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On the lift-off constant for elastically supported plates

Authors: R. F. Bass, J. Horák and P. J. McKenna
Journal: Proc. Amer. Math. Soc. 132 (2004), 2951-2958
MSC (2000): Primary 35J40; Secondary 60J65
Published electronically: June 2, 2004
MathSciNet review: 2063115
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Abstract: In this paper we continue the study begun by Kawohl and Sweers of the precise constant at which the elastic foundation supporting a bending plate can allow lift-off in the case of downward loading. We provide a number of numerical results and a rigorous result on a different counterexample than the one suggested in Kawohl and Sweers (2002). Important open problems are summarized at the conclusion.

References [Enhancements On Off] (What's this?)

  • 1. Bass, R. F., Probabilistic Techniques in Analysis, Springer-Verlag, New York, 1995. MR 96e:60001
  • 2. Bass, R. F., and Burdzy, K., Conditioned Brownian motion in planar domains, Probab. Theory Related Fields 101 (1995) 479-493. MR 96b:60199
  • 3. Boggio, T., Sull'equilibrio delle piastre elastiche incastrate, Rend. Acc. Lincei 10 (1901), 197- 205.
  • 4. Boggio, T., Sulle funzioni di Green d'ordine $m$, Rend. Circ. Mat. Palermo 20 (1905), 97-135.
  • 5. Cranston, M.; Fabes, E.; and Zhao, Z., Potential theory for the Schrödinger equation, Bull. Amer. Math. Soc. (N.S.) 15 (1986), no. 2, 213-216. MR 88d:60197
  • 6. Cranston, M. and McConnell, T. R., The lifetime of conditioned Brownian motion. Z. Wahrsch. Verw. Gebiete 65 (1983), no. 1, 1-11. MR 85d:60150
  • 7. Cranston, M., Lifetime of conditioned Brownian motion in Lipschitz domains. Z. Wahrsch. Verw. Gebiete 70 (1985), no. 3, 335-340. MR 87a:60088
  • 8. Coffman, C. V., On the structure of solutions to $\Delta^2u =\lambda u$which satisfy the clamped plate condition on a right angle, SIAM J. Math. Anal. 13, 746-757 (1982). MR 84a:35015
  • 9. Coffman, C. V. and Duffin, R. J., On the structure of biharmonic functions satisfying the clamped plate conditions on a right angle, Adv. Appl. Math. 1 (1980), 373-389. MR 82e:31004
  • 10. Duffin, R. J., On a question of Hadamard concerning super-biharmonic functions, J. Math. Phys. 27 (1949), 253-258. MR 10:534h
  • 11. Garabedian, P. R., A partial differential equation arising in conformal mapping, Pacific J. Math. 1 (1951), 485-524. MR 13:735a
  • 12. Grunau, H.-C. and Sweers, G., Positivity for perturbations of polyharmonic operators with Dirichlet boundary conditions in two dimensions. Math. Nachr. 179 (1996), 89-102. MR 97f:35040
  • 13. Hadamard, J., Mémoire sur le problème d'analyse relatif à l'équilibre des plaques élastiques incastrées, Mémoires presentes par divers savants à l'Académie des Sciences, Vol. 33 (1908), 1-128.
  • 14. Hadamard, J., Sur certains cas intéressants du problème biharmonique, Atti IVe Congr. Intern. Mat. Rome (1909), 12-14.
  • 15. Kawohl, B., and Sweers, G., On ``anti''-eigenvalues for elliptic systems and a question of McKenna and Walter, Indiana Univ. Math. J. 51 (2002), no. 5, 1023-1040. MR 2003j:35059
  • 16. Lin, Li, and Adams, G. G., Beam on tensionless elastic foundation J. Eng. Mech., 113 (1986), 542-553.
  • 17. McKenna, P. J., and Walter, W., Nonlinear oscillations in a suspension bridge, Arch. Rational Mech. Anal., 98 (1987), 167-177. MR 88a:35160
  • 18. Patil, S. P., Response of infinite railroad track to vibrating mass, J. Eng. Mech., 114 (1988) 688-703.
  • 19. Schröder, Johann, Operator inequalities, Mathematics in Science and Engineering, 147, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1980. MR 84k:65063

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Additional Information

R. F. Bass
Affiliation: Department of Mathematics, University of Connecticut, Storrs, Connecticut 06269

J. Horák
Affiliation: Department of Mathematics, University of Basel, Basel, Switzerland
Address at time of publication: Mathematisches Institut, Universität zu Köln, Weyertal 86-90, D-50923 Köln, Germany

P. J. McKenna
Affiliation: Department of Mathematics, University of Connecticut, Storrs, Connecticut 06269

Keywords: Lift-off constant, vibrating plate, beam, Brownian motion
Received by editor(s): January 7, 2003
Published electronically: June 2, 2004
Additional Notes: The research of the first author was partially supported by NSF grant DMS-9988496
Communicated by: David S. Tartakoff
Article copyright: © Copyright 2004 American Mathematical Society

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