## Continuity and injectivity of optimal maps for non-negatively cross-curved costs

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Figalli, Alessio

Kim, Young-Heon

McCann, Robert J.

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Consider transportation of one distribution of mass onto another, chosen to
optimize the total expected cost, where cost per unit mass transported from x
to y is given by a smooth function c(x,y). If the source density f^+(x) is
bounded away from zero and infinity in an open region U' \subset R^n, and the
target density f^-(y) is bounded away from zero and infinity on its support clV
\subset R^n, which is strongly c-convex with respect to U', and the
transportation cost c is non-negatively cross-curved, we deduce continuity and
injectivity of the optimal map inside U' (so that the associated potential u
belongs to C^1(U')). This result provides a crucial step in the low/interior
regularity setting: in a subsequent paper [15], we use it to establish
regularity of optimal maps with respect to the Riemannian distance squared on
arbitrary products of spheres. The present paper also provides an argument
required by Figalli and Loeper to conclude in two dimensions continuity of
optimal maps under the weaker (in fact, necessary) hypothesis A3w [17]. In
higher dimensions, if the densities f^\pm are H\"older continuous, our result
permits continuous differentiability of the map inside U' (in fact,
C^{2,\alpha}_{loc} regularity of the associated potential) to be deduced from
the work of Liu, Trudinger and Wang [33].

Comment: 37 pages, 7 figures

Comment: 37 pages, 7 figures

##### Keywords

Mathematics - Analysis of PDEs, Mathematics - Differential Geometry, 35J60, 35B65