Optimal Partial Mass Transportation and Obstacle Monge–Kantorovich Equation

Optimal partial mass transport, which is a variant of the optimal transport problem , consists in transporting effectively a prescribed amount of mass from a source to a target. The problem was first studied by Caffarelli and McCann [12, Ann. of Math., 2010] and Figalli [21, Arch. Ration. Mech. Anal., 2010] with a particular attention to the quadratic cost. Our aim here is to study the optimal partial mass transport problem with Finsler distance costs including the Monge cost given by the Euclidian distance. Our approach is different and our results do not follow from previous works. Among our results, we introduce a PDE of Monge–Kantorovich type with a double obstacle to characterize optimal active submeasures, Kantorovich potential and optimal flow for the optimal partial transport problem. This new PDE enables us to study the uniqueness and monotonicity results for the optimal active sub-measures. Another interesting issue of our approach is its convenience for numerical analysis and computations that we develop in a separate paper [23, IMA J. Numer. Anal., 2017]