A note on global regularity in optimal transportion

Abstract

We indicate how recent work of Figalli-Kim-McCann and Vetois can be used to improve previous results of Trudinger and Wang on classical solvability of the second boundary value problem for Monge-Ampère type partial differential equations arising in optimal transportation together with the global regularity of the associated optimal mappings.

On Pogorelov estimates in optimal transportation and geometric optics

“…We then conclude from Theorem 3.2 the following global second derivative estimate which improves Theorem 1.1 in [20] and Theorem 2.1 in [15]. where the constant C depends on n, c, B, , * and J 1 [u].…”

“…4 we focus on applications to optimal transportation and near field geometric optics. In the optimal transportation case, we obtain a different and more direct proof of the improved global regularity result in [15]. Then we conclude by treating some examples of generating functions in geometric optics satisfying our hypotheses, which arise from the reflection and refraction of parallel beams.…”

“…We also mention though that the special case of optimal transportation is given by taking g(x, y, z) = c(x, y) − z, (1.3) where c is a cost function defined on a domain D in R n × R n so that = D × R, g x = c x , g z = −1, E x,y = c x,y and conditions A1 and A2 are equivalent to those in [11,19]. Note that here we follow the same sign convention as in [15,20] so that our cost functions are the negatives of those usually considered. We remark also that the case when Y is independent of u is equivalent to the optimal transportation case.…”

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)).…”

Untitled

“…Under this hypothesis and a geometric condition on the supports of the measures (which is the analogue of the convexity assumption of Caffarelli), Ma, Trudinger, and Wang could prove the following result [87,108,109] (see also [103]): Sketch of the proof. For simplicity here we treat the simpler case when MTW(K) holds for some K > 0.…”