### Abstract

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 |

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**Uniqueness of the polar factorisation and projection of a vector-valued mapping.** / Burton, G R; Douglas, R J.

Research output: Contribution to journal › Article

*Annales De L Institut Henri Poincare: Analyse Non Linéaire*, vol. 20, no. 3, pp. 405-418. https://doi.org/10.1016/s0294-1449(02)00026-4/fla

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TY - JOUR

T1 - Uniqueness of the polar factorisation and projection of a vector-valued mapping

AU - Burton, G R

AU - Douglas, R J

N1 - ID number: ISI:000182756000003

PY - 2003

Y1 - 2003

N2 - 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.

AB - 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.

U2 - 10.1016/s0294-1449(02)00026-4/fla

DO - 10.1016/s0294-1449(02)00026-4/fla

M3 - Article

VL - 20

SP - 405

EP - 418

JO - Annales De L Institut Henri Poincare: Analyse Non Linéaire

JF - Annales De L Institut Henri Poincare: Analyse Non Linéaire

SN - 0294-1449

IS - 3

ER -