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.(#) o 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(#) o s are exactly those, if any, which are closest to u in the L-2-norm. (C) 2003 Editions scientifiques et medicales Elsevier SAS.