In this book, a class of nonconvex variational problems with spherical symmetry is studied. It is motivated by physical models for the elastic behavior of thin films.
As the main result of the first part of the book, existence and symmetry of global minimizers are shown for suitable lower order terms in the energy functional. In the second part, the original energy is regularized by adding a convex term consisting of derivatives of higher order with a small coefficient, which -- physically speaking -- represents a bending energy of the film. In this context, the existence of nontrivial radially symmetric critical points is shown by means of global bifurcation theory. Some of their qualitative properties obtained as a byproduct of the bifurcation analysis are then used in a study of the singular limit, as the coefficient of the bending energy goes to zero.
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