Free boundaries in optimal transport and Monge-Ampère obstacle problems | Annals of Mathematics

Abstract Given compactly supported 0≤f,g∈L1(ℝn), 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|2/2. 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 sptg in the quadratic case. The part of f to be transported increases monotonically with m, and if sptf and sptg 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=fχΩ and g=gχΛ are bounded away from zero and infinity on separated strictly convex domains Ω,Λ⊂Rn, for the quadratic cost this graph is shown to be a C1,αloc 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.

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数学年刊(Annals of Mathematics)

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