Fundamental tones and buckling loads of clamped plates

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1. [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 MR167642
2. [2] M.S. Ashbaugh - R.D. Benguria, More bounds on eigenvalue ratios for Dirichlet Laplacians in N dimensions, SIAM J. Math. Anal., 24 (1993), 1622-1651. Zbl0809.35067MR1241161
3. [3] M.S. Ashbaugh - R.D. Benguria, On Rayleigh's conjecture for the clamped plate and its generalization to three dimensions, Duke Math. J., 78 (1995), 1-17. Zbl0833.35035MR1328749
4. [4] N.W. Bazley - D.W. Fox - J.T. Stadter, Upper and lower bounds for the frequencies of rectangular clamped plates, Z. Angew. Math. Mech., 47 (1967), 191-198. Zbl0161.44204
5. [5] J.H. Bramble - L.E. Payne, Pointwise bounds in the first biharmonic boundary value problem, J. Math. & Phys., 42 (1963), 278-286. Zbl0168.37003MR159135
6. [6] C.V. Coffman, 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. Zbl0498.73010
7. [7] C.V. Coffman - R.J. Duffin, On the fundamental eigenfunctions of a clamped punctured disk, Adv. in Appl. Math., 13 (1992), 142-151. Zbl0762.35073MR1162137
8. [8] S.J. Cox - M. Ross, Extremal eigenvalue problems for starlike planar domains, J. Differential Equations., 120 (1995), 174-197. Zbl0837.35101MR1339673
9. [9] P.L. Duren - D. Khavinson - H.S. Shapiro - C. Sundberg, Contractive zero-divisors in Bergman spaces, Pacific J. Math., 157 (1993), 37-56. Zbl0739.30029MR1197044
10. [10] G. Fichera, Numerical and Quantitative Analysis, Trans. by S. Graffi. Pitman, London, 1978. Zbl0384.65043MR519677
11. [11] D.W. Fox - W.C. Rheinboldt, Computational methods for determining lower bounds for eigenvalues of operators in Hilbert space, SIAM Rev., 8 (1996), 427-462. Zbl0161.35504MR215508
12. [12] L. Fox - P. Henrici - C. Moler, Approximations and bounds for eigenvalues of elliptic operators, SIAM J. Numer. Anal., 4 (1967), 89-103. Zbl0148.39502MR215542
13. [13] B. Kawohl, Rearrangements and Convexity of Level Sets in PDE. Springer-Verlag, Berlin, 1985. Zbl0593.35002MR810619
14. [14] V.A. Kozlov - V.A. Kondrat'ev - V.G. Maz'ya, On sign variation and the absence of "strong " zeros of solutions of elliptic equations, Math. USSR-Izv., 34 (1990), 337-353. Zbl0701.35062MR998299
15. [15] B. Knauer, Untere Schranken für die Eigenwerte selbstadjungierter positiv-definiter Operatoren, Numer. Math., 17 (1971), 166-171. Zbl0206.46004MR283978
16. [16] E. Krahn, 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. Zbl52.0510.03JFM52.0510.03
17. [17] J.R. Kuttler, Upper and lower bounds for eigenvalues by finite differences, Pacific J. Math., 35 (1970), 429-440. Zbl0204.12003MR277105
18. [18] J.R. Kuttler - V.G. Sigillito, Upper and lower bounds for frequencies of clamped rhombical plates, J. Sound Vibration, 68 (1980), 597-607. Zbl0435.73064
19. [19] J.R. Kuttler - V.G. Sigillito, Upper and lower bounds for frequencies of trapezoidal and triangular plates, J. Sound Vibration, 78 (1981), 585-590. 
20. [20] A.W. Leissa, Vibration of Plates. NASA SP-160. Office of Technology Utilization, NASA, Washington, D.C., 1969. 
21. [21] L. Lorch, Monotonicity of the zeros of a cross product of Bessel functions, Methods Applicat. Anal., 1 (1994), 75-80. Zbl0840.33002MR1260384
22. [22] J. McLaurin, Bounding eigenvalues of clamped plates, Z. Angew. Math. Phys., 19 (1968), 676-681. Zbl0187.46502MR235715
23. [23] J.W. McLaurin, Bounds for Vibration Frequencies and Buckling Loads of Clamped Plates. Dissertation No. 4415, ETH, Zürich. Juris-Druck Verlag, Zürich, 1969. 
24. [24] E. Mohr, 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. Zbl0315.35036MR381462
25. [25] N. Nadirashvili, 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. MR1328469
26. [26] S. Osher, On Green's function for the biharmonic equation in a right-angle wedge, J. Math. Anal. Appl., 43 (1973), 705-716. Zbl0263.31005MR324209
27. [27] L.E. Payne, Inequalities for eigenvalues of membranes and plates, J. Rat. Mech. Anal., 4 (1955), 517-529. Zbl0064.34802MR70834
28. [28] G. Pólya, On the characteristicfrequencies of a symmetric membrane, Math. Z., 63 (1955), 331-337. Zbl0065.08703MR73047
29. [29] G. Pólya - G. Szegö, Isoperimetric Inequalities in Mathematical Physics. Princeton University Press, Princeton, N.J., 1951. Zbl0044.38301MR43486
30. [30] Lord Rayleigh ( J.W. Strutt), The Theory of Sound, Vol. 1. Dover Publications, New York, 1945. Zbl0061.45904MR16009
31. [31] Y. Shibaoka, 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. MR85006
32. [32] P.-Y. Shih - H.L. Schreyer, Lower bounds to fundamental frequencies and buckling loads of columns and plates, Internat. J. Solids and Structures, 14 (1978), 1013-1026. Zbl0388.73042MR513765
33. [33] G. Szegö, 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. Zbl0088.17001MR35629
34. [34] G. Talenti, On the first eigenvalue of the clamped plate, Ann. Mat. Pura Appl.,129 (1981), 265-280. Zbl0475.73050MR648335
35. [35] G.I. Taylor, The buckling load for a rectangular plate with four clamped edges, Z. Angew. Math. Mech., 13 (1933), 147-152. Zbl59.0745.03JFM59.0745.03
36. [36] S. Timoshenko - J.M. Gere, Theory of Elastic Stability. 2nd Ed. McGraw-Hill, New York, 1961. MR134026
37. [37] B.A. Troesch, Elliptical membranes with smallest second eigenvalue, Math. Comp., 27 (1973), 767-772. Zbl0271.35018MR421277
38. [38] W. Velte, 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. Zbl0547.70026MR788563
39. [39] G.N. Watson, Theory of Bessel Functions, 2nd Ed. Cambridge University Press, Cambridge, 1944. Zbl0849.33001MR10746
40. [40] H.F. Weinberger, Variational Methods for Eigenvalue Approximation. Regional conference series in applied mathematics, 15. SIAM, Philadelphia, 1974. Zbl0296.49033MR400004
41. [41 ] A. Weinstein - W. Stenger, Methods of Intermediate Problems for Eigenvalues: Theory and Ramifications. Academic Press, New York, 1972. Zbl0291.49034MR477971
42. [42] W.H. Wittrick, Symmetrical buckling of right-angled isosceles triangular plates, Aeronaut. Quart., 5 (1954), 131-143. MR63254

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