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 \(E(u)=\int _B f(x,\nabla u) \,dx\) 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 \(C^1\).