Abstract
In this paper we study constrained variational problems that are principally motivated by nonlinear elasticity theory. We examine in particular the relationship between the positivity of the Jacobian det∇u and the uniqueness and regularity of energy minimizers u that are either twist maps or shear maps. We exhibit ____emph{explicit} twist maps, defined on two-dimensional annuli, that are stationary points of an appropriate energy functional and whose Jacobian vanishes on a set of positive measure in the annulus. Within the class of shear maps we precisely characterize the unique global energy minimizer u σ :Ω→R 2 in a model, two-dimensional case. The shear map minimizer has the properties that (i) det∇u σ is strictly positive on one part of the domain Ω , (ii) det∇u σ =0 necessarily holds on the rest of Ω , and (iii) properties (i) and (ii) combine to ensure that ∇u σ is not continuous on the whole domain.