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 . 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).
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-ap77-1-5,
author = {Joanna Janczewska},
title = {Bifurcation in the solution set of the von K\'arm\'an equations of an elastic disk lying on an elastic foundation},
journal = {Annales Polonici Mathematici},
volume = {77},
year = {2001},
pages = {53-68},
zbl = {0996.35079},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-ap77-1-5}
}
Joanna Janczewska. Bifurcation in the solution set of the von Kármán equations of an elastic disk lying on an elastic foundation. Annales Polonici Mathematici, Tome 77 (2001) pp. 53-68. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-ap77-1-5/