Duality Theory in Nonlinear Buckling Analysis for von Kármán Equations

The nonlinear eigenvalue problem in buckling analysis is studied for von Kármán plates. By using the general duality theory developed by Gao-Strang [1, 2] it is proved that the stability criterion for the bifurcated state depends on a reduced complementary gap function. The duality theory is established for nonlinear bifurcation problems. This theory shows that the nonlinear eigenvalue problem is eventually equivalent to a coupled quadratic dual optimization problem. A series of equivalent variational principles are constructed and a lower bound theorem for the first eigenvalue of the buckling factor is proved.