2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/157136We 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: \begin{eqnarray*} A(x, F'_x)F''_{xx}+B(y, F'_y)F''_{yy}&=&C(x, y, F'_x, F'_y) \end{eqnarray*} 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.ProbabilityAnalysis of PDEs35J15, 35J20, 60A10text