Partial Differential Equations/Probability Theory
On Monge–Kantorovich problem in the plane
Comptes Rendus. Mathématique, Volume 348 (2010) no. 5-6, pp. 267-271.

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:

 $∂∂xA(x,Fx′)+∂∂yB(y,Fy′)=0$
where $Ay′(.,.)>0,Bx′(.,.)>0$ 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 :

 $∂∂xA(x,Fx′)+∂∂yB(y,Fy′)=0$
$Ay′(.,.)>0,Bx′(.,.)>0$, 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.

Accepted:
Published online:
DOI: 10.1016/j.crma.2009.11.022

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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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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