Partial regularity of Brenier solutions of the Monge-Ampère equation

Abstract

Given Ω, Λ ⊂ R n two bounded open sets, and f and g two probability densities concentrated on Ω and Λ respectively, we investigate the regularity of the optimal map ∇ϕ (the optimality referring to the Euclidean quadratic cost) sending f onto g. We show that if f and g are both bounded away from zero and infinity, we can find two open sets Ω ′ ⊂ Ω and Λ ′ ⊂ Λ such that f and g are concentrated on Ω ′ and Λ ′ respectively, and ∇ϕ : Ω ′ → Λ ′ is a (bi-Hölder) homeomorphism. This generalizes the 2-dimensional partial regularity result of [8].

The Monge–Ampère equation and its link to optimal transportation

“…As shown in [52] (see also [50] for a more precise description of the singular set in two dimensions), in this case one can prove that the optimal trasport map is smooth outside a closed set of measure zero. More precisely, the following holds: Theorem 3.4.…”

“…In the quadatic cost case, in [50,52] it was shown that on a big set there is still a good control on the Monge-Ampère measure of the potential u allowing to apply the local theory of classical Monge-Ampère equation, see the sketch of the proof of Theorem 3.4. However, the failure of the MTW condition does not allow us to use any local regularity estimate for the PDE, therefore a completely new strategy with respect to [50,52] has to be used.…”

Untitled

“…As shown in [51] (see also [49] for a more precise description of the singular set in two dimensions), in this case one can prove that the optimal transport map is smooth outside a closed set of measure zero. More precisely, the following holds.…”

Partial regularity for optimal transport maps

“…In addition, since u is close to a parabola, so is v. Hence, by [18] and Caffarelli's regularity theory, v is smooth, and we can use this information to deduce that u is even closer to a second parabola (given by the second order Taylor expansion of v at the origin) inside a small neighborhood around of origin. By rescaling back this neighborhood at scale 1 and iterating this construction, we obtain that u is C 1,β at the origin for some β ∈ (0, 1).…”

“…In [16,18] the authors proved the following result: Theorem 1.2. Let f and g be smooth probability densities, respectively bounded away from zero and infinity on two bounded open sets X and Y , and let T : X → Y denote the unique optimal transport map from f to g for the quadratic cost |x − y| 2 /2.…”