An optimal matching problem

Abstract

Given two measured spaces (X,dx), (Y,dy) and a third space Z, given two functions u(x,z) and v(x,z), we study the problem of finding two maps s from X to Z and t from Y to Z such that the images s(dx) and t(dy) coincide, and the integral of u(x,s(x))dx+v(y,t(y))dy is maximal. We give condition on u and v for which there is a unique solution.