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A note on the Monge-Kantorovich problem in the plane

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  • 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.
    Mathematics Subject Classification: Primary: 35J62; Secondary: 49K20.


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    L. Kantorovich, On the translocation of masses, C. R. (Doklady) Acad. Sci. URSS (N. S.), 37 (1942), 199-201.


    G. Monge, Mémoire sur la théorie des déblais et des remblais, Histoire de l'Académie Royale des Sciences de Paris, avec les Mémoires de Mathématique et de Physique pour la même année, (1781), 666-704.


    S. T. Rachev and L. Rüschendorf, Mass Transportation Problems, Volume I: Theory (Probability and its Applications), Springer-Verlag, 1998.


    S. T. Rachev and L. Rüschendorf, Mass Transportation Problems, Volume II: Applications (Probability and its Applications), Springer-Verlag, 1998.


    Y. F. Shen and W. A. Zheng, On Monge-Kantorovich problem in the plane, C. R. Acad. Sci. Paris, Ser. I, 348 (2010), 267-271.doi: 10.1016/j.crma.2009.11.022.

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