Non-existence of polar factorisations and polar inclusion of a vector-valued mapping

Details Non-existence of polar factorisations and polar inclusion of a vector-valued mapping Non-existence of polar factorisations and polar inclusion of a vector-valued mapping

2007-12-13

arxiv.org/abs/0712.2161v1

Mathematics

Functional Analysis

9 pages, 0 figures, to be published in the International Journal of Pure and Applied Mathematics, IJPAM, 41, no. 3, 2007, 363-

Scientific paper

This paper proves some results concerning the polar factorisation of an integrable vector-valued function u into the composition of the gradient of a convex function with a measure-preserving mapping. Not every integrable function has a polar factorisation; we extend the class of counterexamples. We introduce a generalisation: u has a polar inclusion if u(x) belongs to the subdifferential of the convex function at y for almost every pair (x,y) with respect to a measure-preserving plan. Given a regularity assumption, we show that such measure-preserving plans are exactly the minimisers of a Monge-Kantorovich optimisation problem.

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