Optimal transportation in a discrete setting

P. -E. Jabin, A. Mellet, M. Molina

Published 2019-11-01Version 1

We investigate the regularity properties of Kantorovich potentials for a Monge-Kantorovich problem with quadratic cost between non-absolutely continuous measures. For each measure, we introduce a discrete scale so that the measure behaves as an absolutely continuous measure up to that scale. Our main theorem then proves that the Kantorovich potential cannot exhibit any flat part at a scale larger than the corresponding discrete scales on the measures. This, in turn, implies a $C^1$ regularity result up to the discrete scale. The proof relies on novel explicit estimates directly based on the optimal transport problem, instead of the Monge-Amp\`ere equation.