On Monge–Kantorovich problem in the plane

Yinfang Shen; Weian Zheng

doi:10.1016/j.crma.2009.11.022

Partial Differential Equations/Probability Theory

On Monge–Kantorovich problem in the plane
[Sur le problème de Monge–Kantorovich du plan]

Présenté par : Marc Yor

Yinfang Shen ^1,² ; Weian Zheng ^1,²

¹ SFS, ITCS, East China Normal University, Shanghai, China, 200062
² Department of Mathematics, University of California, Irvine, CA 92697, USA

Comptes Rendus. Mathématique, Volume 348 (2010) no. 5-6, pp. 267-271.

Résumé
Abstract

Nous utilisons une méthode probabiliste pour transformer le célèbre problème de Monge–Kantorovich dans une région bornée du plan Euclidien à celui de Dirichlet associé à une équation aux dérivées partielles quasi-linéaire :

\frac{\partial}{\partial x} A (x, F_{x}^{'}) + \frac{\partial}{\partial y} B (y, F_{y}^{'}) = 0

où

A_{y}^{'} (., .) > 0, B_{x}^{'} (., .) > 0

, et F est une loi de probabilité inconnue. Ainsi, nous avons développé une nouvelle méthode probabiliste pour l'équation de Monge–Ampère associé au problème ci-dessus.

We use a simple probability method to transform the celebrated Monge–Kantorovich problem in a bounded region of Euclidean plane into a Dirichlet boundary problem associated to a quasi-linear elliptic equation with 0-order term missing in its diffusion coefficients:

\frac{\partial}{\partial x} A (x, F_{x}^{'}) + \frac{\partial}{\partial y} B (y, F_{y}^{'}) = 0

where

A_{y}^{'} (., .) > 0, B_{x}^{'} (., .) > 0

and F is an unknown probability distribution function. Thus, we are able to give a probability approach to the famous Monge–Ampère equation, which is known to be associated to the above problem.

Reçu le : 2008-08-19
Accepté le : 2009-11-02
Publié le : 2010-02-20

DOI : 10.1016/j.crma.2009.11.022

Affiliations des auteurs :

Yinfang Shen ^{1, 2} ; Weian Zheng ^{1, 2}

¹ SFS, ITCS, East China Normal University, Shanghai, China, 200062
² Department of Mathematics, University of California, Irvine, CA 92697, USA

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     author = {Yinfang Shen and Weian Zheng},
     title = {On {Monge{\textendash}Kantorovich} problem in the plane},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {267--271},
     publisher = {Elsevier},
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     number = {5-6},
     year = {2010},
     doi = {10.1016/j.crma.2009.11.022},
     language = {en},
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Yinfang Shen; Weian Zheng. On Monge–Kantorovich problem in the plane. Comptes Rendus. Mathématique, Volume 348 (2010) no. 5-6, pp. 267-271. doi : 10.1016/j.crma.2009.11.022. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2009.11.022/

Bibliographie
Cité par

[1] L. Ambrosio, Optimal transport maps in Monge–Kantorovich problem, in: Proceedings of ICM, Beijing 2002, vol. iii, 2002, pp. 131–140

[2] Y. Brenier Décomposition polaire et réarrangement monotone des champs de vecteurs, C. R. Acad. Sci. Paris, Ser. I Math., Volume 305 (1987), pp. 805-808

[3] Y. Brenier Polar factorization and monotone rearrangement of vector-valued functions, Comm. Pure Appl. Math., Volume 44 (1991), pp. 375-417

[4] L. Caffarelli, Nonlinear elliptic theory and the Monge–Ampere equation, in: Proc. ICM, vol. i, 2002, pp. 179–187

[5] L.C. Evans; W. Gangbo Differential equations methods for the Monge–Kantorovich mass transfer problem, Mem. Amer. Math. Soc., Volume 137 (1999) no. 653

[6] D. Gilbarg; N.S. Trudinger Elliptic Partial Differential Equations of Second Order, Springer-Verlag, 1983

[7] S.T. Rachev; L. Rüschendorf; S.T. Rachev; L. Rüschendorf Mass Transportation Problems, vol I: Theory, Probability and Its ApplicationsMass Transportation Problems, vol. II: Applications, Probability and Its Applications, Springer-Verlag, 1998

[8] N. Trudinger, Recent developments in elliptic partial differential equations of Monge–Ampère type, in: Proc. ICM, 2006, invited lecture

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