A variational approach to the regularity theory for optimal transportation: Lecture 1

• Lecture 1 (Lecture notes)
• Lecture 2 (Lecture notes)
• Lecture 3 (Lecture notes)

In this mini-course, we shall explain the variational approach to regularity
theory for optimal transportation introduced in [8]. This approach does
completely bypass the celebrated regularity theory of Caffarelli [2], which is
based on the regularity theory for the Monge-Amp ere equation as a fully
nonlinear elliptic equation with a comparison principle. Nonetheless, one
recovers the same partial regularity theory [5, 4].

The advantage of the variational approach resides in its robustness regarding
the regularity of the measures, which can be arbitrary measures [7][Theorem
1.4], and in terms of the problem formulation, e.g. by its extension to almost
minimizers [10]. The former for instance is crucial in order to tackle the
widely popular matching problem [3, 1] e.g. the optimal transportation between
(random) point clouds, as carried out in [7, 6, 9]. The latter is convenient
when treating more general than square Euclidean cost functions.

The variational approach follows de Giorgi’s philosophy for minimal surfaces.
At its core is the approximation of the displacement by the gradient of a
harmonic function. This approximation is based on the Eulerian formulation of
optimal transportation, which reveals its strict convexity and the proximity to
the $H^{-1}$-norm. In this mini-course, we shall give a pretty self-contained
derivation of this harmonic approximation result, and establish applications to
the matching problem.

References

• [1] L. Ambrosio, F. Stra, D. Trevisan: A PDE approach to a 2-dimensional
  matching problem. Probab. Theory Relat. Fields 173, 433–477 (2019).
• [2] L.A. Caffarelli: The regularity of mappings with a convex potential.
  Journal of the American Mathematical Society 5 (1992), no. 1, 99–104.
• [3] S. Caracciolo, C. Lucibello, G. Parisi, G. Sicuro: Scaling hypothesis for
  the Euclidean bipartite matching problem. Physical Review E, 90(1), 2014.
• [4] G. De Philippis, A. Figalli: Partial regularity for optimal transport
  maps. Publications Mathématiques. Institut de Hautes Études Scientifiques
  121 (2015), 81–112.
• [5] A. Figalli, Y.-H. Kim: Partial regularity of Brenier solutions of the
  Monge-Amépre equation. Discrete and Continuous Dynamical Systems (Series A)
  28 (2010), 559–565.
• [6] M. Goldman, M. Huesmann: A fluctuation result for the displacement in the
  optimal matching problem. arXiv e-prints, May 2021. arXiv:2105.02915.
• [7] M. Goldman, M. Huesmann, F. Otto: Quantitative linearization results for
  the Monge-Amp`ere equation. Communications on Pure and Applied Mathematics
  (2021).
• [8] M. Goldman, F. Otto: A variational proof of partial regularity for optimal
  transportation maps. Annales Scientifiques de l’Ećole Normale Supérieure.
  Quatriéme Série 53 (2020), no. 5, 1209–1233.
• [9] M. Huesmann, F. Mattesini, F. Otto: There is no stationary cyclically
  monotone Poisson matching in 2d. arXiv e-prints, September 2021.
  arXiv:2109.13590.
• [10] F. Otto, M. Prod’homme, T. Ried: Variational approach to regularity of
  optimal transport maps: general cost functions. (English summary) Ann. PDE 7
  (2021), no. 2, Paper No. 17, 74 pp.