Regularity of optimal maps and Monge-Ampère equations

Published/accepted papers

19. On optimal transport maps between 1/d-concave densities (with G. Carlier, and F. Santambrogio)
    Ann. Inst. H. Poincaré Anal. Non Linéaire, to appear.
20. Regularity properties of monotone measure-preserving maps (with Y. Jhaveri)
    Adv. Nonlinear Stud. 23 (2023), no.1, Paper No. 20220057, 11 pp.
21. Regularity of monotone transport maps between unbounded domains (with D. Cordero-Erausquin)
    Discrete Contin. Dyn. Syst., 39 (2019), no. 12, 7101-7112.
22. On the Continuity of Center-Outward Distribution and Quantile Functions
    Nonlinear Anal. 177 (2018), part B, 413-421.
23. Lipschitz changes of variables between perturbations of log-concave measures (with M. Colombo and Y. Jhaveri)
    Ann. Sc. Norm. Super. Pisa Cl. Sci. 17 (2017), no. 4, 1491-1519.
24. Partial W^{2,p} regularity for optimal transport maps (with S. Chen)
    J. Funct. Anal. 272 (2017), no. 11, 4588-4605.
25. Stability results on the smoothness of optimal transport maps with general costs (with S. Chen)
    J. Math. Pures Appl. (9) 106 (2016), no. 2, 280-295.
26. Nonlinear bounds in Hölder spaces for the Monge-Ampère equation (with Y. Jhaveri and C. Mooney)
    J. Funct. Anal. 270 (2016), no. 10, 3808-3827.
27. Boundary ε-regularity in optimal transportation (with S. Chen)
    Adv. Math. 273 (2015), 540-567.
28. Partial regularity for optimal transport maps (with G. De Philippis)
    Publ. Math. Inst. Hautes Études Sci. 121 (2015), 81-112.
29. Sobolev regularity for Monge-Ampère type equations (with G. De Philippis)
    SIAM J. Math. Anal. 45 (2013), no. 3, 1812-1824.
30. Hölder continuity and injectivity of optimal maps (with Y.-H. Kim and R. J. McCann)
    Arch. Ration. Mech. Anal. 209 (2013), no. 3, 747-795.
31. Second order stability for the Monge-Ampère equation and strong Sobolev convergence of optimal transport maps (with G. De Philippis)
    Anal. PDE 6 (2013), no. 4, 993-1000.
32. A note on interior W^2,1+epsilon estimates for the Monge-Ampère equation (with G. De Philippis and O. Savin)
    Math. Ann. 357 (2013), no. 1, 11-22.
33. W^2,1 regularity for solutions of the Monge-Ampère equation (with G. De Philippis)
    Invent. Math. 192 (2013), no. 1, 55-69.
34. Regularity of optimal transport maps on multiple products of spheres (with Y.-H. Kim and R. J. McCann)
    J. Eur. Math. Soc. (JEMS) 15 (2013), no. 4, 1131-1166.
35. Partial regularity of Brenier solutions of the Monge-Ampère equation (with Y.-H. Kim)
    Discrete Contin. Dyn. Syst. 28 (2010), no. 2, 559-565.
36. Regularity properties of optimal maps between nonconvex domains in the plane
    Comm. Partial Differential Equations 35 (2010), no. 3, 465-479.
37. C¹ regularity of solutions of the Monge-Ampère equation for optimal transport in dimension two (with G. Loeper)
    Calc. Var. Partial Differential Equations 35 (2009), no. 4, 537-550.

Surveys and lecture notes

6. On the Monge-Ampère equation
   Séminaire Bourbaki. Vol. 2017/2018. Exposé 1148, to appear.
7. Global existence for the semigeostrophic equations via Sobolev estimates for Monge-Ampère
   Partial differential equations and geometric measure theory, 1-42,
   Lecture Notes in Math., 2211, Fond. CIME/CIME Found. Subser., Springer, Cham (2018).
8. Partial regularity results in optimal transportation (with G. De Philippis)
   Trends in Contemporary Mathematics, Springer INdAM Series Volume 8, (2014), 293-307
9. The Monge-Ampère equation and its link to optimal transportation (with G. De Philippis)
   Bull. Amer. Math. Soc. (N.S.) 51 (2014), no. 4, 527-580.
10. Sobolev regularity for the Monge-Ampère equation, with application to the semigeostrophic equations
    Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 411 (2013), Teoriya Predstavlenii Dinamicheskie Sistemy, Kombinatornye Metody. XXII, 103-118, 242; translation in J. Math. Sci. (N. Y.) 196 (2014), no. 2, 175-183.
11. Regularity of optimal transport maps [after Ma-Trudinger-Wang and Loeper]
    Séminaire Bourbaki. Vol. 2008/2009. Exposés 997-1011. Astérisque 332 (2010), Exp. No. 1009, ix, 341-368.

Books

1. The Monge-Ampère Equation and Its Applications
   Zurich Lectures in Advanced Mathematics. European Mathematical Society (EMS), Zürich, 2017. x+200