  In this note we prove that any $W^{1,2}$ mapping $u$ in the plane that
minimizes an appropriate quasiconvex energy functional subject to the Jacobian
constraint ${\rm det} \na u=1$ a.e., are necessarily Lipschitz. Furthermore we
show that the minimizers corresponding to uniformly convex energy are affine
and give an example of non-affine minimizers subject to affine boundary data
corresponding to a convex energy. We also discuss the regularity issues in
dimension greater than or equal to 3.
