## Optimal mass transportation and Mather theory (2005)

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Venue: | JOURNAL OF THE EUROPEAN MATHEMATICAL SOCIETY |

Citations: | 34 - 4 self |

### BibTeX

@ARTICLE{Bernard05optimalmass,

author = {Patrick Bernard and Boris Buffoni},

title = {Optimal mass transportation and Mather theory},

journal = {JOURNAL OF THE EUROPEAN MATHEMATICAL SOCIETY},

year = {2005}

}

### Years of Citing Articles

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### Abstract

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.

### Citations

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Citation Context ... vectorfield on M × [0,T], and if ν is a finite non-negative measure on M × [0,T], we define the current Z ∧ ν by ∫ Z ∧ ν(ω) := ω(Z)dν. M×[0,T] Every transport current can be written in this way, see =-=[22]-=- or [25]. As a consequence, currents extend as linear forms on the set Ω∞(M × [0,T]) of bounded measurable one-forms. If I is a Borel subset of the interval [0,T], it is therefore possible to define t... |

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Citation Context ... {0,T } for each f ∈ C1 (M, R), with a Lipschitz constant which depends only on ‖df‖∞ · ‖X‖∞. The family µt is then Lipschitz on I ∪ {0,T } for the 1-Wasserstein distance on probability measures, see =-=[39, 17, 4]-=- for example, the Lipschitz constant depending only on ‖X‖∞. It suffices to remember that, on the compact manifold M, the 1-Wasserstein distance on probabilities is topologically equivalent to the wea... |

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Citation Context ... the study of geometric properties of optimal objects. The costs functions we consider are natural generalizations of the cost c(x,y) = d(x,y) 2 considered by Brenier and many other authors. The book =-=[39]-=- gives some ideas of the applications expected from this kind of questions. More precisely, we consider a Lagrangian function L(x,v,t) : TM × R −→ R which is convex in v and satisfies standard hypothe... |

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Citation Context ...n which is induced from a transport map. Part of the result below is that this holds true in the case of the cost cT 0 . The method we use to prove this is an elaboration on ideas due to Brenier, see =-=[12]-=- and developed for instance in [24], (see also [23]) and [16], which is certainly the closest to our needs. Theorem B. Assume that µ0 is absolutely continuous with respect to the Lebesgue class on M. ... |

161 | The geometry of optimal transportation
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Citation Context ... map. Part of the result below is that this holds true in the case of the cost cT 0 . The method we use to prove this is an elaboration on ideas due to Brenier, see [12] and developed for instance in =-=[24]-=-, (see also [23]) and [16], which is certainly the closest to our needs. Theorem B. Assume that µ0 is absolutely continuous with respect to the Lebesgue class on M. Then for each final measure µT, the... |

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Citation Context ... min L(γ(σ), ˙γ(σ),σ)dσ γ s where the minimum is taken on the set of curves γ ∈ C 2 ([s,t],M) satisfying γ(s) = x and γ(t) = y. That this minimum exists is a standard result under our hypotheses, see =-=[33]-=- or [20]. Proposition 1. Let us fix a subinterval [s,t] ⊂ [0,T]. The set E ⊂ C 2 ([s,t],M) of minimizing extremals is compact for the C 2 topology. Let us mention that, for each (x0,s) ∈ M × [0,T], th... |

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Citation Context ...constant depending only on ‖X‖∞. It suffices to remember that, on the compact manifold M, the 1-Wasserstein distance on probabilities is topologically equivalent to the weak topology, see for example =-=[41]-=-, (48.5) or [39]. Smooth transport currents. A regular transport current is said smooth if it can be written on the form (X,1) ∧ λ with a bounded vectorfield X smooth on M×]0,T[ and a measure λ that h... |

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Citation Context ...ed is that there exists a Borel section S : M −→ TM of the canonical projection such that S(x) ∈ F0(φ0,φ1) for each x ∈ M. This follows from general statements of set-valued analysis, see for example =-=[14]-=- or the appendix in [12]. 6 Aubry-Mather theory We explain the relations between the results obtained so far and Mather theory, and prove Theorem C. Up to now, we have worked with fixed measures µ0 an... |

