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) d x 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 . [ABSTRACT FROM AUTHOR]