Variational Problems for Föppl--von Kármán Plates

Some variational problems for a Foppl--von Karman plate subject to general equilibrated loads are studied. The existence of global minimizers is proved under the assumption that the out-of-plane displacement fulfils homogeneous Dirichlet condition on the whole boundary while the in-plane displacement fulfils nonhomogeneous Neumann condition. If the Dirichlet condition is prescribed only on a subset of the boundary, then the energy may be unbounded from below over the set of admissible configurations, as shown by several explicit conterexamples: in these cases the analysis of critical points is addressed through an asymptotic development of the energy functional in a neighborhood of the flat configuration. By a $\Gamma$-convergence approach we show that critical points of the Foppl--von Karman energy can be strongly approximated by uniform Palais--Smale sequences of suitable functionals: this property leads to identifying relevant features for critical points of approximating functionals, e.g., buckled con...