Published/accepted papers

  1. Sharp gradient stability for the Sobolev inequality (with Y. Ru-Ya Zhang)
    Duke Math. J., to appear.
  2. Strong stability for the Wulff inequality with a crystalline norm (with Y. Ru-Ya Zhang)
    Comm. Pure Appl. Math., to appear
  3. A sharp Freiman type estimate for semisums in two and three dimensional Euclidean spaces (with D. Jerison)
    Ann. Sci. Éc. Norm. Supér. (4) 54 (2021), no. 1, 235–257.
  4. Sharp Extinction Rates for Fast Diffusion Equations on Generic Bounded Domains (with M. Bonforte)
    Comm. Pure Appl. Math. 74 (2021), no. 4, 744–789.
  5. Generic regularity of free boundaries for the obstacle problem (with X. Ros-Oton and J. Serra)
    Publ. Math. Inst. Hautes Études Sci. 132 (2020), 181–292.
  6. Stable solutions to semilinear elliptic equations are smooth up to dimension 9 (with X. Cabré, X. Ros-Oton and J. Serra)
    Acta Math. 224 (2020), no. 2, 187–252.
  7. Symmetry results for critical anisotropic p-Laplacian equations in convex cones (with G. Ciraolo and A. Roncoroni)
    Geom. Funct. Anal. 30 (2020), no. 3, 770-803.
  8. On the obstacle problem for the 1D wave equation (with X. Fernández-Real)
    Math. Eng. 2 (2020), no. 4, 584-597.
  9. Optimal regularity and structure of the free boundary for minimizers in cohesive zone models (with L. Caffarelli and F. Cagnetti)
    Arch. Ration. Mech. Anal. 237 (2020), no. 1, 299–345.
  10. On the sharp stability of critical points of the Sobolev inequality (with F. Glaudo)
    Arch. Ration. Mech. Anal. 237 (2020), no. 1, 201–258.
  11. On stable solutions for boundary reactions: a De Giorgi type result in dimension 4+1 (with J. Serra)
    Invent. Math. 219 (2020), no. 1, 153-177.
  12. Regularity of monotone transport maps between unbounded domains (with D. Cordero-Erausquin)
    Discrete Contin. Dyn. Syst. 39 (2019), no. 12, 7101-7112.
  13. Optimal regularity for the convex envelope and semiconvex functions related to supersolutions of fully nonlinear elliptic equations (with J. E. M. Braga and D. Moreira)
    Comm. Math. Phys. 367 (2019), no. 1, 1-32.
  14. On the fine structure of the free boundary for the classical obstacle problem (with J. Serra)
    Invent. Math. 215 (2019), no. 1, 311-366.
  15. A rigorous derivation from the kinetic cucker-smale model to the pressureless euler system with nonlocal alignment (with M.-J. Kang)
    Anal. PDE 12 (2019), no. 3, 843-866.
  16. Gradient stability for the Sobolev inequality: the case p≥2 (with R. Neumayer)
    J. Eur. Math. Soc. (JEMS) 21 (2019), no. 2, 319-354.
  17. On the Continuity of Center-Outward Distribution and Quantile Functions
    Nonlinear Anal. 177 (2018), part B, 413-421.
  18. Sharp boundary behaviour of solutions to semilinear nonlocal elliptic equations (with M. Bonforte and J. L. Vázquez)
    Calc. Var. Partial Differential Equations 57 (2018), no. 2, Art. 57, 34 pp.
  19. The sharp quantitative Euclidean concentration inequality (with F. Maggi and C. Mooney)
    Camb. J. Math. 6 (2018), no. 1, 59-87.
  20. Sharp global estimates for local and nonlocal porous medium-type equations in bounded domains (with M. Bonforte and J. L. Vázquez)
    Anal. PDE 11 (2018), no. 4, 945-982.
  21. An obstacle problem for conical deformations of thin elastic sheets (with C. Mooney)
    Arch. Ration. Mech. Anal. 228 (2018), no. 2, 401-429.
  22. Global well-posedness of the spatially homogeneous Kolmogorov-Vicsek model as a gradient flow (with M.-J. Kang and J. Morales)
    Arch. Ration. Mech. Anal. 227 (2018), no. 3, 869-896.
  23. Free boundary regularity in the parabolic fractional obstacle problem (with B. Barrios and X. Ros-Oton)
    Comm. Pure Appl. Math. 71 (2018), no. 10, 2129-2159.
