On the local geometry of maps with c-convex potentials

Abstract

We identify a condition for regularity of optimal transport maps that requires only three derivatives of the cost function, for measures given by densities that are only bounded above and below. This new condition is equivalent to the weak Ma-Trudinger-Wang condition when the cost is C 4 . Moreover, we only require (non-strict) c-convexity of the support of the target measure, removing the hypothesis of strong c-convexity in a previous result of Figalli, Kim, and McCann, but at the added cost of assuming compact containment of the supports of both the source and target measures.

The Monge–Ampère equation and its link to optimal transportation

“…As proved by the second author together with Kim and McCann [53], such a result is indeed true (see also [55,66,117]):…”

“…Note that Theorem 4.8 as well as the other results in [84,85,38,66,117] only deal with the interior regularity for optimal transport maps. It would be interesting to develop a boundary regularity theory, in the spirit of [21,23], for the class of equations (1.2) arising in optimal transport.…”

Untitled

“…As proved by the second author together with Kim and McCann [52], such a result is indeed true (see also [54,63,113]): Let us mention that further extensions of Theorem 3.3 to this general setting have been obtained in [37,81,82].…”

Pointwise Estimates and Regularity in Geometric Optics and Other Generated Jacobian Equations

“…This condition follows from (G3w) when G is smooth enough, it generalizes the (QQConv) condition introduced in [GK15] for optimal transport (and in that case, it refines Loeper's maximum principle).…”

“…In [GK15] the authors introduced a condition on costs, "quantitative quasiconvexity" (QQConv), and used it to derive analogues of (1.3)-(1.4). This (QQConv) condition is a refinement of Loeper's "maximum principle" [Loe09] but at least for C 4 costs turns out to be equivalent to (A3w) (and thus to Loeper's condition itself).…”