Research

Here is th
e list of my research papers.
PDF files of "Published/accepted papers" and "Lecture notes and reviews" can be downloaded following the links below.
• Published/accepted papers:
1. The Monge problem on non-compact manifolds, Rend. Sem. Mat. Univ. Padova, 117 (2007), 147-166.
2. Existence, uniqueness and regularity of optimal transport maps, SIAM J. Math. Anal., 39 (2007), no. 1, 126-137.
3. High action orbits for Tonelli Lagrangians and superlinear Hamiltonians on compact configuration spaces (with A. Abbondandolo), J. Differential Equations, 234 (2007), no. 2, 626-653.
4. Strong displacement convexity on Riemannian manifolds (with C. Villani), Math. Z., 257 (2007), no. 2, 251-259.
5. On the regularity of the pressure field of Brenier's weak solutions to incompressible Euler equations (with L. Ambrosio), Calc. Var. Partial Differential Equations, 31 (2008), no. 4, 497-509.
6. Existence and uniqueness of martingale solutions for SDEs with rough or degenerate coefficients, J. Funct. Anal., 254 (2008), no. 1, 109-153.
7. A simple proof of the Morse-Sard theorem in Sobolev spaces, Proc. Amer. Math. Soc., 136 (2008), no. 10, 3675-3681.
8. Geodesics in the space of measure-preserving maps and plans (with L. Ambrosio), Arch. Ration. Mech. Anal., 194 (2009), no. 2, 421-462.
9. Synchronized traffic plans and stability of optima (with M. Bernot), ESAIM Control Optim. Calc. Var., 14 (2008), no. 4, 864-878.
10. Optimal transportation on non-compact manifolds (with A. Fathi), Israel J. Math., 175 (2010), no. 1, 1-59.
11. Invariant measures of Hamiltonian systems with prescribed asymptotical Maslov index (with A. Abbondandolo), J. Fixed Point Theory Appl., 3 (2008), no. 1, 95-120.
12. A geometric lower bound on Grad's number, ESAIM Control Optim. Calc. Var., 15 (2009), no. 3, 569-575.
13. On the Hausdorff Dimension of the Mather quotient (with A. Fathi and L. Rifford), Comm. Pure Appl. Math., 62 (2009), no. 4, 445-500.
14. Absolute continuity of Wasserstein geodesics in the Heisenberg group (with N. Juillet), J. Funct. Anal., 255 (2008), no. 1, 133-141.
15. An approximation lemma about the cut locus, with applications in optimal transport theory (with C. Villani), Methods Appl. Anal., 15 (2008), no. 2, 149-154.
16. On flows associated to Sobolev vector fields in Wiener spaces: an approach à la DiPerna-Lions (with L. Ambrosio), J. Funct. Anal., 256 (2009), 179-214.
17. Convergence to the viscous porous medium equation and propagation of chaos (with R. Philipowski), ALEA Lat. Am. J. Probab. Math. Stat., 4 (2008), 185-203.
18. Generalized solutions for the Euler equations in one and two dimensions (with M. Bernot and F. Santambrogio), J. Math. Pures Appl., 91 (2008), no. 2, 137-155.
19. A note on Cheeger sets (with F. Maggi and A. Pratelli), Proc. Amer. Math. Soc., 137 (2009), 2057-2062.
20. The optimal partial transport problem, Arch. Ration. Mech. Anal., 195 (2010), no. 2, 533-560.
21. 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.
22. Mass Transportation on Sub-Riemannian Manifolds (with L. Rifford), Geom. Funct. Anal., 20 (2010), no. 1, 124-159.
23. A note on the regularity of the free boundaries in the optimal partial transport problem, Rend. Circ. Mat. Palermo, 58 (2009), no. 2, 283-286.
24. Continuity of optimal transport maps and convexity of injectivity domains on small deformations of S^2 (with L. Rifford), Comm. Pure Appl. Math., 62 (2009), no. 12, 1670-1706.
