Bulletin of the Belgian Mathematical Society - Simon Stevin

Multiple bifurcation in the solution set of the von Kármán equations with $S^{1}$-symmetries

Joanna Janczewska

Abstract

In this work we study bifurcation of forms of equilibrium of a thin circular elastic plate lying on an elastic base under the action of a compressive force. The forms of equilibrium may be found as solutions of the von Kármán equations with two real positive parameters defined on the unit disk in $\mathbb R^2$ centered at the origin. These equations are equivalent to an operator equation $F(x,p)=0$ in the real Hölder spaces with a nonlinear $S^{1}$-equivariant Fredholm map of index $0$. For the existence of bifurcation at a point $(0,p)$ it is necessary that $\dim\operatorname{Ker}F_{x}^{\prime}(0,p)>0$. The space $\operatorname{Ker}F_{x}^{\prime}(0,p)$ can be at most four-dimensional. We apply the Crandall-Rabinowitz theorem to prove that if $\dim\operatorname{Ker}F_{x}^{\prime}(0,p)=3$ then there is bifurcation of radial solutions at $(0,p)$. What is more, using the Lyapunov-Schmidt finite-dimensional reduction we investigate the number of branches of radial bifurcation at $(0,p)$.

Article information

Source
Bull. Belg. Math. Soc. Simon Stevin, Volume 15, Number 1 (2008), 109-126.

Dates
First available in Project Euclid: 22 February 2008

https://projecteuclid.org/euclid.bbms/1203692450

Digital Object Identifier
doi:10.36045/bbms/1203692450

Mathematical Reviews number (MathSciNet)
MR2406090

Zentralblatt MATH identifier
1185.35279

Citation

Janczewska, Joanna. Multiple bifurcation in the solution set of the von Kármán equations with $S^{1}$-symmetries. Bull. Belg. Math. Soc. Simon Stevin 15 (2008), no. 1, 109--126. doi:10.36045/bbms/1203692450. https://projecteuclid.org/euclid.bbms/1203692450