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

### Reference:

Burton, G. R. and Douglas, R. J., 2003. Uniqueness of the polar factorisation and projection of a vector-valued mapping. *Annales De L Institut Henri Poincare: Analyse Non Linéaire*, 20 (3), pp. 405-418.

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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.(#) 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.

### Details

Item Type | Articles |

Creators | Burton, G. R.and Douglas, R. J. |

DOI | 10.1016/s0294-1449(02)00026-4/fla |

Departments | Faculty of Science > Mathematical Sciences |

Refereed | Yes |

Status | Published |

ID Code | 7345 |

Additional Information | ID number: ISI:000182756000003 |

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