25. On flows of H^{3/2}-vector fields on the circleMath. Ann., 347 (2010), no. 1, 43-57.
26. A refined Brunn-Minkowski inequality for convex sets (with F. Maggi and A. Pratelli), Ann. Inst. H. Poincaré Anal. Non Linéaire, 26  (2009),  no. 6, 2511-2519.
27. Some new well-posedness results for continuity and transport equations, and applications to the chromatography system (with L. Ambrosio, G. Crippa, and L. V. Spinolo), SIAM J. Math. Anal., 41  (2009),  no. 5, 1890-1920.
28. Regularity properties of optimal maps between nonconvex domains in the planeComm. Partial Differential Equations, 35 (2010), no. 3, 465-479.
29. Local semiconvexity of Kantorovich potentials on non-compact manifolds (with N. Gigli), ESAIM Control Optim. Calc. Var., 17 (2011), no. 3, 648-653.
30. A new transportation distance between non-negative measures, with applications to gradients flows with Dirichlet boundary conditions (with N. Gigli), J. Math. Pures Appl., 94 (2010), no. 2, 107-130.
31. On the Ma-Trudinger-Wang curvature on surfaces (with L. Rifford and C. Villani), Calc. Var. Partial Differential Equations, 39 (2010), no. 3-4, 307-332.
32. Almost everywhere well-posedness of continuity equations with measure initial data (with L. Ambrosio), C. R. Math. Acad. Sci. Paris, 348 (2010), no. 5-6, 249-252.
33. A variational method for a class of parabolic PDEs (with W. Gangbo and T. Yolcu), Ann. Scuola Norm. Sup. Pisa Cl. Sci. (5), 10 (2011), no. 1, 207-252.
34. Partial regularity of Brenier solutions of the Monge-Ampère equation (with Y.-H. Kim), Discrete Contin. Dyn. Syst., 28 (2010), no. 2, 559-565.
35. Global in time measure-valued solutions and finite-time aggregation for nonlocal interaction equations (with J. A. Carrillo, M. Di Francesco, T. Laurent, and D. Slepcev), Duke Math. J., 156  (2011),  no. 2, 229-271.
36. A mass transportation approach to quantitative isoperimetric inequalities (with F. Maggi and A. Pratelli), Invent. Math., 182 (2010), no. 1, 167-211.
37. Fine properties of minimizers of mechanical Lagrangians with Sobolev potentials (with V. Mandorino), Discrete Contin. Dyn. Syst., 31 (2011), no. 4, 1325-1346.
38. On the shape of liquid drops and crystals in the small mass regime (with F. Maggi), Arch. Ration. Mech. Anal., 201 (2011), no. 1, 143-207.
39. Surface measures and convergence of the Ornstein-Uhlenbeck semigroup in Wiener spaces (with L. Ambrosio), Ann. Fac. Sci. Toulouse Math. (6), 20 (2011), no. 2, 407-438.
40. When is multidimensional screening a convex program? (with Y.-H. Kim and R. J. McCann), J. Econom. Theory, 146 (2011), 454-478.
41. Nearly round spheres look convex (with L. Rifford and C. Villani), Amer. J. Math., 134  (2012),  no. 1, 109-139.
42. Tangent cut loci on surfaces (with L. Rifford and C. Villani), Differential Geom. Appl., 29 (2011), no. 2, 154-159.
43. Semiclassical limit of quantum dynamics with rough potentials and well posedness of transport equations with measure initial data (with L. Ambrosio, G. Friesecke, J. Giannoulis, and T. Paul), Comm. Pure Appl. Math., 64  (2011),  no. 9, 1199-1242.
44. Non-Local Tug-of-War and the Infinity Fractional Laplacian (with C. Bjorland and L. Caffarelli), Comm. Pure Appl. Math., 65  (2012),  no. 3, 337-380.
45. Necessary and sufficient conditions for continuity of optimal transport maps on Riemannian manifolds (with L. Rifford and C. Villani), Tohoku Math. J. (2), 63 (2011),  no. 4, 855-876.
