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


Reference:

Burton, G. R. and Douglas, R. J., 2003. Uniqueness of the polar factorisation and projection of a vector-valued mapping. Annales De L Institut Henri Poincare: Analyse Non Linéaire, 20 (3), pp. 405-418.

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Abstract

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.

Details

Item Type Articles
CreatorsBurton, G. R.and Douglas, R. J.
DOI10.1016/s0294-1449(02)00026-4/fla
DepartmentsFaculty of Science > Mathematical Sciences
RefereedYes
StatusPublished
ID Code7345
Additional InformationID number: ISI:000182756000003

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