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

G. R. Burton, Robert J. Douglas

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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.
Original languageEnglish
Pages (from-to)405-418
Number of pages14
JournalAnnales de l'Institut Henri Poincaré (C) Analyse Non Linéaire
Issue number3
Early online date27 Nov 2002
Publication statusPublished - 01 May 2003


  • polar factorisation
  • monotone rearrangement
  • measure-preserving mappings
  • L2-projection


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