dc.contributor.author Douglas, Robert J. dc.date.accessioned 2007-12-03T14:39:16Z dc.date.available 2007-12-03T14:39:16Z dc.date.issued 2007 dc.identifier.citation Douglas , R J 2007 , ' Non-existence of polar factorisations and polar inclusion of a vector-valued mapping ' International Journal of Pure and Applied Mathematics , vol 41 , no. 3 . en dc.identifier.other PURE: 73646 dc.identifier.other dspace: 2160/379 dc.identifier.uri http://hdl.handle.net/2160/379 dc.identifier.uri http://math.uctm.edu/journals/index.html en dc.description R.J. DOUGLAS, Non-existence of polar factorisations and polar inclusion of a vector-valued mapping. Intern. Jour. Of Pure and Appl. Math., (IJPAM) 41, no. 3 (2007). 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 $u^{\#} = \nabla \psi$ almost everywhere for some convex function $\psi$, and $s$ is a measure-preserving mapping. Not every integrable function has a polar factorisation; we extend the class of counterexamples. We introduce a generalisation: $u$ has a polar inclusion if $u(x) \in \partial \psi (y)$ for almost every pair $(x,y)$ with respect to a measure-preserving plan. Given a regularity assumption, we show that such measure-preserving plans are exactly the minimisers of a Monge-Kantorovich optimisation problem. en dc.language.iso eng dc.relation.ispartof International Journal of Pure and Applied Mathematics en dc.title Non-existence of polar factorisations and polar inclusion of a vector-valued mapping en dc.type Text en dc.type.publicationtype Article (Journal) en 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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