## Abstract

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.

Original language | English |
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Pages (from-to) | 405-418 |

Number of pages | 14 |

Journal | Annales De L Institut Henri Poincare: Analyse Non Linéaire |

Volume | 20 |

Issue number | 3 |

DOIs | |

Publication status | Published - 2003 |