  24. Global regularity for the free boundary in the obstacle problem for the fractional Laplacian (with B. Barrios and X. Ros-Oton)
    Amer. J. Math. 140 (2018), no. 2, 415-447.
  25. A quantitative analysis of metrics on R^n with almost constant positive scalar curvature, with applications to fast diffusion flows (with G. Ciraolo and F. Maggi)
    Int. Math. Res. Not. IMRN (2018), no. 21, 6780-6797.
  26. 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. 741 (2018), 275-294.
  27. Symplectic G-capacities and integrable systems (with J. Palmer and Á. Pelayo)
    Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 18 (2018), no. 1, 65-103.
  28. On the Lagrangian structure of transport equations: the Vlasov-Poisson system (with L. Ambrosio and M. Colombo)
    Duke Math. J. 166 (2017), no. 18, 3505-3568.
  29. Geometry of minimizers for the interaction energy with mildly repulsive potentials (with J. A. Carrillo and F. S. Patacchini)
    Ann. Inst. H. Poincaré Anal. Non Linéaire 34 (2017), no. 5, 1299-1308.
  30. Lipschitz changes of variables between perturbations of log-concave measures (with M. Colombo and Y. Jhaveri)
    Ann. Sc. Norm. Super. Pisa Cl. Sci., (5) 17 (2017), no. 4, 1491-1519.
  31. Regularity and Bernstein-type results for nonlocal minimal surfaces (with E. Valdinoci)
    J. Reine Angew. Math. 729 (2017), 263-273.
  32. Infinite speed of propagation and regularity of solutions to the fractional porous medium equation in general domains (with M. Bonforte and X. Ros-Oton)
    Comm. Pure Appl. Math. 70 (2017), no. 8, 1472-1508.
  33. Regularity of codimension-1 minimizing currents under minimal assumptions on the integrand
    J. Diff. Geom. 106 (2017), no. 3, 371-391.
  34. Quantitative stability for the Brunn-Minkowski inequality (with D. Jerison)
    Adv. Math. 314 (2017), 1-47.
  35. On the regularity of the free boundary in the p-Laplacian obstacle problem (with B. Krummel and X. Ros-Oton)
    J. Differential Equations 263 (2017), no. 3, 1931-1945.
  36. Partial W^{2,p} regularity for optimal transport maps (with S. Chen)
    J. Funct. Anal. 272 (2017), no. 11, 4588-4605.
  37. Rigidity and stability of Caffarelli's log-concave perturbation theorem (with G. De Philippis)
    Nonlinear Anal. 154 (2017), 59-70.
  38. Quantitative stability of the Brunn-Minkowski inequality for sets of equal volume (with D. Jerison)
    Chin. Ann. Math. Ser. B 38 (2017), no. 2, 393-412.
  39. Characterization of isoperimetric sets inside almost-convex cones (with E. Baer)
    Discrete Contin. Dyn. Syst. 37 (2017), no. 1, 1-14.
  40. Universality in several-matrix models via approximate transport maps (with A. Guionnet)
    Acta Math. 217 (2016), no. 1, 81-176.
  41. Weak KAM Theory for a Weakly Coupled System of Hamilton-Jacobi Equations (with D. Gomes and D. Marcon)
    Calc. Var. Partial Differential Equations 55 (2016), no. 4, 55-79.
  42. 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.
  43. 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.
  44. On the density function on moduli spaces of toric 4-manifolds (with Á. Pelayo)
    Adv. Geom. 16 (2016), no. 3, 291-300.
  45. BMO-type norms related to the perimeter of sets (with L. Ambrosio, J. Bourgain, and H. Brezis)
    Comm. Pure Appl. Math. 69 (2016), no. 6, 1062-1086.
  46. 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.
  47. Transport maps for β-matrix models and universality (with F. Bekerman and A. Guionnet)
    Comm. Math. Phys. 338 (2015), no. 2, 589-619.
  48. A note on the dimension of the singular set in free interface problems (with G. De Philippis)
    Differential Integral Equations 28 (2015), 523-536.
  49. Boundary ε-regularity in optimal transportation (with S. Chen)
    Adv. Math. 273 (2015), 540-567.
  50. 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.