46. 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.
47. Isoperimetric-type inequalities on constant curvature manifolds (with Y. Ge), Adv. Calc. Var., 5 (2012), no. 3, 251-284.
48. Confinement in nonlocal interaction equations (with J. A. Carrillo, M. Di Francesco, T. Laurent, and D. Slepcev), Nonlinear Anal., 75  (2012),  no. 2, 550-558.
49. A sharp stability result for the relative isoperimetric inequality inside convex cones (with E. Indrei), J. Geom. Anal., 23 (2013), no. 2, 938-969.
50. Semiclassical limit for mixed states with singular and rough potentials (with M. Ligabò and T. Paul), Indiana Univ. Math. J., 61 (2012), no. 1, 193-222.
51. Regularity of solutions to the parabolic fractional obstacle problem (with L. Caffarelli), J. Reine Angew. Math., 680 (2013), 191-233.
52. Total Variation Flow and Signed Fast Diffusion in one dimension (with M. Bonforte), J. Differential Equations, 252 (2012), no. 8, 4455-4480.
53. A geometric approach to correlation inequalities in the plane (with F. Maggi and A. Pratelli), Ann. Inst. H. Poincaré Probab. Stat., 50 (2014), no. 1, 1-14.
54. Existence of Eulerian solutions to the semigeostrophic equations in physical space: the 2-dimensional periodic case (with L. Ambrosio, M.Colombo, and G. De Philippis), Comm. Partial Differential Equations, 37 (2012), no. 12, 2209-2227.
55. Non-Local Gradient Dependent Operators (with C. Bjorland and L. Caffarelli), Adv. Math., 230 (2012), no. 4-6, 1859-1894.
56. W^{2,1} regularity for solutions of the Monge-Ampère equation (with G. De Philippis), Invent. Math., 192 (2013), no. 1, 55-69.
57. Sharp stability theorems for the anisotropic Sobolev and log-Sobolev inequalities on functions of bounded variation (with F. Maggi and A. Pratelli), Adv. Math., 242 (2013), 80-101.
58. Stability for a GNS inequality and the Log-HLS inequality, with application to the critical mass Keller-Segel equation (with E. Carlen), Duke Math. J., 162 (2013), no. 3, 579-625.
59. On the isoperimetric problem for radial log-convex densities (with F. Maggi), Calc. Var. Partial Differential Equations, 48 (2013), no. 3-4, 447-489.
60. Asymptotics of the s-perimeter as s --> 0 (with S. Dipierro, G. Palatucci, and E. Valdinoci), Discrete Contin. Dyn. Syst., 33 (2013), no. 7, 2777-2790.
61. Bootstrap regularity for integro-differential operators, and its application to nonlocal minimal surfaces (with B. Barrios Barrera and E. Valdinoci), Ann. Scuola Norm. Sup. Pisa Cl. Sci., to appear.
62. 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.
63. 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.
64. A global existence result for the semigeostrophic equations in three dimensional convex domains (with L. Ambrosio, M. Colombo, and G. De Philippis), Discrete Contin. Dyn. Syst., 34 (2014), no. 4, 1251-1268.
65. On sets of finite perimeter in Wiener spaces: reduced boundary and convergence to half-spaces (with L. Ambrosio and E. Runa), Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl., 24 (2013), no. 1, 111-122.
66. WKB analysis of Bohmian dynamics (with C. Klein, P. Markowich, and C. Sparber), Comm. Pure Appl. Math., 67 (2014), no. 4, 581-620.
67. 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.
68. Regularity results for very degenerate elliptic equations (with M. Colombo), J. Math. Pures Appl. (9), 101 (2014), no. 1, 94-117.
69. Sobolev regularity for Monge-Ampère type equations (with G. De Philippis), SIAM J. Math. Anal., 45 (2013), no. 3, 1812-1824.
70. How to recognize convexity of a set from its marginals (with D. Jerison), J. Funct. Anal., 266 (2014), no. 3, 1685-1701.
