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

Joanna Janczewska
2001 Annales Polonici Mathematici  
We investigate bifurcation in the solution set of the von Kármán equations on a disk Ω ⊂ R 2 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
more » ... ing the implicit function theorem we obtain the following necessary condition for bifurcation: if (0, p) is a bifurcation point then dim Ker F 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). 2000 Mathematics Subject Classification: 35Q72, 46T99.
doi:10.4064/ap77-1-5 fatcat:u5qrtw647revhkqswogfmjlkty