Geodesic
The number of buckled states of circular plates
Applications of Mathematics, Tome 34 (1989) no. 2, pp. 113-132
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This paper deals with the exact number of solutions of von Kármán equations for a rotationally symmetric buckling of a thin elastic plate. The plate of constant thickness is in static equilibrium under a uniform compressive thrust applied along its edge in the plane of the plate. The theory of M. G. Crandall, P. H. Rabinowitz [4], is used and the theory of M. S. Berger [1], [3] and M. S. Berger and P. C. Fife [2] is adapted. This work is a part of [6].
This paper deals with the exact number of solutions of von Kármán equations for a rotationally symmetric buckling of a thin elastic plate. The plate of constant thickness is in static equilibrium under a uniform compressive thrust applied along its edge in the plane of the plate. The theory of M. G. Crandall, P. H. Rabinowitz [4], is used and the theory of M. S. Berger [1], [3] and M. S. Berger and P. C. Fife [2] is adapted. This work is a part of [6].
DOI : 10.21136/AM.1989.104340
Classification : 34B15, 35B32, 35J65, 37G99, 58E07, 58F14, 73C50, 73H05, 73K10, 74G60, 74K20
Keywords: Fredholm operator; static equilibrium; plate of constant thickness; Fredholm map of index zero; singular point; rotationally symmetric buckled states; von Kármán plate equations; operator equation; proper Sobolev space; local bifurcation behavior; nodal properties 
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Marko, Ľubomír. The number of buckled states of circular plates. Applications of Mathematics, Tome 34 (1989) no. 2, pp. 113-132. doi: 10.21136/AM.1989.104340

[1] M. S. Berger: On von Kármán's Equations and the Buckling of a Thin Elastic Plate, I The Clamped Plate. Comm. on Pure and Appl. Math. vol. XX, 1967, 687-719. | DOI | MR | Zbl

[2] M. S. Berger P. C. Fife: Von Kármán's Equations and the Buckling of a Thin Elastic Plate, II Plate with General Edge Conditions. Comm. on Pure and Appl. Math., vol. XXI, 1968, 227-241. | MR

[3] M. S. Berger: Nonlinear Problems with Exactly Three Solutions. Indiana University Math. Journal, vol. 28, No 4 (1979), 689-698. | DOI | MR | Zbl

[4] M. G. Crandall P. H. Rabinowitz: Bifurcation from Simple Eigenvalue. J. Funct. Anal. 8 (1971), 321-340. | DOI | MR

[5] A. Kufner: Weighted Sobolev Spaces. Teubner, Leipzig, 1980. | MR | Zbl

[6] Ľ. Marko: Buckled States of Circular Plates. thesis, 1985 (Slovak).

[7] L. Nirenberg: Topics in Nonlinear Functional Analysis. Russian translation, Mir, Moscow 1977. | MR | Zbl

[8] A. S. Voľmir: Elastic Plates and Shells. (Russian). GITTL, Moscow 1956.

[9] J. H. Wolkowisky: Existence of Buckled States of Circular Plates. Comm. on Pure and Appl. Math. vol. XX, 1967, 549-560. | MR | Zbl

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