Inequalities for eigenvalues of supported and free plates
Author:
L. E. Payne
Journal:
Quart. Appl. Math. 16 (1958), 111-120
MSC:
Primary 73.00
DOI:
https://doi.org/10.1090/qam/96440
MathSciNet review:
96440
Full-text PDF Free Access
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Additional Information
N. Aronszajn, Studies in eigenvalue problems, Tech. Rep. 3, Oklahoma A. and M. College (1951)
N. Aronszajn and W. Donoghue, Studies in eigenvalue problems, Tech. Rep. 12, University of Kansas (1954)
R. Courant and D. Hilbert, Methoden der mathematischen Physik, vol. 1, Berlin, 1931
G. Faber, Beweis, dass unter allen homogenen Membranen von gleicher Fläche und gleicher Spannung die kreisförmige den tiefsten Grundton gibt, Sitz. ber. Bayr. Akad, Wiss., 169 (1923)
- Tosio Kato, On some approximate methods concerning the operators $T^*T$, Math. Ann. 126 (1953), 253–262. MR 58131, DOI https://doi.org/10.1007/BF01343163
- E. T. Kornhauser and I. Stakgold, A variational theorem for $\nabla ^2u+\lambda u=0$ and its applications, J. Math. Physics 31 (1952), 45–54. MR 0047236
E. Krahn, Über eine von Rayleigh formulierte Minimaleigenschaft des Kreises, Math. Ann. 94, 97 (1924)
- Yoshimoto Nakata and Hiroshi Fujita, On upper and lower bounds of the eigenvalues of a free plate, J. Phys. Soc. Japan 10 (1955), 823–824. MR 75780, DOI https://doi.org/10.1143/JPSJ.10.823
- L. E. Payne, Inequalities for eigenvalues of membranes and plates, J. Rational Mech. Anal. 4 (1955), 517–529. MR 70834, DOI https://doi.org/10.1512/iumj.1955.4.54016
L. E. Payne and H. F. Weinberger, Two inequalities for eigenvalues of membranes, Tech. Note BN–65, Univ. of Maryland (1955)
- 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
- Franz Rellich, Darstellung der Eigenwerte von $\Delta u+\lambda u=0$ durch ein Randintegral, Math. Z. 46 (1940), 635–636 (German). MR 2456, DOI https://doi.org/10.1007/BF01181459
- G. Szegö, Inequalities for certain eigenvalues of a membrane of given area, J. Rational Mech. Anal. 3 (1954), 343–356. MR 61749, DOI https://doi.org/10.1512/iumj.1954.3.53017
- H. F. Weinberger, An isoperimetric inequality for the $N$-dimensional free membrane problem, J. Rational Mech. Anal. 5 (1956), 633–636. MR 79286, DOI https://doi.org/10.1512/iumj.1956.5.55021
A. Weinstein, Étude des spectres des équations aux dérivées partielles de la théorie des plaques élastiques, Mémorial de Sciences Math. 88, Paris (1937)
N. Aronszajn, Studies in eigenvalue problems, Tech. Rep. 3, Oklahoma A. and M. College (1951)
N. Aronszajn and W. Donoghue, Studies in eigenvalue problems, Tech. Rep. 12, University of Kansas (1954)
R. Courant and D. Hilbert, Methoden der mathematischen Physik, vol. 1, Berlin, 1931
G. Faber, Beweis, dass unter allen homogenen Membranen von gleicher Fläche und gleicher Spannung die kreisförmige den tiefsten Grundton gibt, Sitz. ber. Bayr. Akad, Wiss., 169 (1923)
T. Kato, On some approximation methods concerning the operator $T*T$, Math. Ann. 126, 253 (1953)
E. T. Kornhauser and T. Stakgold, A variational theorem for ${\nabla ^2}u + \lambda u = 0$ and its applications, J. Math. Phys. 31, 45 (1952)
E. Krahn, Über eine von Rayleigh formulierte Minimaleigenschaft des Kreises, Math. Ann. 94, 97 (1924)
Y. Nakata and H. Fujita, On upper and lower bounds of the eigenvalues of a free plate, J. Phys. Soc. Japan, 10, 823 (1955)
L. E. Payne, Inequalities for eigenvalues of membranes and plates, J. Ratl. Mech. Anal. 4, 517 (1955)
L. E. Payne and H. F. Weinberger, Two inequalities for eigenvalues of membranes, Tech. Note BN–65, Univ. of Maryland (1955)
G. Polya and G. Szegö, Isoperimetric inequalities in mathematical physics, Ann. Math. Studies 27, Princeton (1951)
F. Rellich, Darstellung der Eigenwerte von $\Delta u + \lambda u$ durch ein Randintegral, Math. Z. 46, 635 (1940)
G. Szegö, Inequalities for certain eigenvalues of a membrane of given area, J. Ratl. Mech. Anal. 3, 343 (1954)
H. F. Weinberger, An isoperimetric inequality for the free membrane problem, J. Ratl. Mech. Anal. 5, 633 (1956)
A. Weinstein, Étude des spectres des équations aux dérivées partielles de la théorie des plaques élastiques, Mémorial de Sciences Math. 88, Paris (1937)
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Article copyright:
© Copyright 1958
American Mathematical Society
