Free boundaries in optimal transport and Monge-Ampere obstacle problems

• 成果类型: Article
• 署名作者: Caffarelli, Luis A.; McCann, Robert J.
• 刊物名称: ANNALS OF MATHEMATICS
• ISSN/ISSBN: 0003-486X
• DOI: 10.4007/annals.2010.171.673
• 发表日期: 2010
• 页码: 673-730
• 关键词: polar factorization; optimal maps; mass transportation; REGULARITY; EXISTENCE; EQUATIONS; rearrangement; uniqueness

摘要

Given compactly supported 0 <= f; g is an element of L-1 (R-n), the problem of transporting a fraction m <= min{parallel to f parallel to(L)1, parallel to g parallel to (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) = vertical bar x - y vertical bar(2)/2. This question is shown to be equivalent to a double obstacle problem for the Monge-Ampere 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 = f X Omega and g = gX Lambda are bounded away from zero and infinity on separated strictly convex domains Omega, Lambda subset of R-n, for the quadratic cost this graph is shown to be a C-lox(1,alpha) hypersurface in Omega whose normal coincides with the direction transported; the optimal map between f and g is shown to be Holder continuous up to this free boundary, and to those parts of the fixed boundary partial derivative Omega which map to locally convex parts of the path-connected target region.