Calculus of Variations and Geometric Measure Theory

A. Figalli

Existence, uniqueness and regularity of optimal transport maps

created by figalli on 10 Oct 2006


Accepted Paper

Inserted: 10 oct 2006

Journal: SIAM Journal on Mathematical Analysis
Year: 2006


Adapting some techniques and ideas of McCann, we extend a recent result with Fathi to yield existence and uniqueness of a unique transport map in very general situations, without any integrability assumption on the cost function.\In particular this result applies for the optimal transportation problem on a $n$-dimensional non-compact manifold $M$ with a cost function induced by a $C^2$-Lagrangian, provided that the source measure vanishes on sets with $\sigma$-finite $(n-1)$-dimensional Hausdorff measure. Moreover we prove that, in the case $c(x,y)=d^2(x,y)$, the transport map is approximatively differentiable a.e. with respect to the volume measure, and we extend some results of Cordero-Erasquin, McCann and Schmuckenschlager about concavity estimates and displacement convexity.

Keywords: optimal transportation, existence, uniqueness, displacement convexity