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Pré-Publication, Document De Travail Année : 2015

Optimal transportation between hypersurfaces bounding some strictly convex domains

Résumé

Let $M,N$ be two smooth compact hypersurfaces of $\R^n$ 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 $(\phi:m \to \R,\psi:N \to \R)$ be some functions which achieve the supremum in the Kantorovich formulation of the problem and which satisfy $$ \psi (y) = \inf_{z\in M} \Bigl( \frac{1}{2}|y-z|^2 -\varphi(z)\Bigr); \\ \varphi (x)=\inf_{z\in N} \Bigl( \frac{1}{2}|x-z|^2 -\psi(z)\Bigr).$$ Define for $y \in N$, $$\varphi^\Box(y) = \sup_{z\in M} \Bigl( \frac{1}{2}|y-z|^2 -\varphi(z)\Bigr).$$ In this short paper, we exhibit a relationship between the regularity of $\varphi^\Box$ and the existence of a solution to the Monge problem.
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Dates et versions

hal-01271012 , version 1 (11-02-2016)

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Emmanuel Humbert, Luc Molinet. Optimal transportation between hypersurfaces bounding some strictly convex domains. 2015. ⟨hal-01271012⟩
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