• approximate differential
• displacement convexity
• Mané potential
• mass transportation
• non-compact manifolds
• Tonelli Lagrangians

We consider the optimal transportation problem on non-compact manifolds.
We yield existence and uniqueness of a unique transport map in the case of cost functions 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 about concavity estimates and displacement convexity.
As a corollary of this existence-uniqueness result, we prove the equivalence between the notions of "dispacement convexity" and "weak displacement convexity" on non-compact Riemannian manifolds.
Finally we study the optimal transportation problem with distance-like costs, proving the existence of an optimal transport map on non-compact manifolds for
a large class of cost functions that includes the case $c(x,y)=d(x,y)$, under the only hypothesis that
the source measure is absolutely continuous with respect to the volume measure.
In particular, we assume compactness neither of the support of the source measure nor of that of the target measure.