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 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.

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