Optimal mass transportation and Mather theory

By P. Bernard, Boris Buffoni. 2004

158 citations recorded by the source corpus.

We study the Monge transportation problem when the cost is the action associated to a Lagrangian function on a compact manifold. We show that the transportation can be interpolated by a Lipschitz lamination. We describe several direct variational problems the minimizers of which are these Lipschitz laminations. We prove the existence of an optimal transport map when the transported measure is absolutely continuous. We explain the relations with Mather's minimal measures.

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/e40a31f345857a0073b1681d4e8ef2906b03d271

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In Applied Mathematics

In Geometry and Topology

Citations in the corpus

An introduction to sampling via measure transport (2016, background): “Generalizations of this result accounting for diﬀerent cost functions and spaces can be found in [19, 2, 27, 11].”
Young measures, superposition and transport (2007, unknown): “D S] 9 J ul 2 00 Published in: Indiana University Math Journal, 57 No 1 (2008) pp 247–276.”
A User’s Guide to Optimal Transport (2013, background): “Fathi [35], the paper being inspired by Bernand-Buffoni [13].”
Optimal transportation on non-compact manifolds (2007, methodology): “In this work we study Monge’s problem on manifolds for a large class of cost functions induced by Lagrangians like in [7], where the authors consider the case of compact manifolds.”
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]).”
Optimal transportation under nonholonomic constraints (2007, background): “Recently, Bernard and Buffoni [7] generalized this further to the case where the cost c is the action associated to a”
On the sufficiency of c-cyclical monotonicity for optimality of transport plans (2008, background): “The literature about mass transport problems is extremely wide, the interested reader may look for instance at [1–5,8,15,19].”
Optimal transportation, topology and uniqueness (2010, background): “The first problem has become classical in the calculus of variations; it has deep connections to analysis [30,51,73,74,98], geometry [29,63,70,71,79,83,95,104], dynamics [7,10,59,82] and nonlinear partial differential equations [18,20,21,36,42, 72,102], as well as applications in physics [38,75,97],”
Optimal transport and Perelman’s reduced volume (2008, background): “This sort of transport problem was introduced by Topping [26], as described below, with the purpose of constructing certain monotonic quantities for a Ricci flow.”
Five lectures on optimal transportation: Geometry, regularity and applications (2010, background): “On the pure side, these include connections to inequalities [92] [131] [94] [32] [90] [52], geometry (including sectional [85] [73], Ricci [88] [87] [127] [99] and mean [75] curvature), nonlinear partial differential equations [14] [18] [16] [135] [89], and dynamical systems (weak KAM theory [9]; no”
Transport Equation and Cauchy Problem for Non-Smooth Vector Fields (2008, background): “The Lecture Notes [84] (see also the Appendix of [69]) contain, among several other things, a comprehensive treatment of the topic of measures in the space of actionminimizing curves, including at the same time the optimal transport and the dynamical systems case (this unified treatment was inspired”
Generalized Ricci curvature bounds for three dimensional contact subriemannian manifolds (2009, background): “If ϕ1 is the optimal map between the measures � and ν, then ϕt is optimal between � and ϕt�� . All the above results also hold when the distance squared cost d 2 is replaced by costs defined by Lagrange’s problem (see [ 5 , 3]).”
Continuity equations and ODE flows with non-smooth velocity* (2014, background): “The Lecture Notes [105] (see also the Appendix of [88]) contain, among several other things, a comprehensive treatment of the topic of measures in the space of action-minimizing curves, including at the same time the optimal transport and the dynamical systems case (this unified treatment was inspir”