Existence, Uniqueness, and Regularity of Optimal Transport Maps

Adapting some techniques and ideas of McCann [Duke Math. J., 80 (1995), pp. 309–323], we extend a recent result with Fathi [Optimal Transportation on Manifolds, preprint] 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 an n-dimensional noncompact manifold M with a cost function induced by a C2-Lagrangian, provided that the source measure vanishes on sets with σ-finite (n − 1)-dimensional Hausdorff measure. Moreover we prove that in the case c(x, y) = d2(x, y), the transport map is approximatively differentiable a.e. with respect to the volume measure, and we extend some results of [D. Cordero-Erasquin, R. J. McCann, and M. Schmuckenschlager, Invent. Math., 146 (2001), pp. 219–257] about concavity estimates and displacement convexity.