Optimal transportation on non-compact manifolds

By A. Fathi, Alessio Figalli. 2007

119 citations recorded by the source corpus.

In this work, we show how to obtain for non-compact manifolds the results that have already been done for Monge Transport Problem for costs coming from Tonelli Lagrangians on compact manifolds. In particular, the already known results for a cost of the type dr, r > 1, where d is the Riemannian distance of a complete Riemannian manifold, hold without any curvature restriction.

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/56eb66e2b7c6cc176a56becdbae5a9d6c504a080

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Applied Mathematics

In Geometry and Topology

Citations in the corpus

Optimal transportation under nonholonomic constraints (2007, background): “The infimum of the problem in (5) can be equivalently characterized by taking infimum over all admissible controls u(·) such that the corresponding admissible paths start at the point x1, end at the point x0 of the manifold and satisfy the following control system ẋ(s) = −F (x(s), u(s)).”
Multimarginal Optimal Transport Maps for One–dimensional Repulsive Costs (2015, background): “As regards existence, in the case of two marginals, it holds under the so called “twist condition” (see [4, 9, 21]).”
Optimal transportation, topology and uniqueness (2010, background): “Our discussion is predicated on global differentiability of the cost, since a wide variety of existence and uniqueness results concerning optimal solutions to the Monge–Kantorovich problem have been established for costs with singular sets—including distances in Riemannian [28,46,77], sub-Riemannian”
Partial regularity for optimal transport maps (2012, background): “, it is locally semiconvex when seen in any chart) [13, 17].”
Five lectures on optimal transportation: Geometry, regularity and applications (2010, result): “Part of the discussion is therefore impressionistic, and some of the results are stated under the tacit requirement that the supports of the measures μ± be compact, with the understanding that they extend to non-compactly supported measures under appropriate hypotheses [93] [59] [45] [46] concerning”
Finsler interpolation inequalities (2009, methodology): “we will give a self-contained proof along the same lines as McCann’s [22], this part follows also from Villani’s [42, Chapter 10] (see also [6,7, 13 ]).”
A User’s Guide to Optimal Transport (2013, background): “Fathi [35], the paper being inspired by Bernand-Buffoni [13].”
The Monge–Ampère equation and its link to optimal transportation (2013, background): “dierentiable a.e. Remark 3.5. If csatises (C0) and Y is bounded, then it follows from (3.7) that is semiconvex (i.e., there exists a constant C >0 such that + Cjxj2=2 is convex, see for instance [46]). In particular, by Theorem 3.2, c-convex functions are twice dierentiable a.e. The following is ”