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)=∫ _B f(x,∇ u) dx in a suitably prepared class of incompressible, planar maps u: B →ℝ^2 . Here, B is the unit disk and f(x,ξ ) is quadratic and convex in ξ . 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,ξ ) , depending smoothly on ξ but discontinuously on x , whose unique global minimizer is the so-called N- covering map, which is Lipschitz but not C^1 .