The necessary and sufficient condition for bifurcation in the von K�rm�n equations

Abstract

The paper is devoted to the study of bifurcation in the von Kármán equations with two parameters α, β ∈ R+ that describe the behaviour of a thin round elastic plate lying on an elastic base under the action of a compressing force. The problem appears in the mechanics of elastic constructions. We prove the necessary and sufficient condition for bifurcation at points of the set of trivial solutions. Our proof is based on reducing the von Kármán equations to an operator equation in Banach spaces with a nonlinear Fredholm map of index 0 and applying the Crandall-Rabinowitz theorem on simple bifurcation points or a finite-dimensional reduction and degree theory.2000 Mathematics Subject Classification: 35Q72, 46T99.

Symmetry-Breaking Bifurcation for Free Elastic Shell of Biological Cluster, Part 2.

“…Moreover, we want to adapt Friedman and collaborator's approach to symmetry breaking bifurcations in free boundary problems (see [4,8,9,10]). The scheme of application of the CrandallRabinowitz theorem is similar to that in [2,3,11,12,13,14,16].…”

Description of the solution set of the von Kármán equations for a circular plate in a small neighbourhood of a simple bifurcation point

“…13, 2006 Description of the solution set of the von Kármán equations 341 used throughout this work. For the proofs we refer the reader to [JJancz] and [JJancz3]. Let X and Y be defined as in the previous Section.…”

“…We have not found any mathematical works of others connecting with this problem. In our earlier works on this subject (see [JJancz]- [JJancz3]) we were interested in the problem of the existence of bifurcation. In [JJancz] by the use of Crandall-Rabinowitz bifurcation theorem we proved the existence of simple bifurcation points of (1.2).…”

“…In [JJancz] by the use of Crandall-Rabinowitz bifurcation theorem we proved the existence of simple bifurcation points of (1.2). In [JJancz1] and [JJancz3] applying the Krasnosielski theorem and Z 2 -symmetries we showed that there are multiple bifurcation points of (1.2). Furthermore, in [JJancz3] we formulated a necessary and sufficient condition for bifurcation at a point (0, 0, α 0 , β 0 ) ∈ Γ.…”

“…In [JJancz1] and [JJancz3] applying the Krasnosielski theorem and Z 2 -symmetries we showed that there are multiple bifurcation points of (1.2). Furthermore, in [JJancz3] we formulated a necessary and sufficient condition for bifurcation at a point (0, 0, α 0 , β 0 ) ∈ Γ. In this paper we give a full description of the solution set of (1.2) in a small neighbourhood of a simple bifurcation point.…”