[Paper Review] Continuity of maps solutions of optimal transportation problems

Grégoire Loeper|arXiv (Cornell University)|Apr 7, 2005

Optimization and Variational Analysis2 references8 citations

TL;DR

This paper establishes C¹ and C¹,α regularity for optimal transport maps by analyzing the potential function solving a Monge-Ampère equation involving a cost-dependent Hessian term. Under condition A3 on the cost function and integrability f ∈ Lᵖ with p > n, the authors achieve improved C¹,α regularity compared to the standard Monge-Ampère case, demonstrating enhanced regularity due to structural properties of the cost function.

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,α 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 in [5]. The optimal potential will solve a Monge-Ampère equation of the form det(M(x, ∇φ) + D 2 φ) = 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-Ampère equation det D 2 φ = f, in particular we will obtain here C 1,α regularity for φ under the condition f ∈ L p,p> n.

Motivation & Objective

• To establish higher regularity for optimal transport maps under a structural condition on the cost function.
• To analyze the potential function solving a Monge-Ampère equation with a cost-dependent Hessian term.
• To compare the regularity of solutions under condition A3 to that in the classical Monge-Ampère setting.
• To demonstrate that improved regularity, specifically C¹,α, is achievable when the density f belongs to Lᵖ with p > n.

Proposed method

• The authors derive a Monge-Ampère equation of the form det(M(x, ∇φ) + D²φ) = f, where M depends on the cost function.
• They apply a priori C² estimates from [5], relying on condition A3 to control the structure of the Hessian term.
• The analysis focuses on the potential φ whose gradient generates the optimal transport map.
• Regularity is established via a priori estimates leading to C¹ and C¹,α bounds on φ.
• The method leverages the structure of the cost function to improve regularity beyond the standard det D²φ = f case.
• The proof relies on the interplay between the cost function’s geometry and the integrability of the density f.

Experimental results

Research questions

• RQ1 Under what conditions on the cost function can optimal transport maps be shown to be C¹,α regular?
• RQ2 How does condition A3 influence the regularity of solutions to the Monge-Ampère equation in optimal transport?
• RQ3 Can improved regularity be achieved in optimal transport when f ∈ Lᵖ with p > n, compared to the classical setting?
• RQ4 What role does the Hessian term M(x, ∇φ) play in enhancing the regularity of the potential function?
• RQ5 How does the structure of the cost function affect the C¹,α regularity of the transport map?

Key findings

• The potential function φ solving the Monge-Ampère equation det(M(x, ∇φ) + D²φ) = f is shown to be C¹,α regular under condition A3.
• C¹,α regularity is achieved when the density f belongs to Lᵖ with p > n, improving upon classical results.
• The regularity result is stronger than in the standard Monge-Ampère equation det D²φ = f due to the structure imposed by the cost function.
• Condition A3 ensures the necessary a priori estimates for C² and subsequently C¹,α regularity of the potential.
• The Hessian term M(x, ∇φ) contributes to enhanced regularity by modifying the ellipticity structure of the equation.
• The results demonstrate that the cost function’s geometry directly enables better regularity than the classical case.