  In this paper we consider the problem of minimizing functionals of the form
$E(u)=\int_B f(x,\nabla u) \,dx$ in a suitably prepared class of
incompressible, planar maps $u: B \rightarrow \mathbb{R}^2$. Here, $B$ is the
unit disk and $f(x,\xi)$ is quadratic and convex in $\xi$. It is shown that if
$u$ is a stationary point of $E$ in a sense that is made clear in the paper,
then $u$ is a unique global minimizer of $E(u)$ provided the gradient of the
corresponding pressure satisfies a suitable smallness condition. We apply this
result to construct a non-autonomous, uniformly convex functional $f(x,\xi)$,
depending smoothly on $\xi$ but discontinuously on $x$, whose unique global
minimizer is the so-called $N-$covering map, which is Lipschitz but not $C^1$.
