  We characterize Young measures generated by gradients of bi-Lipschitz
orientation-preserving maps in the plane. This question is motivated by
variational problems in nonlinear elasticity where the orientation preservation
and injectivity of the admissible deformations are key requirements. These
results enable us to derive new weak$^*$ lower semicontinuity results for
integral functionals depending on gradients. As an application, we show the
existence of a minimizer for an integral functional with nonpolyconvex energy
density among bi-Lipschitz homeomorphisms.
