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

Abstract

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 .√ N e R (N θ), where N ∈ N \ {0} and 0 ≤ θ ≤ 2π.