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Variational approach to regularity of optimal transport maps: general cost functions. (English) Zbl 1496.49025

The paper continues the line of research started with [M. Goldman and F. Otto, Ann. Sci. Éc. Norm. Supér. (4) 53, No. 5, 1209–1233 (2020; Zbl 1465.35263)] and [M. Goldman et al., Commun. Pure Appl. Math. 74, No. 12, 2483–2560 (2021; Zbl 1480.35082)]. The aim is the study of the regularity theory for optimal transport maps with a purely variational approach.
Here the authors deal with general cost functions and Hölder continuous densities, obtaining a slightly more quantitative result than the one in the celebrated paper [G. De Philippis and A. Figalli, Publ. Math., Inst. Hautes Étud. Sci. 121, 81–112 (2015; Zbl 1325.49051)]. Besides the differences in the statement of the main result compared to the one of De Philippis and Figalli, the interest is in the different approach. Indeed, the authors do not rely on the regularity theory for the Monge-Ampère equation and use arguments similar to De Giorgi’s strategy for the \(\varepsilon\)-regularity of minimal surfaces.
The result can be also applied to the study of the optimal transport problem on Riemannian manifolds with cost given by the square of the Riemannian distance.

MSC:

49Q22 Optimal transportation
35B65 Smoothness and regularity of solutions to PDEs
53C21 Methods of global Riemannian geometry, including PDE methods; curvature restrictions
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