Free boundaries in optimal transport and Monge-Ampère obstacle problems

Given compactly supported 0 ≤ f, g ∈ L1(R), the problem of transporting a fraction m ≤ min{‖f‖L1 , ‖g‖L1} of the mass of f onto g as cheaply as possible is considered, where cost per unit mass transported is given by a cost function c, typically quadratic c(x,y) = |x − y|/2. This question is shown to be equivalent to a double obstacle problem for the Monge-Ampère equation, for which sufficient conditions are given to guarantee uniqueness of the solution, such as f vanishing on spt g in the quadratic case. The part of f to be transported increases monotonically with m, and if spt f and spt g are separated by a hyperplane H, then this part will separated from the balance of f by a semiconcave Lipschitz graph over the hyperplane. If f = fχΩ and g = gχΛ are bounded away from zero and infinity on separated strictly convex domains Ω,Λ ⊂ R, for the quadratic cost this graph is shown to be a C loc hypersurface in Ω whose normal coincides with the direction transported; the optimal map between f and g is shown to be Hölder continuous up to this free boundary, and to those parts of the fixed boundary ∂Ω which map to locally convex parts of the path-connected target region.