On Monge - Kantorovich Problem in the Plane

We transfer the celebrating Monge-Kontorovich problem in a bounded domain of Euclidean plane into a Dirichlet boundary problem associated to a quasi-linear elliptic equation with 0−order term missing in its diffusion coefficients: A(x, F ′ x )F ′′ xx +B(y, F ′ y )F ′′ yy = C(x, y, F ′ x , F ′ y ) where A(., .) > 0, B(., .) > 0 and C are functions based on the initial distributions, F is an unknown probability distribution function and therefore closed the former problem. The mass transport problem was first formulated by Monge in 1781, and concerned finding the optimal way, in the sense of minimal transportation cost of moving a pile of soil from one site to another. This problem was given a modern formulation in the work of Kantorovitch and so is now known as the Monge-Kontorovich problem. This type of problem has appeared in economics, automatic control, transportation, fluid dynamics, statistical physics, shape optimization, expert system, meteorology and financial mathematics. For example, for the general tracking problem, a robust and reliable object and shape recognition system is of major importance. A key way to carry this out is via template matching. which is the matching of some some object to another within a given catalogue of objects. Typically, the match will not be exact and hence some criterion is necessary to measure the “goodness of fit”. Many mathematicians from different fields are interested in Monge-Kontorovich problem. This classical problem was revived in the mid eighties by the work of Y.Brenier([6], [7]), who characterized the optimal transfer plans in terms of gradients of convex functions. In the last decades, this problem has been recovered to ∗Research partially supported N.S.F.C. Grant 10771070