@article{ASNSP_1996_4_23_2_383_0, author = {Ashbaugh, Mark S. and Laugesen, Richard S.}, title = {Fundamental tones and buckling loads of clamped plates}, journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze}, pages = {383--402}, publisher = {Scuola normale superiore}, volume = {Ser. 4, 23}, number = {2}, year = {1996}, zbl = {0891.73028}, mrnumber = {1433428}, language = {en}, url = {archive.numdam.org/item/ASNSP_1996_4_23_2_383_0/} }
Ashbaugh, Mark S.; Laugesen, Richard S. Fundamental tones and buckling loads of clamped plates. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 4, Tome 23 (1996) no. 2, pp. 383-402. http://archive.numdam.org/item/ASNSP_1996_4_23_2_383_0/
[1] M. ABRAMOWITZ - I.A. STEGUN, Eds., Handbook of Mathematical Functions. National Bureau of Standards Applied Mathematics Series, Vol. 55. U.S. Government Printing Office, Washington, D.C., 1964 | MR 167642
[2] More bounds on eigenvalue ratios for Dirichlet Laplacians in N dimensions, SIAM J. Math. Anal., 24 (1993), 1622-1651. | MR 1241161 | Zbl 0809.35067
- ,[3] On Rayleigh's conjecture for the clamped plate and its generalization to three dimensions, Duke Math. J., 78 (1995), 1-17. | MR 1328749 | Zbl 0833.35035
- ,[4] Upper and lower bounds for the frequencies of rectangular clamped plates, Z. Angew. Math. Mech., 47 (1967), 191-198. | Zbl 0161.44204
- - ,[5] Pointwise bounds in the first biharmonic boundary value problem, J. Math. & Phys., 42 (1963), 278-286. | MR 159135 | Zbl 0168.37003
- ,[6] On the structure of solutions to ΔΔu = λu which satisfy the clamped plate conditions on a right angle, SIAM J. Math. Anal., 13 (1982), 746-757. | Zbl 0498.73010
,[7] On the fundamental eigenfunctions of a clamped punctured disk, Adv. in Appl. Math., 13 (1992), 142-151. | MR 1162137 | Zbl 0762.35073
- ,[8] Extremal eigenvalue problems for starlike planar domains, J. Differential Equations., 120 (1995), 174-197. | MR 1339673 | Zbl 0837.35101
- ,[9] Contractive zero-divisors in Bergman spaces, Pacific J. Math., 157 (1993), 37-56. | MR 1197044 | Zbl 0739.30029
- - - ,[10] Numerical and Quantitative Analysis, Trans. by S. Graffi. Pitman, London, 1978. | MR 519677 | Zbl 0384.65043
,[11] Computational methods for determining lower bounds for eigenvalues of operators in Hilbert space, SIAM Rev., 8 (1996), 427-462. | MR 215508 | Zbl 0161.35504
- ,[12] Approximations and bounds for eigenvalues of elliptic operators, SIAM J. Numer. Anal., 4 (1967), 89-103. | MR 215542 | Zbl 0148.39502
- - ,[13] Rearrangements and Convexity of Level Sets in PDE. Springer-Verlag, Berlin, 1985. | MR 810619 | Zbl 0593.35002
,[14] On sign variation and the absence of "strong " zeros of solutions of elliptic equations, Math. USSR-Izv., 34 (1990), 337-353. | MR 998299 | Zbl 0701.35062
- - ,[15] Untere Schranken für die Eigenwerte selbstadjungierter positiv-definiter Operatoren, Numer. Math., 17 (1971), 166-171. | MR 283978 | Zbl 0206.46004
,[16] Über Minimaleigenschaften der Kugel in drei und mehr Dimensionen, Acta Comm. Univ. Tartu(Dorpat), A9 (1926), 1-44. English translation in: Edgar Krahn 1894-1961 : A Centenary Volume. Ed: Ü. Lumiste and J. Peetre, pp.139-174. IOS Press, Amsterdam, 1994. | JFM 52.0510.03
,[17] Upper and lower bounds for eigenvalues by finite differences, Pacific J. Math., 35 (1970), 429-440. | MR 277105 | Zbl 0204.12003
,[18] Upper and lower bounds for frequencies of clamped rhombical plates, J. Sound Vibration, 68 (1980), 597-607. | Zbl 0435.73064
- ,[19] Upper and lower bounds for frequencies of trapezoidal and triangular plates, J. Sound Vibration, 78 (1981), 585-590.
