Variational Approach to Regularity of Optimal Transport Maps: General Cost Functions

Felix Otto, Maxime Prod'homme, and Tobias Ried

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Submission date: 19. Aug. 2021
Bibtex
MSC-Numbers: 49Q22, 35B65, 53C21
Keywords and phrases: Optimal transportation, epsilon-regularity, partial regularity, General cost functions, Almost-minimality
Link to arXiv: See the arXiv entry of this preprint.
DOI number (of the published article): 10.1007/s40818-021-00106-1

Abstract:
We extend the variational approach to regularity for optimal transport maps initiated by Goldman and the first author to the case of general cost functions. Our main result is an 𝜖-regularity result for optimal transport maps between Hölder continuous densities slightly more quantitative than the result by De Philippis-Figalli. One of the new contributions is the use of almost-minimality: if the cost is quantitatively close to the Euclidean cost function, a minimizer for the optimal transport problem with general cost is an almost-minimizer for the one with quadratic cost. This further highlights the connection between our variational approach and De Giorgi’s strategy for 𝜖-regularity of minimal surfaces.