Uniqueness of the polar factorisation and projection of a vector-valued mapping

This paper proves some results concerning the polar factorisation of an integrable vector-valued function $u$ into the composition $u = u^{\#} \circ s$, where $u^{\#}$ is equal almost everywhere to the gradient of a convex function, and $s$ is a measure-preserving mapping. It is shown that the factorisation is unique (i.e. the measure-preserving mapping $s$ is unique) precisely when $u^{\#}$ is almost injective. Not every integrable function has a polar factorisation; we introduce a class of counterexamples. It is further shown that if $u$ is square integrable, then measure-preserving mappings $s$ which satisfy $u = u^{\#} \circ s$ are exactly those, if any, which are closest to $u$ in the $L^2$-norm.