Rearrangements and polar factorisation of countably degenerate functions

G. R. Burton*, R. J. Douglas

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

7 Citations (SciVal)


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.

Original languageEnglish
Pages (from-to)671-681
Number of pages11
JournalRoyal Society of Edinburgh - Proceedings A
Issue number4
Publication statusPublished - 1998


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