A Note on the Monge-Kantorovich Problem in the Plane

TL;DR

This paper revisits the Monge-Kantorovich mass-transportation problem in the plane, providing a clearer, detailed proof of a probabilistic approach that transforms it into a nonlinear elliptic boundary value problem.

Contribution

It offers a simplified, detailed proof of Shen and Zheng's probabilistic method for the Monge-Kantorovich problem, making their original results more accessible.

Findings

Validated the probabilistic transformation approach

Provided a detailed, easy-to-follow proof

Confirmed the original results of Shen and Zheng

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 (2010) proposed 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.