Non-existence of polar factorisations and polar inclusion of a vector-valued mapping

This paper proves some results concerning the polar factorisation of an integrable vector-valued function u into the composition u = u ◦ s, where u = ∇ψ almost everywhere for some convex function ψ, and s is a measure-preserving mapping. Not every integrable function has a polar factorisation; we extend the class of counterexamples. We introduce a generalisation: u has a polar inclusion if u(x) ∈ ∂ψ(y) for almost every pair (x, y) with respect to a measure-preserving plan. Given a regularity assumption, we show that such measure-preserving plans are exactly the minimisers of a Monge-Kantorovich optimisation problem.