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Rearrangements and polar factorisation of countably degenerate functions

Published online by Cambridge University Press:  14 November 2011

G. R. Burton  and
R. J. Douglas
G. R. Burton
Affiliation:
Department of Mathematical Sciences, University of Bath, Claverton Down, Bath BA2 7AY, U.K.
R. J. Douglas
Affiliation:
Department of Mathematics, University of Reading, Whiteknights, P.O. Box 220, Reading RG6 6AX, U.K.
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Abstract

This paper proves some extensions of Brenier's theorem that an integrable vector-valued function u, satisfying a nondegeneracy condition, admits a unique polar factorisation u = u# ° s. Here u# is the monotone rearrangement of u, equal to the gradient of a convex function almost everywhere on a bounded connected open set Y with smooth boundary, and s is a measure-preserving mapping. We show that two weaker alternative hypotheses are sufficient for the existence of the factorisation; that u# be almost injective (in which case s is unique), or that u be countably degenerate (which allows u to have level sets of positive measure). We allow Y to be any set of finite positive Lebesgue measure. Our construction of the measure-preserving map s is especially simple.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 128 , Issue 4 , 1998 , pp. 671 - 681
Copyright
Copyright © Royal Society of Edinburgh 1998

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