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

Abstract

AbstractIn this paper we consider the problem of minimizing functionals of the form $$E(u)=\int _B f(x,\nabla u) \,dx$$ E ( u ) = ∫ B f ( x , ∇ u ) d x in a suitably prepared class of incompressible, planar maps $$u: B \rightarrow \mathbb {R}^2$$ u : B → R 2 . Here, B is the unit disk and $$f(x,\xi )$$ f ( x , ξ ) is quadratic and convex in $$\xi $$ 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 $$f(x,\xi )$$ f ( x , ξ ) , depending smoothly on $$\xi $$ ξ 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 .