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On Rayleigh’s conjecture for the clamped plate and its generalization to three dimensions
Mark S. Ashbaugh and Rafael D. Benguria
Source: Duke Math. J. Volume 78, Number 1
(1995), 1-17.
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Permanent link to this document: http://projecteuclid.org/euclid.dmj/1077285546
Mathematical Reviews number (MathSciNet): MR1328749
Zentralblatt MATH identifier: 0833.35035
Digital Object Identifier: doi:10.1215/S0012-7094-95-07801-6
References
[1] Abramowitz, M. and Stegun, I. A., eds., Handbook of mathematical functions with formulas, graphs, and mathematical tables, National Bureau of Standards Applied Mathematics Series, vol. 55, For sale by the Superintendent of Documents, U.S. Government Printing Office, Washington, D.C., 1964.
Mathematical Reviews (MathSciNet): MR29:4914
Zentralblatt MATH: 0171.38503
[2] R. A. Adams, Sobolev spaces, Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers], New York-London, 1975.
Mathematical Reviews (MathSciNet): MR56:9247
Zentralblatt MATH: 0314.46030
[3] M. S. Ashbaugh and R. S. Laugesen, Fundamental tones and buckling loads of clamped plates, preprint, 1994.
Mathematical Reviews (MathSciNet): MR1433428
Zentralblatt MATH: 0891.73028
[4] Gábor Szegő, Collected papers. Vol. 3, Contemporary Mathematicians, Birkhäuser Boston, Mass., 1982.
Mathematical Reviews (MathSciNet): MR84d:01082c
Zentralblatt MATH: 0491.01016
[5] C. Bandle, Isoperimetric inequalities and applications, Monographs and Studies in Mathematics, vol. 7, Pitman (Advanced Publishing Program), Boston, Mass., 1980.
Mathematical Reviews (MathSciNet): MR81e:35095
Zentralblatt MATH: 0436.35063
[6] J. H. Bramble and L. E. Payne, Pointwise bounds in the first biharmonic boundary value problem, J. Math. and Phys. 42 (1963), 278–286.
Mathematical Reviews (MathSciNet): MR28:2353
Zentralblatt MATH: 0168.37003
[7] H. Brezis, Analyse fonctionnelle, Collection Mathématiques Appliquées pour la Maîtrise. [Collection of Applied Mathematics for the Master's Degree], Masson, Paris, 1983.
Mathematical Reviews (MathSciNet): MR85a:46001
Zentralblatt MATH: 0511.46001
[8] C. V. Coffman and R. J. Duffin, On the fundamental eigenfunctions of a clamped punctured disk, Adv. in Appl. Math. 13 (1992), no. 2, 142–151.
Mathematical Reviews (MathSciNet): MR93c:35113
Zentralblatt MATH: 0762.35073
Digital Object Identifier: doi:10.1016/0196-8858(92)90006-I
[9] R. J. Duffin, Nodal lines of a vibrating plate, J. Math. Phys. 31 (1953), 294–299.
Mathematical Reviews (MathSciNet): MR14,601f
Zentralblatt MATH: 0050.19006
[10] R. J. Duffin and D. H. Shaffer, On the modes of vibration of a ring-shaped plate, Bull. Amer. Math. Soc. 58 (1952), 652.
[11] G. H. Hardy, J. E. Littlewood, and G. Pólya, Inequalities, 2nd ed., Cambridge, at the University Press, 1952.
Mathematical Reviews (MathSciNet): MR13,727e
Zentralblatt MATH: 0047.05302
[12] S. Kesavan, Topics in functional analysis and applications, John Wiley & Sons Inc., New York, 1989.
Mathematical Reviews (MathSciNet): MR90m:46002
Zentralblatt MATH: 0666.46001
[13] L. Lorch, Monotonicity of the zeros of a cross product of Bessel functions, Methods Appl. Anal. 1 (1994), no. 1, 75–80.
Mathematical Reviews (MathSciNet): MR95b:33008
Zentralblatt MATH: 0840.33002
[14] 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. (4) 104 (1975), 85–122.
Mathematical Reviews (MathSciNet): MR52:2359
Zentralblatt MATH: 0315.35036
Digital Object Identifier: doi:10.1007/BF02417012
[15] N. S. Nadirashvili, An isoperimetric inequality for the main frequency of a clamped plate, Dokl. Akad. Nauk 332 (1993), no. 4, 436–439, (In Russian); English translation to appear in Soviet Phys. Dokl.