71. On supporting hyperplanes to convex bodies (with Y.-H. Kim and R. J. McCann), Methods Appl. Anal., 20 (2013), no. 3, 261-271.
72. Closing Aubry sets I (with L. Rifford), Comm. Pure Appl. Math., 68 (2015), no. 2, 210-285.
73. Closing Aubry sets II (with L. Rifford), Comm. Pure Appl. Math., 68 (2015), no. 3, 345-412.
74. Optimal regularity of the convex envelope (with G. De Philippis), Trans. Amer. Math. Soc., 367 (2015), no. 6, 4407-4422.
75. An excess-decay result for a class of degenerate elliptic equations (with M. Colombo), Discrete Contin. Dyn. Syst. Ser. S, 7 (2014), no. 4, 631-652.
76. Higher integrability for minimizers of the Mumford-Shah functional (with G. De Philippis), Arch. Ration. Mech. Anal., 213 (2014), no. 2, 491-502.
77. A general class of free boundary problems for fully nonlinear elliptic equations (with H. Shahgholian), Arch. Ration. Mech. Anal., 213 (2014), no. 1, 269-286.
78. A general class of free boundary problems for fully nonlinear parabolic equations (with H. Shahgholian), Ann. Mat. Pura Appl. (4), 194 (2015), no. 4, 1123-1134.
79. Strongly nonlocal dislocation dynamics in crystals (with S. Dipierro and E. Valdinoci), Comm. Partial Differential Equations, 39 (2014), no. 12, 2351-2387.
80. Quantitative stability for sumsets in R^n (with D. Jerison), J. Eur. Math. Soc. (JEMS), 17 (2015), no. 5, 1079-1106.
81. Generic hyperbolicity of Aubry sets on surfaces (with G. Contreras and L. Rifford), Invent. Math., 200 (2015), no. 1, 201-261.
82. Isoperimetry and stability properties of balls with respect to nonlocal energies (with N. Fusco, F. Maggi, V. Millot, and M. Morini), Comm. Math. Phys., 336 (2015), no. 1, 441-507.
83. Partial regularity for optimal transport maps (with G. De Philippis), Publ. Math. Inst. Hautes Études Sci., 121 (2015), 81-112.
84. On the convexity of injectivity domains on nonfocal manifolds (with T. Gallouët and L. Rifford), SIAM J. Math. Anal., 47 (2015), no. 2, 969-1000.
85. Boundary ε-regularity in optimal transportation (with S. Chen), Adv. Math., 273 (2015), 540-567.
86. Regularity and Bernstein-type results for nonlocal minimal surfaces (with E. Valdinoci), J. Reine Angew. Math., to appear.
87. A note on the dimension of the singular set in free interface problems (with G. De Philippis), Differential Integral Equations, 28 (2015), 523-536.
88. BMO-type norms related to the perimeter of sets (with L. Ambrosio, J. Bourgain, and H. Brezis), Comm. Pure Appl. Math., to appear.
89. Transport maps for β-matrix models and universality (with F. Bekerman and A. Guionnet), Comm. Math. Phys., 338 (2015), no. 2, 589-619.
90. Existence and uniqueness of maximal regular flows for non-smooth vector fields (with L. Ambrosio and M. Colombo), Arch. Ration. Mech. Anal., 218 (2015), no. 2, 1043-1081.
91. On the density function on moduli spaces of toric 4-manifolds (with A. Pelayo), Adv. Geom., to appear.
92. Rigidity and sharp stability estimates for hypersurfaces with constant and almost-constant nonlocal mean curvature (with G. Ciraolo, F. Maggi, and M. Novaga), J. Reine Angew. Math., to appear.
