On Monge-Kantorovich Problem in the Plane

TL;DR

This paper reformulates the Monge-Kantorovich optimal transport problem in the Euclidean plane as a boundary value problem for a quasi-linear elliptic PDE, linking optimal transport to PDE theory.

Contribution

It introduces a novel PDE formulation of the Monge-Kantorovich problem in the plane, connecting optimal transport with elliptic boundary value problems.

Findings

Establishes a PDE representation of the Monge-Kantorovich problem.

Provides conditions on the coefficients for the PDE formulation.

Links optimal transport solutions to elliptic PDE solutions.

Abstract

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: \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.