Mathematical Reviews (MathSciNet): MR95d:35115
Zentralblatt MATH: 0816.73027
[16] N. S. Nadirashvili, New isoperimetric inequalities in mathematical physics, Partial differential equations of elliptic type (Cortona, 1992) eds. A. Alvino and E. Fabes and G. Talenti, Symposia Mathematica, XXXV, Cambridge Univ. Press, Cambridge, 1994, pp. 197–203.
Mathematical Reviews (MathSciNet): MR1297780
Zentralblatt MATH: 0814.35083
[17] N. S. Nadirashvili, Rayleigh's conjecture on the principal frequency of the clamped plate, Arch. Rational Mech. Anal., to appear.
Mathematical Reviews (MathSciNet): MR1328469
Zentralblatt MATH: 0826.73035
Digital Object Identifier: doi:10.1007/BF00375124
[18] L. E. Payne, Inequalities for eigenvalues of membranes and plates, J. Rational Mech. Anal. 4 (1955), 517–529.
Mathematical Reviews (MathSciNet): MR17,42a
Zentralblatt MATH: 0064.34802
[19] G. Pólya and G. Szegö, Isoperimetric Inequalities in Mathematical Physics, Annals of Mathematics Studies, vol. 27, Princeton University Press, Princeton, N. J., 1951.
Mathematical Reviews (MathSciNet): MR13,270d
Zentralblatt MATH: 0044.38301
[20] J. W. S. Rayleigh and B. Strutt, The Theory of Sound, Dover Publications, New York, N. Y., 1945, 2nd edition revised and enlarged (in two volumes), (republication of the 1894/1896 edition).
Mathematical Reviews (MathSciNet): MR7,500e
Zentralblatt MATH: 0061.45904
[21] G. F. Simmons, Differential equations with applications and historical notes, International Series in Pure and Applied Mathematics, McGraw-Hill Book Co., New York, 1972.
Mathematical Reviews (MathSciNet): MR58:17258
Zentralblatt MATH: 0231.34001
[22] H. C. Simpson and S. J. Spector, Some monotonicity results for ratios of modified Bessel functions, Quart. Appl. Math. 42 (1984), no. 1, 95–98.
Mathematical Reviews (MathSciNet): MR85d:33021
Zentralblatt MATH: 0549.33008
[23] R. P. Soni, On an inequality for modified Bessel functions, J. Math. and Phys. 44 (1965), 406–407, reprinted in [4], 187–193.
Mathematical Reviews (MathSciNet): MR32:2634
Zentralblatt MATH: 0137.04801
[24] G. Szegö, On membranes and plates, Proc. Nat. Acad. Sci. U. S. A. 36 (1950), 210–216, reprinted in 4, 481–483.
Mathematical Reviews (MathSciNet): MR11,757g
Zentralblatt MATH: 0039.20604
Digital Object Identifier: doi:10.1073/pnas.36.3.210
JSTOR: links.jstor.org
[25] G. Szegö, Note to my paper “On membranes and plates”, Proc. Nat. Acad. Sci. U.S.A. 44 (1958), 314–316.
Mathematical Reviews (MathSciNet): MR20:2924
Zentralblatt MATH: 0088.17001
Digital Object Identifier: doi:10.1073/pnas.44.4.314
JSTOR: links.jstor.org
[26] G. Szegö, On the vibrations of a clamped plate, Atti del Quarto Congresso dell'Unione Matematica Italina, Taormina, 1951, vol. II, Casa Editrice Perrella, Roma, 1953, Reprinted in [4], 311–315, pp. 573–577.
Mathematical Reviews (MathSciNet): MR15,130a
Zentralblatt MATH: 0051.16701
[27] G. Talenti, On the first eigenvalue of the clamped plate, Ann. Mat. Pura Appl. (4) 129 (1981), 265–280 (1982).
Mathematical Reviews (MathSciNet): MR83b:73048
Zentralblatt MATH: 0475.73050
Digital Object Identifier: doi:10.1007/BF01762146
[28] G. N. Watson, A Treatise on the Theory of Bessel Functions 2nd ed., Cambridge University Press, Cambridge, England, 1944.
Mathematical Reviews (MathSciNet): MR6,64a
Zentralblatt MATH: 0063.08184
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