  This paper mainly addresses the Monge mass transfer problem in the 1-D case.
Through an ingenious approximation mechanism, one transforms the Monge problem
into a sequence of minimization problems, which can be converted into a
sequence of nonlinear differential equations with constraints by variational
method. The existence and uniqueness of the solution for each equation can be
demonstrated by applying the canonical duality method. Moreover, the duality
method gives a sequence of perfect dual maximization problems. In the final
analysis, one constructs the approximation of optimal mapping for the Monge
problem according to the theoretical results.
