Optimal Transportation Between Unequal Dimensions
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Optimal Transportation Between Unequal Dimensions Robert J. McCann & Brendan Pass Communicated by A. Figalli
Abstract We establish that solving an optimal transportation problem in which the source and target densities are defined on spaces with different dimensions, is equivalent to solving a new nonlocal analog of the Monge–Ampère equation, introduced here for the first time. Under suitable topological conditions, we also establish that solutions are smooth if and only if a local variant of the same equation admits a smooth and uniformly elliptic solution. We show that this local equation is elliptic, and C 2,α solutions can therefore be bootstrapped to obtain higher regularity results, assuming smoothness of the corresponding differential operator, which we prove under simplifying assumptions. For one-dimensional targets, our sufficient criteria for regularity of solutions to the resulting ODE are considerably less restrictive than those required by earlier works. 1. Introduction Since the 1980s [13,26,36] and the celebrated work of Brenier [2,3], it has been well-understood [32] that for the quadratic cost c(x, y) = 21 |x − y|2 on Rn , solving The authors are grateful to Toronto’s Fields’ Institute for the Mathematical Sciences for its kind hospitality during part of this work, and to two anonymous referees, for their careful reading and crucial comments. RJM acknowledges partial support of this research by Natural Sciences and Engineering Research Council of Canada Grant 217006-15, by a Simons Foundation Fellowship, and by the US National Science Foundation under Grant No. DMS-14401140 while in residence at the Mathematical Sciences Research Institute in Berkeley CA during January and February of 2016. He thanks S.-Y. Alice Chang for a stimulating conversation. BP is pleased to acknowledge support from Natural Sciences and Engineering Research Council of Canada Grants 412779-2012 and 04658-2018, as well as a University of Alberta start-up grant. He is also grateful to the Pacific Institute for the Mathematical Sciences, in Vancouver, BC, Canada, for its generous hospitality during his visit in February and March of 2017
R. J. McCann, B. Pass
the Monge–Kantorovich optimal transportation problem is equivalent to solving a degenerate elliptic Monge–Ampère equation: that is, given two probability densities ˜ is given by a f and g on Rn , the unique optimal map between them, F = D u, convex solution u˜ to the boundary value problem g ◦ D u˜ det D 2 u˜ = f [a.e.], D u˜ ∈ sptg [a.e.],
(1) (2)
where sptg ⊂ Rn is the smallest closed set of full mass for g. Similarly, its inverse is given by the gradient of the convex solution v˜ to the boundary value problem f ◦ D v˜ det D 2 v˜ = g [a.e.],
(3)
D v˜ ∈ spt f
(4)
[a.e.].
Notice the quadratic cost implicitly requires x and y to live in the same space. Subsequent work of Ma, Trudinger and Wang [31] leads to an analogous result for other cost functions c(x, y) = −s(x, y) satisfying suitable conditions, still requiring x and y to live in spaces of the same dimensio
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