Existence, Uniqueness, and Regularity of Optimal Transport Maps

Adapting some techniques and ideas of McCann [8], we extend a recent result with Fathi [6] 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 C2-Lagrangian, provided that the source measure vanishes on sets with σnite (n − 1)-dimensional Hausdor measure. Moreover we prove that, in the case c(x, y) = d(x, y), the transport map is approximatively di erentiable a.e. with respect to the volume measure, and we extend some results of [4] about concavity estimates and displacement convexity.