# American Institute of Mathematical Sciences

March  2015, 14(2): 517-525. doi: 10.3934/cpaa.2015.14.517

## A note on the Monge-Kantorovich problem in the plane

 1 Department of Applied Mathematics, The Hong Kong Polytechnic University, Kowloon, Hong Kong, China 2 Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, China

Received  March 2014 Revised  September 2014 Published  December 2014

The Monge-Kantorovich mass-transportation problem has been shown to be fundamental for various basic problems in analysis and geometry in recent years. Shen and Zheng propose a probability method to transform the celebrated Monge-Kantorovich problem in a bounded region of the Euclidean plane into a Dirichlet boundary problem associated to a nonlinear elliptic equation. Their results are original and sound, however, their arguments leading to the main results are skipped and difficult to follow. In the present paper, we adopt a different approach and give a short and easy-followed detailed proof for their main results.
Citation: Zuo Quan Xu, Jia-An Yan. A note on the Monge-Kantorovich problem in the plane. Communications on Pure & Applied Analysis, 2015, 14 (2) : 517-525. doi: 10.3934/cpaa.2015.14.517
##### References:
 [1] L. Kantorovich, On the translocation of masses,, \emph{C. R. (Doklady) Acad. Sci. URSS (N. S.)}, 37 (1942), 199.   Google Scholar [2] G. Monge, Mémoire sur la théorie des déblais et des remblais,, \emph{Histoire de l'Acad\'emie Royale des Sciences de Paris, (1781), 666.   Google Scholar [3] S. T. Rachev and L. Rüschendorf, Mass Transportation Problems, Volume I: Theory (Probability and its Applications),, Springer-Verlag, (1998).   Google Scholar [4] S. T. Rachev and L. Rüschendorf, Mass Transportation Problems, Volume II: Applications (Probability and its Applications),, Springer-Verlag, (1998).   Google Scholar [5] Y. F. Shen and W. A. Zheng, On Monge-Kantorovich problem in the plane,, \emph{C. R. Acad. Sci. Paris, 348 (2010), 267.  doi: 10.1016/j.crma.2009.11.022.  Google Scholar

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##### References:
 [1] L. Kantorovich, On the translocation of masses,, \emph{C. R. (Doklady) Acad. Sci. URSS (N. S.)}, 37 (1942), 199.   Google Scholar [2] G. Monge, Mémoire sur la théorie des déblais et des remblais,, \emph{Histoire de l'Acad\'emie Royale des Sciences de Paris, (1781), 666.   Google Scholar [3] S. T. Rachev and L. Rüschendorf, Mass Transportation Problems, Volume I: Theory (Probability and its Applications),, Springer-Verlag, (1998).   Google Scholar [4] S. T. Rachev and L. Rüschendorf, Mass Transportation Problems, Volume II: Applications (Probability and its Applications),, Springer-Verlag, (1998).   Google Scholar [5] Y. F. Shen and W. A. Zheng, On Monge-Kantorovich problem in the plane,, \emph{C. R. Acad. Sci. Paris, 348 (2010), 267.  doi: 10.1016/j.crma.2009.11.022.  Google Scholar
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