The Optimal Partial Transport Problem

Given two densities f and g, we consider the problem of transporting a fraction $${m \in [0,\min\{\|f\|_{L^1},\|g\|_{L^1}\}]}$$ 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 $${m \in [\|f\wedge g\|_{L^1},\min\{\|f\|_{L^1},\|g\|_{L^1}\}]}$$ . This extends the result of Caffarelli and McCann in Ann Math (in print), 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 $${{\Omega,\Lambda \subset \mathbb {R}^n}}$$ , and bounded away from zero and infinity on their respective supports, $${C^{0,\alpha}_{\rm loc}}$$ regularity of the optimal transport map and local C1 regularity of the free boundaries away from $${{\Omega\cap \Lambda}}$$ are shown. Finally, the optimal transport map extends to a global homeomorphism between the active regions.