Results 1  10
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35
Optimal transportation on noncompact manifolds
"... In this work, we show how to obtain for noncompact 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 distan ..."
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Cited by 22 (6 self)
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In this work, we show how to obtain for noncompact 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. 1
Optimal transport and Perelman’s reduced volume
, 2008
"... We show that a certain entropylike function is convex, under an optimal transport problem that is adapted to Ricci flow. We use this to reprove the monotonicity of Perelman’s reduced volume. ..."
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Cited by 15 (0 self)
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We show that a certain entropylike function is convex, under an optimal transport problem that is adapted to Ricci flow. We use this to reprove the monotonicity of Perelman’s reduced volume.
The Monge Problem for supercritical Mañé Potentials on compact manifolds
 ADVANCES IN MATHEMATICS, 207 N
, 2006
"... We prove the existence of optimal transport maps for the Monge problem when the cost is a Finsler distance on a compact manifold. Our point of view consists in considering the distance as a Mañé potential, and to rely on recent developments in the theory of viscosity solutions of the HamiltonJacobi ..."
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Cited by 10 (1 self)
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We prove the existence of optimal transport maps for the Monge problem when the cost is a Finsler distance on a compact manifold. Our point of view consists in considering the distance as a Mañé potential, and to rely on recent developments in the theory of viscosity solutions of the HamiltonJacobi equation.
A.Figalli: Geodesics in the space of measurepreserving maps and plans
"... We study Brenier’s variational models for incompressible Euler equations. These models give rise to a relaxation of the Arnold distance in the space of measurepreserving maps and, more generally, measurepreserving plans. We analyze the properties of the relaxed distance, we show a close link betwe ..."
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Cited by 6 (2 self)
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We study Brenier’s variational models for incompressible Euler equations. These models give rise to a relaxation of the Arnold distance in the space of measurepreserving maps and, more generally, measurepreserving plans. We analyze the properties of the relaxed distance, we show a close link between the Lagrangian and the Eulerian model, and we derive necessary and sufficient optimality conditions for minimizers. These conditions take into account a modified Lagrangian induced by the pressure field. Moreover, adapting some ideas of Shnirelman, we show that, even for nondeterministic final conditions, generalized flows can be approximated in energy by flows associated to measurepreserving maps. 1
Minimal measures, onedimensional currents and the MongeKantorovich problem
 Calc. Var
"... Abstract. In recent works L.C. Evans has noticed a strong analogy between Mather’s theory of minimal measures in Lagrangian dynamic and the theory developed in the last years for the optimal mass transportation (or MongeKantorovich) problem. In this paper we start to investigate this analogy by pro ..."
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Cited by 6 (2 self)
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Abstract. In recent works L.C. Evans has noticed a strong analogy between Mather’s theory of minimal measures in Lagrangian dynamic and the theory developed in the last years for the optimal mass transportation (or MongeKantorovich) problem. In this paper we start to investigate this analogy by proving that to each minimal measure it is possible to associate, in a natural way, a family of curves on the space of probability measures. These curves are absolutely continuous with respect to the metric structure related to the optimal mass transportation problem. Some minimality properties of such curves are also addressed. Keywords. Mather’s minimal measures, MongeKantorovich problem, optimal transport problems, normal 1currents. MSC 2000. 37J50, 49Q20, 49Q15. 1.
Optimal transport under nonholonomic constraints
 Transactions of the Amer. Math. Soc
"... Abstract. We study Monge’s optimal transportation problem, where the cost is given by optimal control cost. We prove the existence and uniqueness of an optimal map under certain regularity conditions on the Lagrangian, absolute continuity of the measures with respect to Lebesgue, and most importantl ..."
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Cited by 5 (1 self)
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Abstract. We study Monge’s optimal transportation problem, where the cost is given by optimal control cost. We prove the existence and uniqueness of an optimal map under certain regularity conditions on the Lagrangian, absolute continuity of the measures with respect to Lebesgue, and most importantly the absence of sharp abnormal minimizers. In particular, this result is applicable in the case of subriemannian manifolds with a 2generating distribution and cost given by d 2, where d is the subriemannian distance. Also, we discuss some properties of the optimal plan when abnormal minimizers are present. Finally, we consider some examples of displacement interpolation in the case of Grushin plane. 1.
Weak KAM Pairs and MongeKantorovich Duality Advanced studies in pure math, asymptotic analysis and singularity
"... The dynamics of globally minimizing orbits of Lagrangian systems can be studied using the Barrier function, as Mather first did, or using the pairs of weak KAM solutions introduced by Fathi. The central observation of the present paper is that Fathi weak KAM pairs are precisely the admissible pairs ..."
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Cited by 5 (0 self)
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The dynamics of globally minimizing orbits of Lagrangian systems can be studied using the Barrier function, as Mather first did, or using the pairs of weak KAM solutions introduced by Fathi. The central observation of the present paper is that Fathi weak KAM pairs are precisely the admissible pairs for the Kantorovich problem dual to the Monge transportation problem with the Barrier function as cost. We exploit this observation to recover several relations between the Barrier functions and the set of weak KAM pairs in an axiomatic and elementary way. Let M be a compact connected manifold and consider a C 2 Lagrangian function L: TM × R → R that satisfies the standard hypotheses of the calculus of variations,
FIVE LECTURES ON OPTIMAL TRANSPORTATION: GEOMETRY, REGULARITY AND APPLICATIONS
"... Abstract. In this series of lectures we introduce the MongeKantorovich problem of optimally transporting one distribution of mass onto another, where optimality is measured against a cost function c(x, y). Connections to geometry, inequalities, and partial differential equations will be discussed, ..."
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Cited by 5 (2 self)
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Abstract. In this series of lectures we introduce the MongeKantorovich problem of optimally transporting one distribution of mass onto another, where optimality is measured against a cost function c(x, y). Connections to geometry, inequalities, and partial differential equations will be discussed, focusing in particular on recent developments in the regularity theory for MongeAmpère type equations. An application to microeconomics will also be described, which amounts to finding the equilibrium price distribution for a monopolist marketing a multidimensional line of products to a population of anonymous agents whose preferences are known only statistically.
Mass Transportation on SubRiemannian Manifolds
"... We study the optimal transport problem in subRiemannian manifolds where the cost function is given by the square of the subRiemannian distance. Under appropriate assumptions, we generalize BrenierMcCann’s Theorem proving existence and uniqueness of the optimal transport map. We show the absolute ..."
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Cited by 5 (0 self)
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We study the optimal transport problem in subRiemannian manifolds where the cost function is given by the square of the subRiemannian distance. Under appropriate assumptions, we generalize BrenierMcCann’s Theorem proving existence and uniqueness of the optimal transport map. We show the absolute continuity property of Wassertein geodesics, and we address the regularity issue of the optimal map. In particular, we are able to show its approximate differentiability a.e. in the Heisenberg group (and under some weak assumptions on the measures the differentiability a.e.), which allows to write a weak form of the MongeAmpère equation. 1