Free boundaries in optimal transport and Monge-Ampère obstacle problems

Luis Caffarelli, Robert McCann
2010 Annals of Mathematics  
Given compactly supported 0 ≤ f, g ∈ L 1 (R n ), the problem of transporting a fraction m ≤ min { f L 1 , g L 1 } 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 /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
more » ... 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 = f χ Ω and g = gχ Λ are bounded away from zero and infinity on separated strictly convex domains Ω, Λ ⊂ R n , for the quadratic cost this graph is shown to be a C 1,α 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.
doi:10.4007/annals.2010.171.673 fatcat:zsozcaebfngefkn4zkopeem6r4