Free boundaries in optimal transport and Monge-Ampère obstacle problems

Given compactly supported 0 f; g 2 L 1 . n /, the problem of transporting a fraction m minfkf k L 1 ; kgk L 1 g 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/ D jx yj 2 =2. This question is shown to be equivalent to a double obstacle problem for the Monge-Ampre 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 be separated from the balance of f by a semiconcave Lipschitz graph over the hyperplane. If f D f and g D g are bounded away from zero and infinity on separated strictly convex domains ; R n , for the quadratic cost this graph is shown to be a C 1;l oc hypersurface in whose normal coincides with the direction transported; the optimal map between f and g is shown to be Hlder continuous up to this free boundary, and to those parts of the fixed boundary @ which map to locally convex parts of the path-connected target region.