  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 $\det \nabla u$ 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 $u_{\sigma}: \Omega\to \mathbb{R}^2$ in a model,
two-dimensional case. The shear map minimizer has the properties that (i) $\det
\nabla u_{\sigma}$ is strictly positive on one part of the domain $\Omega$,
(ii) $\det \nabla u_{\sigma} = 0$ necessarily holds on the rest of $\Omega$,
and (iii) properties (i) and (ii) combine to ensure that $\nabla u_{\sigma}$ is
not continuous on the whole domain.
