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

G. R. Burton, Robert J. Douglas

Allbwn ymchwil: Cyfraniad at gyfnodolynErthygladolygiad gan gymheiriaid

2 Dyfyniadau(SciVal)
190 Wedi eu Llwytho i Lawr (Pure)

Crynodeb

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.
Iaith wreiddiolSaesneg
Tudalennau (o-i)405-418
Nifer y tudalennau14
CyfnodolynAnnales de l'Institut Henri Poincaré (C) Analyse Non Linéaire
Cyfrol20
Rhif cyhoeddi3
Dyddiad ar-lein cynnar27 Tach 2002
Dynodwyr Gwrthrych Digidol (DOIs)
StatwsCyhoeddwyd - 01 Mai 2003

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