57 | Semiconcave functions, Hamilton–Jacobi equations and optimal control
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Citation Context ... semi-concave functions It is certainly useful to recall the main properties of viscosity solutions in connection with semiconcave functions. We will not give proofs, and instead refer to [20], [21], =-=[14]-=-, as well as the appendix in [8]. We will consider the Hamilton-Jacobi equation ∂tu + H(x,∂xu,t) = 0. (HJ) The function u : M × [0,T] −→ M is called K-semi-concave if, for each chart θ ∈ Θ (see append... |

53 |
Gradient Flows
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Citation Context ... {0,T } for each f ∈ C1 (M, R), with a Lipschitz constant which depends only on ‖df‖∞ · ‖X‖∞. The family µt is then Lipschitz on I ∪ {0,T } for the 1-Wasserstein distance on probability measures, see =-=[39, 17, 4]-=- for example, the Lipschitz constant depending only on ‖X‖∞. It suffices to remember that, on the compact manifold M, the 1-Wasserstein distance on probabilities is topologically equivalent to the wea... |

39 | On the optimal mapping of distributions - Knott, Smith - 1984 |

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31 |
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Citation Context ...ns and semi-concave functions It is certainly useful to recall the main properties of viscosity solutions in connection with semiconcave functions. We will not give proofs, and instead refer to [20], =-=[21]-=-, [14], as well as the appendix in [8]. We will consider the Hamilton-Jacobi equation ∂tu + H(x,∂xu,t) = 0. (HJ) The function u : M × [0,T] −→ M is called K-semi-concave if, for each chart θ ∈ Θ (see ... |

31 |
Cartesian currents in the Calculus of Variations
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Citation Context ...ield on M × [0,T], and if ν is a finite non-negative measure on M × [0,T], we define the current Z ∧ ν by ∫ Z ∧ ν(ω) := ω(Z)dν. M×[0,T] Every transport current can be written in this way, see [22] or =-=[25]-=-. As a consequence, currents extend as linear forms on the set Ω∞(M × [0,T]) of bounded measurable one-forms. If I is a Borel subset of the interval [0,T], it is therefore possible to define the restr... |

29 |
Existence and stability results in the L 1 theory of optimal transportation
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Citation Context ...which depends only on L, such that the measure m0 is supported on the graph of a K-Lipschitz vectorfield. Proof. Let us fix a probability measure µ 1 on M such that C 1 0 (µ1 ,µ 1 ) = α1. Let X : M × =-=[0,2]-=- −→ TM be a vectorfield associated to the transport problem C 2 0 (µ1 ,µ 1 ) by Theorem A. Note that X1 is Lipschitz on M with a Lipschitz constant K which does not depend on µ1. We choose X once and ... |

23 | Currents, flows and diffeomorphisms. Topology 14 - Ruelle, Sullivan - 1975 |

21 | Dynamics of pseudographs in convex Hamiltonian systems
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Citation Context ...tainly useful to recall the main properties of viscosity solutions in connection with semiconcave functions. We will not give proofs, and instead refer to [20], [21], [14], as well as the appendix in =-=[8]-=-. We will consider the Hamilton-Jacobi equation ∂tu + H(x,∂xu,t) = 0. (HJ) The function u : M × [0,T] −→ M is called K-semi-concave if, for each chart θ ∈ Θ (see appendix), the function (x,t) ↦−→ u(θ(... |

19 | On a problem of - Kantorovich - 1948 |

16 | Connecting orbits of time dependent Lagrangian systems - Bernard |

15 |
V.,Minimal measures and minimizing closed normal onecurrents, GAFA
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Citation Context ...µ0,µT ) A(m0) � min m∈M(µ0,µT) A(m) This formulation finds its roots on one hand in the works of Benamou and Brenier, see [6], and then Brenier, see [13], and on the other hand in the work of Bangert =-=[5]-=-. Let Ω 0 (M × [0,T]) be the set of continuous one-forms on M × [0,T], endowed with the uniform norm. We will often decompose forms ω ∈ Ω 0 (M × [0,T]) as ω = ω x + ω t dt, where ω x is a time-depende... |

