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

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 dc.contributor.author Douglas, Robert J. dc.contributor.author Burton, G. R. dc.date.accessioned 2007-12-03T14:47:44Z dc.date.available 2007-12-03T14:47:44Z dc.date.issued 2003-05 dc.identifier.citation Douglas , R J & Burton , G R 2003 , ' Uniqueness of the polar factorisation and projection of a vector-valued mapping ' Annales de l'Institut Henri Poincare (C) Non Linear Analysis , vol 20 , no. 3 , pp. 405-418 . en dc.identifier.issn 0294-1449 dc.identifier.other PURE: 73670 dc.identifier.other dspace: 2160/380 dc.identifier.uri http://hdl.handle.net/2160/380 dc.description G.R. BURTON and R.J. DOUGLAS, Uniqueness of the polar factorisation and projection of a vector-valued mapping. Ann. I.H. Poincare â A.N. 20 (2003), 405-418. en dc.description.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. en dc.format.extent 14 en dc.language.iso eng dc.relation.ispartof Annales de l'Institut Henri Poincare (C) Non Linear Analysis en dc.title Uniqueness of the polar factorisation and projection of a vector-valued mapping en dc.type Text en dc.type.publicationtype Article (Journal) en dc.identifier.doi http://dx.doi.org/10.1016/S0294-1449(02)00026-4 dc.contributor.institution Institute of Mathematics & Physics (ADT) en dc.contributor.institution Mathematical Modelling of Structures, Solids and Fluids en dc.description.status Peer reviewed en
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