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Application of topological degree to the study of bifurcation in von Kármán equations. (English) Zbl 1006.35093
The author studies a bifurcation problem in von Kármán equations for thin elastic circular plate lying on the elastic base under the action of a compressing force. The problem is described by the equations \[ \Delta \Delta w-[w,\sigma] +2\alpha\Delta w+\beta w-\gamma w^3=0 \text{ in } \Omega,\tag{1} \] \[ \Delta\Delta \sigma+0.5[w,w]=0\text{ in }\Omega,\tag{2} \] where \(\Omega\) is the unit ball in \(\mathbb{R}^2\), \(\Delta\) the Laplace operator, \(w\) the deflection, \(\sigma\) the stress function, \(\alpha\) the applied compressing force, \(\beta,\gamma\) the parameters of elastic foundation, and \[ [h,g]= {\partial^2h \over\partial x^2} {\partial^2g\over \partial y^2}-2{\partial^2 h\over \partial x\partial y}{\partial^2 g\over\partial x\partial y}+{\partial^2 h\over\partial y^2}{\partial^2g\over\partial x^2}. \] The von Kármán equations (1), (2) are reformulated into an operator equation \(F(x,p)=0\) in a real Banach space, \(x=(w,\sigma)\), \(p=(\alpha,\beta)\). It is shown that the necessary condition for bifurcation at \((0,p)\) is that \(N(p)=\dim(\text{Ker} F_x'(0,p))>0\). The author proves \(N(p)<4\). To solve the bifurcation problem homotopy and degree theory are used.

35Q72 Other PDE from mechanics (MSC2000)
74K20 Plates
74G60 Bifurcation and buckling
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