Optimal transportation between hypersurfaces bounding some strictly convex domains

Let $M,N$ be two smooth compact hypersurfaces of ${\mathbb{R}}^n$ which bound strictly convex domains equipped with two absolutely continuous measures $\mu$ and $\nu$ (with respect to the volume measures of $M$ and $N$). We consider the optimal transportation from $\mu$ to $\nu$ for the quadratic cost. Let $(\phi:m \to $\mathbb{R}$,\psi:N \to $\mathbb{R}$)$ be some functions which achieve the supremum in the Kantorovich formulation of the problem and which satisfy $$\psi (y)=\underset{z\in M}{\inf }(\frac{1}{2}\mathrm{\mid }y-z{\mathrm{\mid }}^2-\phi (z));\phi (x)=\underset{z\in N}{\inf }(\frac{1}{2}\mathrm{\mid }x-z{\mathrm{\mid }}^2-\psi (z))\mathrm{.}$$ Define for $y\in N$, $${\phi }^{\mathrm{\square }}(y)=\underset{z\in M}{\sup }(\frac{1}{2}\mathrm{\mid }y-z{\mathrm{\mid }}^2-\phi (z))\mathrm{.}$$ In this short paper, we exhibit a relationship between the regularity of ${\phi }^{\mathrm{\square }}$ and the existence of a solution to the Monge problem.