Some problems in the calculus of variations related to thin elastic sheets

Ginzburg-Landau theory may be thought of as the relaxation of
the Dirichlet energy for maps into the circle. Sticking to this
interpretation, what I will present in this talk are some results on the
relaxation of the Dirichlet energy of O(2,3)-valued maps with a gradient
constraint, and variants of this functional. Here O(2,3) denotes the set
of linear isometries from two- to three-dimensional Euclidean space.
These variational functionals can be interpreted as models for thin
elastic sheets, and in particular crumpled structures.