On Monotonicity of Some Functionals Under Monotone Rearrangement with Respect To One Variable

2017 ◽  
Vol 224 (3) ◽  
pp. 385-390
Author(s):  
S. V. Bankevich
Author(s):  
G. R. Burton ◽  
R. J. Douglas

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.


2004 ◽  
Vol 17 (3) ◽  
pp. 353-355
Author(s):  
J.E Rakotoson ◽  
J.M Rakotoson

2018 ◽  
Vol 6 (4) ◽  
pp. 1503-1531 ◽  
Author(s):  
Donsub Rim ◽  
Kyle T. Mandli

2020 ◽  
Vol 25 (0) ◽  
Author(s):  
Julio Backhoff-Veraguas ◽  
Mathias Beiglböck ◽  
Gudmund Pammer

1986 ◽  
Vol 97 (4) ◽  
pp. 626-626 ◽  
Author(s):  
Anthony Horsley ◽  
Andrzej J. Wrobel

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