A note on the Monge-Kantorovich problem  in the plane

Zuo Quan Xu; Jia-An Yan

doi:10.3934/cpaa.2015.14.517

Article Contents

2015, Volume 14, Issue 2: 517-525. Doi: 10.3934/cpaa.2015.14.517

This issue Previous Article Sharp existence criteria for positive solutions of Hardy--Sobolev type systems Next Article An integral equation involving Bessel potentials on half space

A note on the Monge-Kantorovich problem in the plane

Zuo Quan Xu^1, and
Jia-An Yan^2,

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

Received: February 28, 2014

Revised: August 31, 2014

Published: February 2015

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Abstract

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.

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

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References

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