A variational proof of partial regularity for optimal transportation maps

2017/04/18 by Goldman, Michael, Otto, F · 2 citations
#Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.1704.05339

Abstract

We provide a new proof of the known partial regularity result for the optimal transportation map (Brenier map) between two sets. Contrary to the existing regularity theory for the Monge-Ampère equation, which is based on the maximum principle, our approach is purely variational. By constructing a competitor on the level of the Eulerian (Benamou-Brenier) formulation, we show that locally, the velocity is close to the gradient of a harmonic function provided the transportation cost is small. We then translate back to the Lagrangian description and perform a Campanato iteration to obtain an ε-regularity result.

Cited by

• C^1,α-regularity for Schrödinger potentials of Shannon-entropy regularized Optimal Transport
• Partial regularity of optimal transport with Coulomb cost

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