  51. Partial regularity for optimal transport maps (with G. De Philippis)
    Publ. Math. Inst. Hautes Études Sci. 121 (2015), 81-112.
  52. 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.
  53. Generic hyperbolicity of Aubry sets on surfaces (with G. Contreras and L. Rifford)
    Invent. Math. 200 (2015), no. 1, 201-261.
  54. Quantitative stability for sumsets in Rn (with D. Jerison)
    J. Eur. Math. Soc. (JEMS) 17 (2015), no. 5, 1079-1106.
  55. 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.
  56. Optimal regularity of the convex envelope (with G. De Philippis)
    Trans. Amer. Math. Soc. 367 (2015), no. 6, 4407-4422.
  57. Closing Aubry sets II (with L. Rifford)
    Comm. Pure Appl. Math. 68 (2015), no. 3, 345-412.
  58. Closing Aubry sets I (with L. Rifford)
    Comm. Pure Appl. Math. 68 (2015), no. 2, 210-285.
  59. Strongly nonlocal dislocation dynamics in crystals (with S. Dipierro and E. Valdinoci)
    Comm. Partial Differential Equations 39 (2014), no. 12, 2351-2387.
  60. 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.
  61. Higher integrability for minimizers of the Mumford-Shah functional (with G. De Philippis)
    Arch. Ration. Mech. Anal. 213 (2014), no. 2, 491-502.
  62. 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.
  63. How to recognize convexity of a set from its marginals (with D. Jerison)
    J. Funct. Anal. 266 (2014), no. 3, 1685-1701.
  64. Regularity results for very degenerate elliptic equations (with M. Colombo)
    J. Math. Pures Appl. (9) 101 (2014), no. 1, 94-117.
  65. WKB analysis of Bohmian dynamics (with C. Klein, P. Markowich, and C. Sparber)
    Comm. Pure Appl. Math. 67 (2014), no. 4, 581-620.
  66. 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.
  67. 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. (5) 13 (2014), no. 3, 609-639.
  68. 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.
  69. On supporting hyperplanes to convex bodies (with Y.-H. Kim and R. J. McCann)
    Methods Appl. Anal. 20 (2013), no. 3, 261-271.
  70. Sobolev regularity for Monge-Ampère type equations (with G. De Philippis)
    SIAM J. Math. Anal. 45 (2013), no. 3, 1812-1824.
  71. 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.
  72. 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.
  73. 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.
  74. A note on interior W2,1+epsilon estimates for the Monge-Ampère equation (with G. De Philippis and O. Savin)
    Math. Ann. 357 (2013), no. 1, 11-22.
  75. 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.
  76. On the isoperimetric problem for radial log-convex densities (with F. Maggi)
    Calc. Var. Partial Differential Equations 48 (2013), no. 3-4, 447-489.
  77. 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.
  78. 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.
  79. W2,1 regularity for solutions of the Monge-Ampère equation (with G. De Philippis)
    Invent. Math. 192 (2013), no. 1, 55-69.
  80. Regularity of solutions to the parabolic fractional obstacle problem (with L. Caffarelli)
    J. Reine Angew. Math. 680 (2013), 191-233.
  81. A sharp stability result for the relative isoperimetric inequality inside convex cones (with E. Indrei)
    J. Geom. Anal. 23 (2013), no. 2, 938-969.
  82. 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.
  83. Non-Local Gradient Dependent Operators (with C. Bjorland and L. Caffarelli)
    Adv. Math. 230 (2012), no. 4-6, 1859-1894.
  84. 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.
  85. Total Variation Flow and Signed Fast Diffusion in one dimension (with M. Bonforte)
    J. Differential Equations 252 (2012), no. 8, 4455-4480.
  86. 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.
  87. 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.
  88. Isoperimetric-type inequalities on constant curvature manifolds (with Y. Ge)
    Adv. Calc. Var. 5 (2012), no. 3, 251-284.
  89. 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.
  90. Nearly round spheres look convex (with L. Rifford and C. Villani)
    Amer. J. Math. 134 (2012), no. 1, 109-139.
  91. 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.
  92. 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.
  93. Tangent cut loci on surfaces (with L. Rifford and C. Villani)
    Differential Geom. Appl. 29 (2011), no. 2, 154-159.
  94. When is multidimensional screening a convex program? (with Y.-H. Kim and R. J. McCann)
    J. Econom. Theory 146 (2011), no. 2, 454-478.
  95. 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.
