A note on the Monge-Kantorovich problem in the plane

Zuoquan Xu, Jia An Yan

Research output: Journal article publicationJournal articleAcademic researchpeer-review


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.
Original languageEnglish
Pages (from-to)517-525
Number of pages9
JournalCommunications on Pure and Applied Analysis
Issue number2
Publication statusPublished - 1 Jan 2015


  • Calculus of variations
  • Comonotonic random variable
  • Dirichlet boundary problem
  • Monge-Kantorovich problem
  • Transportation problem

ASJC Scopus subject areas

  • Analysis
  • Applied Mathematics


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