Calculus of Variations and Geometric Measure Theory

S. Bianchini - L. Caravenna

On the extremality, uniqueness and optimality of transference plans

created by caravenna on 18 Sep 2009
modified on 30 Oct 2009


Submitted Paper

Inserted: 18 sep 2009
Last Updated: 30 oct 2009

Year: 2009


In this paper we consider the following standard problems appearing in optimal transportation theory: \begin{itemize} \item when a transference plan is extremal, \item when a transference plan is the unique transference plan concentrated on a set $A$, \item when a transference plan is optimal. \end{itemize} We show that these three problems can be studied with a general approach: \begin{enumerate} \item choose some necessary conditions, depending on the problem we are considering; \item find a partition into sets $B_t$ where these necessary conditions become also sufficient; \item show that all the transference plans are concentrated on the union of $B_t$. \end{enumerate} Explicit procedures are provided in the three cases above, the principal one being that the problem has an hidden structure of linear preorder with Borel graph.

As by sides results, we study the disintegration theorem w.r.t. family of equivalence relations, the construction of optimal potentials, a natural relation obtained from $c$-cyclical monotonicity.