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 $μ$ and $ν$ (with respect to the volume measures of $M$ and $N$). We consider the optimal transportation from $μ$ to $ν$ for the quadratic cost. Let $(ϕ:m \to \mathbb{R},ψ:N \to \mathbb{R})$ be some functions which achieve the supremum in the Kantorovich formulation of the problem and which satisfy $ψ(y) = \inf_{z\in M} \Bigl( \frac{1}{2}|y-z|^2 -φ(z)\Bigr); φ(x)=\inf_{z\in N} \Bigl( \frac{1}{2}|x-z|^2 -ψ(z)\Bigr).$ Define for $y \in N$, $φ^\Box(y) = \sup_{z\in M} \Bigl( \frac{1}{2}|y-z|^2 -φ(z)\Bigr).$ In this short paper, we exhibit a relationship between the regularity of $φ^\Box$ and the existence of a solution to the Monge problem.

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