Transport problems and disintegration maps
Received: 6 July 2012
Revised: 26 September 2012
By disintegration of transport plans it is introduced the notion of transport class. This allows to consider the Monge problem as a particular case of the Kantorovich transport problem, once a transport class is fixed. The transport problem constrained to a fixed transport class is equivalent to an abstract Monge problem over a Wasserstein space of probability measures. Concerning solvability of this kind of constrained problems, it turns out that in some sense the Monge problem corresponds to a lucky case.
Mathematics Subject Classification: 37J50 / 49Q20 / 49Q15
Key words: Optimal mass transportation theory / Monge − Kantorovich problem / calculus of variations / shape analysis / geometric measure theory
© EDP Sciences, SMAI, 2013