M, Yang H.Bounds on eigenvalues of Dirichlet Laplacian. Mathematische Annalen, 2007, 337(1): 159-175
[11] Du F, Mao J, Wang Q L, Wu C X.Eigenvalue inequalities for the buckling problem of the drifting Laplacian on Ricci solitons. Journal of Differential Equations, 2016, 260(7): 5533-5564
[12] Evans L C. Partial Differential Equations.Providence, RI: American Mathematical Soc, 2010
[13] Gilbarg D, Trudinger N S.Elliptic Partial Differential Equations of Second Order. Berlin: Springer, 1977
[14] Hile G N, Protter M H. Inequalities for eigenvalues of the Laplacian. Indiana University Mathematics Journal, 1980, 29(4): 523-538 %12
[15] Krahn E.Über eine von Rayleigh formulierte Minimaleigenschaft des Kreises. Mathematische Annalen, 1925, 94(1): 97-100
[16] Levitin M, Parnovski L.Commutators spectral trace identities, and universal estimates for eigenvalues. J Funct Anal, 2002, 192(2): 425-445
[17] Lu W, Mao J, Wu C X, Zeng L Z.Eigenvalue estimates for the drifting Laplacian and the $p$-Laplacian on submanifolds of warped products. Applicable Analysis, 2021, 100(11): 2275-2300
[18] Mao J.Functional inequalities and manifolds with nonnegative weighted Ricci curvature. Czechoslovak Mathematical Journal, 2020, 70(1): 213-233
[19] Mao J.The Gagliardo-Nirenberg inequalities and manifolds with non-negative weighted Ricci curvature. Kyushu Journal of Mathematics, 2016, 70(1): 029-046
[20] Mao J, Tu R, Zeng K.Eigenvalue estimates for submanifolds in Hadamard manifolds and product manifolds $ N\times\mathbb R$. Hiroshima Mathematical Journal, 2020, 50(1): 17-42
[21] Payne L E, Pólya G, Weinberger H F.On the ratio of consecutive eigenvalues. Journal of Mathematical Physics, 1956, 35(1-4): 289-298
[22] Pólya G.On the eigenvalues of vibrating membranes. Proceedings of the London Mathematical Society, 1961, 11(3): 419-433
[23] Rosales C, Canete A, Bayle V, Morgan F.On the isoperimetric problem in Euclidean space with density. Calculus of Variations and Partial Differential Equations, 2008, 31(1): 27-46
[24] Thompson C J.On the ratio of consecutive eigenvalues in $n$-dimensions. Studies in Applied Mathematics, 1969, 48(3): 281-283
[25] Wei G F, Wylie W.Comparison geometry for the Bakry-Émery Ricci tensor. Journal of Differential Geometry, 2009, 83(2): 337-405
[26] Yang H C.An estimate of the difference between consecutive eigenvalues. International Centre for Theoretical Physics, 1991
[27] Zeng L Z, Sun H J.Eigenvalues of the drifting Laplacian on smooth metric measure spaces. Pacific J Math, 2022, 319(2): 439-470
[28] Zhao Y, Wu C, Mao J, Du F.Eigenvalue comparisons in Steklov eigenvalue problem and some other eigenvalue estimates. Revista Matemática Complutense, 2020, 33: 389-414 |