Regularity of Potential Functions of the Optimal Transportation Problem

The Monge–Ampère equation and its link to optimal transportation

“…Under this hypothesis, and a geometric condition on the supports of the measures (which is the analogous of the convexity assumption of Caffarelli), Ma, Trudinger, and Wang could prove the following result [90,111,112] (see also [106]): Then u ∈ C ∞ (X) and T : X → Y is a smooth diffeomorphism, where T (x) = c-exp x (∇u(x)).…”

“…The breakthrough in the study of regularity of optimal transport maps came with the paper of Ma, Trudinger, and Wang [90] (whose roots lie in an earlier work of Wang on the reflector antenna problem [116]), where the authors found a mysterious fourth-order condition on the cost functions, which turned out to be sufficient to prove the regularity of u. The idea was to differentiate twice equation (4.3) in order to get a linear PDE for the second derivatives of u, and then to try to show an a priori estimate on the second derivatives of u, compare with Theorem 2.12.…”

“…For simplicity here we treat the simpler case when MTW(K) holds for some K > 0. This stronger MTW condition is actually the one originally used in [90,112]. The general case K = 0 is treated in [111], where the authors relax the stronger assumption by applying a barrier argument.…”

“…In [90], Ma, Trudinger, and Wang showed that, in analogy with the quadratic case (see Section 3.2), the convexity of D x c(x, Y ) for all x ∈ X is necessary for regularity. By slightly modifying their argument, Loeper [87] showed that the connectedness of the c-subdifferential is a necessary condition for the smoothness of optimal transport (see also [114,Theorem 12.7]): Theorem 4.5.…”

“…Apart from having its own interest, this general theory of Monge-Ampère type equations satisfying the MTW condition turns out to be useful also in the classical case. Indeed the MTW condition can be proven to be coordinate invariant [90,87,76]. This implies that, if u solves an equation of the form (1.2) with A satisfying the MTW condition (see (4.4)) and Φ is a smooth diffeomorphism, then u•Φ satisfies an equation of the same form with a new matrixà which still satisfies the MTW condition.…”