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

Presented by: 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.

Abstract (Original language)
Résumé

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.

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.

Received: 2008-08-19
Accepted: 2009-11-02
Published online: 2010-02-20

DOI: 10.1016/j.crma.2009.11.022

Author's affiliations:

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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     title = {On {Monge{\textendash}Kantorovich} problem in the plane},
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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/

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