Optimal transportation between unequal dimensions

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.