  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.
