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We have decided to discontinue the publication of preprints on our preprint server as of 1 March 2024. The publication culture within mathematics has changed so much due to the rise of repositories such as ArXiV (www.arxiv.org) that we are encouraging all institute members to make their preprints available there. An institute's repository in its previous form is, therefore, unnecessary. The preprints published to date will remain available here, but we will not add any new preprints here.

MiS Preprint
20/2021

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

Felix Otto, Maxime Prod'homme and Tobias Ried

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 $\epsilon$-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 $\epsilon$-regularity of minimal surfaces.

Received:
19.08.21
Published:
22.08.21
MSC Codes:
49Q22, 35B65, 53C21
Keywords:
Optimal transportation, epsilon-regularity, partial regularity, General cost functions, Almost-minimality

Related publications

inJournal
2021 Journal Open Access
Felix Otto, Maxime Prod'homme and Tobias Ried

Variational approach to regularity of optimal transport maps : general cost functions

In: Annals of PDE, 7 (2021) 2, p. 17