- ,[20] Vibration of Plates. NASA SP-160. Office of Technology Utilization, NASA, Washington, D.C., 1969.
,[21] Monotonicity of the zeros of a cross product of Bessel functions, Methods Applicat. Anal., 1 (1994), 75-80. | MR 1260384 | Zbl 0840.33002
,[22] Bounding eigenvalues of clamped plates, Z. Angew. Math. Phys., 19 (1968), 676-681. | MR 235715 | Zbl 0187.46502
,[23] Bounds for Vibration Frequencies and Buckling Loads of Clamped Plates. Dissertation No. 4415, ETH, Zürich. Juris-Druck Verlag, Zürich, 1969.
,[24] Über die Rayleighsche Vermutung: Unter allen Platten von gegebener Fläche und konstanter Dichte und Elastizität hat die kreisförmige den tiefsten Grundton, Ann. Mat. Pura Appl., 104 (1975), 85-122. | MR 381462 | Zbl 0315.35036
,[25] Rayleigh's conjecture on the principal frequency of the clamped plate, Arch. Ration. Mech. Anal., 129 (1995), 1-10. See also New isoperimetric inequalities in mathematical physics. In: Partial Differential Equations of Elliptic Type. Ed: A. Alvino, E. Fabes and G. Talenti. Cambridge University Press, Cambridge, 1994. | MR 1328469
,[26] On Green's function for the biharmonic equation in a right-angle wedge, J. Math. Anal. Appl., 43 (1973), 705-716. | MR 324209 | Zbl 0263.31005
,[27] Inequalities for eigenvalues of membranes and plates, J. Rat. Mech. Anal., 4 (1955), 517-529. | MR 70834 | Zbl 0064.34802
,[28] On the characteristicfrequencies of a symmetric membrane, Math. Z., 63 (1955), 331-337. | MR 73047 | Zbl 0065.08703
,[29] Isoperimetric Inequalities in Mathematical Physics. Princeton University Press, Princeton, N.J., 1951. | MR 43486 | Zbl 0044.38301
- ,[30] The Theory of Sound, Vol. 1. Dover Publications, New York, 1945. | MR 16009 | Zbl 0061.45904
( ),[31] On the buckling of an elliptic plate with clamped edge, II, J. Phys. Soc. Japan, 12 (1957), 529-532. See also On the buckling of an elliptic plate with clamped edge, I. J. Phys. Soc. Japan, 11 (1957), 1088-1091. | MR 85006
,[32] Lower bounds to fundamental frequencies and buckling loads of columns and plates, Internat. J. Solids and Structures, 14 (1978), 1013-1026. | MR 513765 | Zbl 0388.73042
- ,[33] On membranes and plates, Proc.' Nat. Acad. Sci. U.S.A., 36 (1950), 210-216. See also Note to my paper "On membranes and plates," Proc. Nat. Acad. Sci. U.S.A., 44 (1958), 314-316. | MR 35629 | Zbl 0088.17001
,[34] On the first eigenvalue of the clamped plate, Ann. Mat. Pura Appl.,129 (1981), 265-280. | MR 648335 | Zbl 0475.73050
,[35] The buckling load for a rectangular plate with four clamped edges, Z. Angew. Math. Mech., 13 (1933), 147-152. | JFM 59.0745.03
,[36] Theory of Elastic Stability. 2nd Ed. McGraw-Hill, New York, 1961. | MR 134026
- ,[37] Elliptical membranes with smallest second eigenvalue, Math. Comp., 27 (1973), 767-772. | MR 421277 | Zbl 0271.35018
,[38] Bounds for critical values and eigenfrequencies of mechanical systems, in: Proceedings of the German Italian Symposium "Applications of Mathematics in Technology", held in Rome, 1984, pp. 469-484. B.G. Teubner, Stuttgart, 1984. | MR 788563 | Zbl 0547.70026
,[39] Theory of Bessel Functions, 2nd Ed. Cambridge University Press, Cambridge, 1944. | MR 10746 | Zbl 0849.33001
,[40] Variational Methods for Eigenvalue Approximation. Regional conference series in applied mathematics, 15. SIAM, Philadelphia, 1974. | MR 400004 | Zbl 0296.49033
,[41 ] Methods of Intermediate Problems for Eigenvalues: Theory and Ramifications. Academic Press, New York, 1972. | MR 477971 | Zbl 0291.49034
- ,[42] Symmetrical buckling of right-angled isosceles triangular plates, Aeronaut. Quart., 5 (1954), 131-143. | MR 63254
,