The optimal partial transport problem

Given two densities f and g, we consider the problem of transporting a fraction m ∈ [0, min{‖f‖L1 , ‖g‖L1}] of the mass of f onto g minimizing a transportation cost. If the cost per unit of mass is given by |x − y|2, we will see that uniqueness of solutions holds for m ∈ [‖f ∧ g‖L1 , min{‖f‖L1 , ‖g‖L1}]. This extends the result of Caffarelli and McCann in [8], where the authors consider two densities with disjoint supports. The free boundaries of the active regions are shown to be (n− 1)-rectifiable (provided the supports of f and g have Lipschitz boundaries), and under some weak regularity assumptions on the geometry of the supports they are also locally semiconvex. Moreover, assuming f and g supported on two bounded strictly convex sets Ω, Λ ⊂ R, and bounded away from zero and infinity on their respective supports, C loc regularity of the optimal transport map and local C 1 regularity of the free boundaries away from Ω∩Λ are shown. Finally, the optimal transport map extends to a global homeomorphism between the active regions.