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## Annales Polonici Mathematici

2001 | 77 | 1 | 53-68
Tytuł artykułu

### Bifurcation in the solution set of the von Kármán equations of an elastic disk lying on an elastic foundation

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EN
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EN
We investigate bifurcation in the solution set of the von Kármán equations on a disk Ω ⊂ ℝ² with two positive parameters α and β. The equations describe the behaviour of an elastic thin round plate lying on an elastic base under the action of a compressing force. The method of analysis is based on reducing the problem to an operator equation in real Banach spaces with a nonlinear Fredholm map F of index zero (to be defined later) that depends on the parameters α and β. Applying the implicit function theorem we obtain the following necessary condition for bifurcation: if (0,p) is a bifurcation point then $dim KerF'_{x}(0,p) > 0$. Next, we give a full description of the kernel of the Fréchet derivative of F. We study in detail the situation when the dimension of the kernel is one. We prove that (0,p) is a bifurcation point by the use of the Lyapunov-Schmidt finite-dimensional reduction and the Crandall-Rabinowitz theorem. For a one-dimensional bifurcation point, analysing the Lyapunov-Schmidt branching equation we determine the number of families of solutions, their directions and asymptotic behaviour (shapes).
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Tom
Numer
Strony
53-68
Opis fizyczny
Daty
wydano
2001
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autor
• Institute of Mathematics, University of Gdańsk, Wita Stwosza 57, 80-952 Gdańsk, Poland
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