A note on Monge–Kantorovich problem

Shen and Zheng (2010) and Xu and Yan (2013) considered the Monge–Kantorovich problem in the plane and proved that the optimal coupling for the problem has a form (X1,g(X1,Y2)X1,g(X1,Y2), h(X1,Y2),Y2h(X1,Y2),Y2), and then they assumed (X1,Y2)(X1,Y2) has a density pp and gave the equation which pp should satisfy. In this article, we prove that (X1,Y2)(X1,Y2) naturally has a density under more weak conditions. We again prove a similar result in dimension 3 and give an exact form (X1,g1(X1,Y2,Y3)X1,g1(X1,Y2,Y3), g2(X1,Y2,Y3),h(X1,Y2,Y3),Y2,Y3g2(X1,Y2,Y3),h(X1,Y2,Y3),Y2,Y3) depending on a certain convex function.