# Regularity of optimal maps and Monge-Ampère equations

### Published/accepted papers

- Regularity properties of monotone measure-preserving maps (with Y. Jhaveri)
*Adv. Nonlinear Stud.*23 (2023), no. 1, ans–2022–0057. - Regularity of monotone transport maps between unbounded domains (with D. Cordero-Erausquin)
*Discrete Contin. Dyn. Syst.,*39 (2019), no. 12, 7101-7112. - On the Continuity of Center-Outward Distribution and Quantile Functions
*Nonlinear Anal.*177 (2018), part B, 413-421. - 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. - Partial W^{2,p} regularity for optimal transport maps (with S. Chen)

*J. Funct. Anal.*272 (2017), no. 11, 4588-4605. - 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. - 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. - Boundary ε-regularity in optimal transportation (with S. Chen)

*Adv. Math.*273 (2015), 540-567. - Partial regularity for optimal transport maps (with G. De Philippis)

*Publ. Math. Inst. Hautes Études Sci.*121 (2015), 81-112. - Sobolev regularity for Monge-Ampère type equations (with G. De Philippis)

*SIAM J. Math. Anal.*45 (2013), no. 3, 1812-1824. - 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. - 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. - 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. - W
^{2,1}regularity for solutions of the Monge-Ampère equation (with G. De Philippis)

*Invent. Math.*192 (2013), no. 1, 55-69. - 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. - Partial regularity of Brenier solutions of the Monge-Ampère equation (with Y.-H. Kim)

*Discrete Contin. Dyn. Syst.*28 (2010), no. 2, 559-565. - Regularity properties of optimal maps between nonconvex domains in the plane

*Comm. Partial Differential Equations*35 (2010), no. 3, 465-479. - C
^{1}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

- On the Monge-Ampère equation

*Séminaire Bourbaki.*Vol. 2017/2018. Exposé 1148, to appear. - 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).* - Partial regularity results in optimal transportation (with G. De Philippis)

Trends in Contemporary Mathematics,*Springer INdAM Series*Volume 8, (2014), 293-307 - 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. - 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. - 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

- The Monge-Ampère Equation and Its Applications

Zurich Lectures in Advanced Mathematics. European Mathematical Society (EMS), Zürich, 2017. x+200