TY - JOUR

T1 - Rearrangements and polar factorisation of countably degenerate functions

AU - Burton, G. R.

AU - Douglas, R. J.

N1 - Funding Information:
The authors are grateful for the hospitality and support of the Isaac Newton Institute for Mathematical Sciences, where this research was conducted during the programme Mathematics of Atmosphere and Ocean Dynamics. The second author's research was supported by EPSRC Research Fellow Grant no. 21409 MTA S08.
Copyright:
Copyright 2017 Elsevier B.V., All rights reserved.

PY - 1998

Y1 - 1998

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=22044456113&partnerID=8YFLogxK

U2 - 10.1017/S0308210500021703

DO - 10.1017/S0308210500021703

M3 - Article

AN - SCOPUS:22044456113

SN - 0308-2105

VL - 128

SP - 671

EP - 681

JO - Proceedings of the Royal Society of Edinburgh, Section A: Mathematics

JF - Proceedings of the Royal Society of Edinburgh, Section A: Mathematics

IS - 4

ER -