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

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Type Article
Original language English 405-418 14 Annales de l'Institut Henri Poincaré (C) Analyse Non Linéaire 20 3 27 Nov 2002 https://doi.org/10.1016/S0294-1449(02)00026-4 Published - 01 May 2003

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Abstract

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.

Keywords

• polar factorisation, monotone rearrangement, measure-preserving mappings, L2-projection