15 | Abstract cyclical monotonicity and Monge solutions for the general MongeKantorovich problem - Levin - 1999 |

13 |
Extended Monge-Kantorovich theory, Optimal transportation and applications (Martina Franca
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Citation Context ...ld in view of Proposition 6. 2.2 Currents min m0∈I(µ0,µT ) A(m0) � min m∈M(µ0,µT) A(m) This formulation finds its roots on one hand in the works of Benamou and Brenier, see [6], and then Brenier, see =-=[13]-=-, and on the other hand in the work of Bangert [5]. Let Ω 0 (M × [0,T]) be the set of continuous one-forms on M × [0,T], endowed with the uniform norm. We will often decompose forms ω ∈ Ω 0 (M × [0,T]... |

9 | The Monge problem for supercritical Mañé potentials on compact manifolds
- Bernard, Buffoni
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Citation Context ...en on the set of curves γ : [0,1] −→ M satisfying γ(0) = x and γ(1) = y. Note that this class of costs does not contain the very natural cost c(x,y) = d(x,y). Such costs are studied in a second paper =-=[9]-=-. Our main result is that the optimal transports can be interpolated by measured Lipschitz laminations, or geometric currents in the sense of Ruelle and Sullivan. Interpolations of transport have alre... |

9 |
Weak KAM theorem in Lagrangian dynamics. Preliminary version
- Fathi
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Citation Context ... (x,t) ∈ M × [0,T], the function v ↦−→ L(x,v,t) is convex with positive definite Hessian at each point. superlinearity For each (x,t) ∈ M × [0,T], we have L(x,v,t)/‖v‖ −→ ∞ as ‖v‖ −→ ∞. Arguing as in =-=[20]-=- (Lemma 3.2.2), this implies that for all α > 0 there exists C > 0 such that L(x,v,t) � α‖v‖ − C for all (x,v,t) ∈ TM × [0,T]. completeness For each (x,v,t) ∈ TM × [0,T], there exists one and only one... |

8 | Duality and existence for a class of mass transportation problems and economic applications
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Citation Context ...elow is that this holds true in the case of the cost cT 0 . The method we use to prove this is an elaboration on ideas due to Brenier, see [12] and developed for instance in [24], (see also [23]) and =-=[16]-=-, which is certainly the closest to our needs. Theorem B. Assume that µ0 is absolutely continuous with respect to the Lebesgue class on M. Then for each final measure µT, there exists one and only one... |

7 | Minimal measures, one-dimensional currents and the Monge-Kantorovich problem
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Citation Context ...n, some of which overlap partly the present work. For example, at the moment of submitting the paper, we have been informed of the existence of the recent preprints of De Pascale, Gelli and Granieri, =-=[15]-=-, and of Granieri, [26]. We had also been aware of a manuscript by Wolansky [40] for a few weeks, which, independently, and by somewhat different methods, obtains results similar to ours. Note however... |

7 | Linear programming interpretations of Mather’s variational principle
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Citation Context ...s connections between Mather-Fathi theory, optimal mass transportation and HamiltonJacobi equations have recently been discussed, mainly at a formal level, in the literature, see for example [39], or =-=[19]-=-, where they are all presented as infinite dimensional linear programming problems. This have motivated a lot of activity around the interface between Aubry-Mather theory and optimal transportation, s... |

7 |
On action minimizing measures for the Monge-Kantorovitch problem, preprint
- Granieri
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Citation Context ...partly the present work. For exemple, 1at the moment of submitting the paper, we have been informed of the existence of the recent preprints of De Pascal, Stella and Granieri, [31], and of Granieri, =-=[22]-=-. We had also been aware of a preprint by Wolansky [35] for a few weeks, which, independently, and by somewhat different methods, studies questions very similar to the ones we are interested in. This ... |

