On frequencies of strings and deformations of beams

Abramovich, Shoshana

doi:10.1090/S0033-569X-05-00959-5

Author: Shoshana Abramovich
Journal: Quart. Appl. Math. 63 (2005), 291-299
MSC (2000): Primary 34B24, 34L15, 74B05
DOI: https://doi.org/10.1090/S0033-569X-05-00959-5
Published electronically: February 23, 2005
MathSciNet review: 2150775
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: The paper deals with the ratio of the first two frequencies of the vibrating string and a monotonicity property of deformations of beams under symmetrization method applied to its loads.

Shoshana Abramovich, Monotonicity of eigenvalues under symmetrization, SIAM J. Appl. Math. 28 (1975), 350–361. MR 382774, DOI https://doi.org/10.1137/0128030
S. Abramovich, The gap between the first two eigenvalues of a one-dimensional Schrödinger operator with symmetric potential, Proc. Amer. Math. Soc. 111 (1991), no. 2, 451–453. MR 1036981, DOI https://doi.org/10.1090/S0002-9939-1991-1036981-X
Shoshana Abramovich, Monotonicity of buckling loads under symmetrization, Quart. Appl. Math. 44 (1987), no. 4, 621–627. MR 872814, DOI https://doi.org/10.1090/S0033-569X-1987-0872814-3
Mark S. Ashbaugh and Rafael Benguria, Best constant for the ratio of the first two eigenvalues of one-dimensional Schrödinger operators with positive potentials, Proc. Amer. Math. Soc. 99 (1987), no. 3, 598–599. MR 875408, DOI https://doi.org/10.1090/S0002-9939-1987-0875408-4
Mark S. Ashbaugh and Rafael D. Benguria, Optimal bounds for ratios of eigenvalues of one-dimensional Schrödinger operators with Dirichlet boundary conditions and positive potentials, Comm. Math. Phys. 124 (1989), no. 3, 403–415. MR 1012632
Mark S. Ashbaugh and Rafael Benguria, Optimal lower bound for the gap between the first two eigenvalues of one-dimensional Schrödinger operators with symmetric single-well potentials, Proc. Amer. Math. Soc. 105 (1989), no. 2, 419–424. MR 942630, DOI https://doi.org/10.1090/S0002-9939-1989-0942630-X
Mark S. Ashbaugh and Rafael D. Benguria, Eigenvalue ratios for Sturm-Liouville operators, J. Differential Equations 103 (1993), no. 1, 205–219. MR 1218744, DOI https://doi.org/10.1006/jdeq.1993.1047
Vincenzo Ferone and Bernd Kawohl, Rearrangements and fourth order equations, Quart. Appl. Math. 61 (2003), no. 2, 337–343. MR 1976374, DOI https://doi.org/10.1090/qam/1976374
R. D. Gentry and D. O. Banks, Bounds for functions of eigenvalues of vibrating systems, J. Math. Anal. Appl. 51 (1975), 100–128. MR 372312, DOI https://doi.org/10.1016/0022-247X%2875%2990144-4
G. H. Hardy, J. E. Littlewood, and G. Pólya, Inequalities, Cambridge, at the University Press, 1952. 2d ed. MR 0046395
Miklós Horváth, On the first two eigenvalues of Sturm-Liouville operators, Proc. Amer. Math. Soc. 131 (2003), no. 4, 1215–1224. MR 1948113, DOI https://doi.org/10.1090/S0002-9939-02-06637-6
Min-Jei Huang, On the eigenvalue ratio for vibrating strings, Proc. Amer. Math. Soc. 127 (1999), no. 6, 1805–1813. MR 1621941, DOI https://doi.org/10.1090/S0002-9939-99-05015-7
Yu Ling Huang and C. K. Law, Eigenvalue ratios for the regular Sturm-Liouville system, Proc. Amer. Math. Soc. 124 (1996), no. 5, 1427–1436. MR 1328351, DOI https://doi.org/10.1090/S0002-9939-96-03396-5
Joseph B. Keller, The minimum ratio of two eigenvalues, SIAM J. Appl. Math. 31 (1976), no. 3, 485–491. MR 422751, DOI https://doi.org/10.1137/0131042
Richard Lavine, The eigenvalue gap for one-dimensional convex potentials, Proc. Amer. Math. Soc. 121 (1994), no. 3, 815–821. MR 1185270, DOI https://doi.org/10.1090/S0002-9939-1994-1185270-4
T. J. Mahar and B. E. Willner, An extremal eigenvalue problem, Comm. Pure Appl. Math. 29 (1976), no. 5, 517–529. MR 425244, DOI https://doi.org/10.1002/cpa.3160290505
G. Pólya and G. Szegö, Isoperimetric Inequalities in Mathematical Physics, Annals of Mathematics Studies, No. 27, Princeton University Press, Princeton, N. J., 1951. MR 0043486