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 $u_{\sigma }: \Omega \to {\open R}^2$ in a model, two-dimensional case. We exploit the Jacobian constraint $\det \nabla u_{\sigma} \gt 0$ a.e. to obtain regularity results that apply ‘up to the boundary’ of domains with corners. It is shown that the unique shear map minimizer has the properties that (i) $\det \nabla u_{\sigma }$ is strictly positive on one part of the domain Ω, (ii) $\det \nabla u_{\sigma } = 0$ necessarily holds on the rest of Ω, and (iii) properties (i) and (ii) combine to ensure that $\nabla u_{\sigma }$ is not continuous on the whole domain.