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arXiv · 1805.11187

Optimal transportation between unequal dimensions

Robert J McCann·Brendan Pass
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Abstract

We establish that solving an optimal transportation problem in which the source and target densities are defined on manifolds with different dimensions, is equivalent to solving a new nonlocal analog of the Monge-Amp\`ere equation, introduced here for the first time. Under suitable topological conditions, we also establish that solutions are smooth if and only if a local variant of the same equation admits a smooth and uniformly elliptic solution. We show that this local equation is elliptic, and $C^{2,\alpha}$ solutions can therefore be bootstrapped to obtain higher regularity results, assuming smoothness of the corresponding differential operator, which we prove under simplifying assumptions. For one-dimensional targets, our sufficient criteria for regularity of solutions to the resulting ODE are considerably less restrictive than those required by earlier works.

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Robert J McCann, Brendan Pass. 2018-05-28. Optimal transportation between unequal dimensions. https://arxiv.org/abs/1805.11187

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