• Submitted papers:
1. Quantitative stability for the Brunn-Minkowski inequality (with D. Jerison)
2. Universality in several-matrix models via approximate transport maps (with A. Guionnet)
3. Regularity of codimension-1 minimizing currents under minimal assumptions on the integrand
4. On the Lagrangian structure of transport equations: the Vlasov-Poisson system (with L. Ambrosio and M. Colombo)
5. Global regularity for the free boundary in the obstacle problem for the fractional Laplacian (with B. Barrios and X. Ros-Oton)
6. Nonlinear bounds in Hölder spaces for the Monge-Ampère equation (with Y. Jhaveri and C. Mooney)
7. Global well-posedness of the spatially homogeneous Kolmogorov-Vicsek model as a gradient flow (with M.-J. Kang and J. Morales)
8. Quantitative stability of the Brunn-Minkowski inequality for sets of equal volume (with D. Jerison)
9. Stability results on the smoothness of optimal transport maps with general costs (with S. Chen)
10. Gradient stability for the Sobolev inequality: the case p$\geq$2 (with R. Neumayer)
11. Lipschitz changes of variables between perturbations of log-concave measures (with M. Colombo and Y. Jhaveri)
• Lecture notes and reviews:
1. Optimal Transport and Curvature (with C. Villani), Nonlinear PDE's and applications,  171-217, Lecture Notes in Math., 2028, Springer, Heidelberg, 2011.
2. 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.
3. Cédric Villani reçoit un prix de la Société Mathématique Européenne (French) [Cédric Villani, 2008 EMS Prize] (with L. Desvillettes), Gaz. Math., no. 120 (2009), 76-81.
4. Optimal Transport. Old and New. [book review], Bull. Amer. Math. Soc. (N.S.), 47 (2010), no. 4, 723-727.
5. Almost everywhere well-posedness of continuity equations with measure initial data (with L. Ambrosio), C. R. Math. Acad. Sci. Paris 348 (2010), no. 5-6, 249-252.
6. Lecture notes on variational models for incompressible Euler equations (with L. Ambrosio), Optimal transportation, 58-71, London Math. Soc. Lecture Note Ser., 413, Cambridge Univ. Press, Cambridge, 2014.
7. Quantitative isoperimetric inequalities, with applications to the stability of liquid drops and crystals, Concentration, functional inequalities and isoperimetry, 77-87, Contemp. Math., 545, Amer. Math. Soc., Providence, RI, 2011.
8. Existence and uniqueness results for the continuity equation and applications to the chromatography system (with L. Ambrosio, G. Crippa, and L. V. Spinolo), Nonlinear conservation laws and applications, 195-204, IMA Vol. Math. Appl., 153, Springer, New York, 2011.
9. Variational models for the incompressible Euler equations (with S. Daneri), AIMS Book Series, Applied Mathematics, to appear.
10. Stability in geometric and functional inequalities, Proceedings of the 6th European Congress of Mathematics, to appear.
11. 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.
12. Aubry sets, Hamilton-Jacobi equations, and the Mañé Conjecture (with L. Rifford), Geometric analysis, mathematical relativity, and nonlinear partial differential equations, 83-104, Contemp. Math., 599, Amer. Math. Soc., Providence, RI, 2013.
13. 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.
14. Partial regularity results in optimal transportation (with G. De Philippis), Springer INdAM series, to appear.
15. Quantitative stability results for the Brunn-Minkowski inequality, Proceedings of the ICM 2014, to appear.
16. Stability results for the Brunn-Minkowski inequality, Colloquium De Giorgi 2013-2014, to appear.
17. Perimeter of sets and BMO-type norms (with L. Ambrosio, J. Bourgain, and H. Brezis), C. R. Math. Acad. Sci. Paris, 352 (2014), no. 9, 697-698.
18. An overview of unconstrained free boundary problems (with H. Shahgholian), Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., to appear
19. Global existence for the semigeostrophic equations via Sobolev estimates for Monge-Ampère, CIME Lecture Notes, Springer, to appear.
• Books:
1. Optimal transportation and action-minimizing measures (author), Publications of the Scuola Normale Superiore, 2008.
2. Autour des Inégalités Isopérimétriques (editor and supervisor, in French), Publications of the Ecole Polytechnique, 2011.