  96. 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.
  97. Fine properties of minimizers of mechanical Lagrangians with Sobolev potentials (with V. Mandorino)
    Discrete Contin. Dyn. Syst. 31 (2011), no. 4, 1325-1346.
  98. 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.
  99. 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.
  100. Local semiconvexity of Kantorovich potentials on non-compact manifolds (with N. Gigli)
    ESAIM Control Optim. Calc. Var. 17 (2011), no. 3, 648-653.
  101. A mass transportation approach to quantitative isoperimetric inequalities (with F. Maggi and A. Pratelli)
    Invent. Math. 182 (2010), no. 1, 167-211.
  102. Partial regularity of Brenier solutions of the Monge-Ampère equation (with Y.-H. Kim)
    Discrete Contin. Dyn. Syst. 28 (2010), no. 2, 559-565.
  103. 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.
  104. On the Ma-Trudinger-Wang curvature on surfaces (with L. Rifford and C. Villani)
    Calc. Var. Partial Differential Equations 39 (2010), no. 3, 307-332.
  105. 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.
  106. Regularity properties of optimal maps between nonconvex domains in the plane
    Comm. Partial Differential Equations 35 (2010), no. 3, 465-479.
  107. On flows of H3/2-vector fields on the circle
    Math. Ann. 347 (2010), no. 1, 43-57.
  108. Mass Transportation on Sub-Riemannian Manifolds (with L. Rifford)
    Geom. Funct. Anal. 20 (2010), no. 1, 124-159.
  109. The optimal partial transport problem
    Arch. Ration. Mech. Anal. 195 (2010), no. 2, 533-560.
  110. Optimal transportation on non-compact manifolds (with A. Fathi)
    Israel J. Math. 175 (2010), no. 1, 1-59.
  111. 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.
  112. 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.
  113. Continuity of optimal transport maps and convexity of injectivity domains on small deformations of S2 (with L. Rifford)
    Comm. Pure Appl. Math. 62 (2009), no. 12, 1670-1706.
  114. 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.
  115. 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.
  116. A note on Cheeger sets (with F. Maggi and A. Pratelli)
    Proc. Amer. Math. Soc. 137 (2009), no.6, 2057-2062.
  117. On flows associated to Sobolev vector fields in Wiener spaces: an approach à la DiPerna-Lions (with L. Ambrosio)
    J. Funct. Anal. 256 (2009), no.1, 179-214.
  118. On the Hausdorff Dimension of the Mather quotient (with A. Fathi and L. Rifford)
    Comm. Pure Appl. Math. 62 (2009), no. 4, 445-500.
  119. A geometric lower bound on Grad's number
    ESAIM Control Optim. Calc. Var. 15 (2009), no. 3, 569-575.
  120. Geodesics in the space of measure-preserving maps and plans (with L. Ambrosio)
    Arch. Ration. Mech. Anal. 194 (2009), no. 2, 421-462.
  121. 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.
  122. 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.
  123. 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.
  124. Absolute continuity of Wasserstein geodesics in the Heisenberg group (with N. Juillet)
    J. Funct. Anal. 255 (2008), no. 1, 133-141.
  125. Invariant measures of Hamiltonian systems with prescribed asymptotic Maslov index (with A. Abbondandolo)
    J. Fixed Point Theory Appl. 3 (2008), no. 1, 95-120.
  126. Synchronized traffic plans and stability of optima (with M. Bernot)
    ESAIM Control Optim. Calc. Var. 14 (2008), no. 4, 864-878.
  127. A simple proof of the Morse-Sard theorem in Sobolev spaces
    Proc. Amer. Math. Soc. 136 (2008), no. 10, 3675-3681.
  128. Existence and uniqueness of martingale solutions for SDEs with rough or degenerate coefficients
    J. Funct. Anal. 254 (2008), no. 1, 109-153.
  129. 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.
  130. Strong displacement convexity on Riemannian manifolds (with C. Villani)
    Math. Z. 257 (2007), no. 2, 251-259.
  131. 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.
  132. Existence, uniqueness and regularity of optimal transport maps
    SIAM J. Math. Anal. 39 (2007), no. 1, 126-137.
  133. The Monge problem on non-compact manifolds
    Rend. Sem. Mat. Univ. Padova 117 (2007), 147-166.