A uniqueness criterion and a counterexample to regularity in an incompressible variational problem

In this paper we consider the problem of minimizing functionals of the form \(E(u)=\int _B f(x,\nabla u) \,dx\) $E(u)={\int }_{B}f(x,\mathrm{\nabla }u)dx$ in a suitably prepared class of incompressible, planar maps \(u: B \rightarrow \mathbb {R}^2\) $u:B\to {\mathbb{R}}^2$ . Here, B is the unit disk and \(f(x,\xi )\) $f(x,\xi )$ is quadratic and convex in \(\xi \) $\xi$ . 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 \(f(x,\xi )\) $f(x,\xi )$ , depending smoothly on \(\xi \) $\xi$ but discontinuously on x, whose unique global minimizer is the so-called \(N-\) $N-$ covering map, which is Lipschitz but not \(C^1\) $C^1$ .