Continuity of maps solutions of optimal transportation problems
G. Loeper
TL;DR
The paper studies the regularity and continuity of optimal transport maps, expressing them as the gradient of a c-convex potential solving a Monge-Ampere type equation $\det(M(x, ∇φ) + D^2φ) = f$, under cost conditions A1–A3. Under A1–A3 and mild density conditions, the authors prove $C^{1}$ and $C^{1,α}$ regularity of the potential; in particular φ ∈ $C^{1,β}$ with β = α/(4n-2+α) and α = 1−n/p when f ∈ L^p, p>n, and the gradient's modulus is controlled by the small-scale behavior of ρ0. The approach relies on the c-convex potential and a Monge-Ampere equation with a cost-dependent matrix M, and the discussion in Section 2 provides a geometric interpretation of A3 via c-segments and a surrogate function to derive lower bounds, culminating in a modulus-of-continuity estimate for ∇φ. The results show improved regularity compared to the standard Monge-Ampere equation det D^2φ = f under A3, with quantitative bounds linking gradient continuity to the density ρ0, advancing the understanding of optimal transport map continuity under nontrivial cost structures.
Abstract
In this paper we investigate the continuity of maps solutions of optimal transportation problems. These maps are expressed through the gradient of a potential for which we establish $C^{1}$ and $C^{1, \alpha}$ regularity. Our results hold assuming a condition on the cost function (condition A3 below), that was the one used for $C^2$ a priori estimates by Ma, Trudinger and Wang. The optimal potential will solve a Monge-Amp\`ere equation of the form $$\det (M(x, \nabla\phi) + D^2\phi) = f$$ where $M$ depends on the cost function. One of the interesting outcome is that under the condition A3, the regularity obtained is better than the one obtained in the case of the 'usual' Monge-Amp\`ere equation $\det D^2\phi = f$, in particular we will obtain here $C^{1, \alpha}$ regularity for $\phi$ under the condition $f\in L^p, p>n$.