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

Joanna Janczewska¹

¹ Institute of Mathematics University of Gdańsk Wita Stwosza 57 80-952 Gdańsk, Poland

Annales Polonici Mathematici, Tome 77 (2001) no. 1, pp. 53-68

Résumé

We investigate bifurcation in the solution set of the von Kármán equations on a disk ${\mit \Omega }\subset {\mathbb R}^{2}$ with two positive parameters $\alpha $ and $\beta $. 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 $\alpha $ and $\beta $. Applying the implicit function theorem we obtain the following necessary condition for bifurcation: if $(0,p) $ is a bifurcation point then $\mathop {\rm dim}\nolimits \mathop {\rm Ker}F_{x}^{\prime }(0,p) >0$. 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).

Détail

DOI : 10.4064/ap77-1-5

Keywords: investigate bifurcation solution set von equations disk mit omega subset mathbb positive parameters alpha beta equations describe behaviour elastic thin round plate lying elastic base under action compressing force method analysis based reducing problem operator equation real banach spaces nonlinear fredholm map index zero defined later depends parameters alpha beta applying implicit function theorem obtain following necessary condition bifurcation bifurcation point mathop dim nolimits mathop ker prime full description kernel chet derivative study detail situation dimension kernel prove bifurcation point lyapunov schmidt finite dimensional reduction crandall rabinowitz theorem one dimensional bifurcation point analysing lyapunov schmidt branching equation determine number families solutions their directions asymptotic behaviour shapes