5 | Optimal transportation in the presence of a prescribed pressure field
- Wolansky
(Show Context)
Citation Context ...f submitting the paper, we have been informed of the existence of the recent preprints of De Pascal, Stella and Granieri, [31], and of Granieri, [22]. We had also been aware of a preprint by Wolansky =-=[35]-=- for a few weeks, which, independently, and by somewhat different methods, studies questions very similar to the ones we are interested in. This paper emanates from the collaboration of the Authors du... |

4 | Munkres, Topology (second edition - R - 2000 |

4 |
Equivalence between some definitions for the optimal mass transport problem and for transport density on manifolds
- PRATELLI
(Show Context)
Citation Context ...r to ours. Note however that Lipschitz regularity, which we consider as one of our most important results, was not obtained in this preliminary version of [40]. It is worth also mentioning the papers =-=[36]-=- of Pratelli and [31] of Loeper. This paper emanates from the collaboration of the Authors during the end of the stay of the first author in EPFL for the academic year 2002-2003, granted by the Swiss ... |

2 |
A computational fluid dynamics solution to the Monge-Kantorovich mass transfer problem
- Benamou, Brenier
(Show Context)
Citation Context ...ualities C T 0 (µ0,µT) � hold in view of Proposition 6. 2.2 Currents min m0∈I(µ0,µT ) A(m0) � min m∈M(µ0,µT) A(m) This formulation finds its roots on one hand in the works of Benamou and Brenier, see =-=[6]-=-, and then Brenier, see [13], and on the other hand in the work of Bangert [5]. Let Ω 0 (M × [0,T]) be the set of continuous one-forms on M × [0,T], endowed with the uniform norm. We will often decomp... |

2 | The reconstruction problem for the Euler-Poisson system in cosmology
- Loeper
(Show Context)
Citation Context ...er that Lipschitz regularity, which we consider as one of our most important results, was not obtained in this preliminary version of [40]. It is worth also mentioning the papers [36] of Pratelli and =-=[31]-=- of Loeper. This paper emanates from the collaboration of the Authors during the end of the stay of the first author in EPFL for the academic year 2002-2003, granted by the Swiss National Science Foun... |

1 |
Lecture notes on tranport equation and Cauchy problem for BV vector fields and applications
- Ambrosio
(Show Context)
Citation Context ...t, and µt is the continuous family of probability measures such that µχ = µt ⊗ dt. This statement follows from standard representation results for solutions of the transport equation, see for example =-=[3]-=- or [4]. Transport current induced from a transport measure. To a transport measure m, we associate the transport current χm defined by ∫ ( ) x t χm(ω) = ω (x,t) · v + ω (x,t) dm(x,v,t) TM×[0,T] where... |

1 |
habilitation thesis
- Gangbo
(Show Context)
Citation Context ...e result below is that this holds true in the case of the cost cT 0 . The method we use to prove this is an elaboration on ideas due to Brenier, see [12] and developed for instance in [24], (see also =-=[23]-=-) and [16], which is certainly the closest to our needs. Theorem B. Assume that µ0 is absolutely continuous with respect to the Lebesgue class on M. Then for each final measure µT, there exists one an... |

1 | The Monge transport problem in Lagrangian dynamics, preprint - Bernard, Buffoni |

1 |
Convergence of Probability Measures–2nd ed., Wiley-Interscience publication
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- 1999
(Show Context)
Citation Context ... satisfy A T 0 (m0) � a, are compact. As a consequence, there exist optimal initial transport measures, and optimal transport measures. Proof. This is an easy application of the Prohorov theorem, see =-=[8]-=-. Now we have seen that the problem of finding optimal transport measures is well-posed, let us describe its solutions. Theorem 1. We have The mapping C T 0 (µ0,µT) = min A(m) = m∈M(µ0,µT ) min ∫ T m0... |