Research
Here is the
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:
 The
Monge
problem
on noncompact manifolds, Rend. Sem. Mat. Univ. Padova, 117 (2007),
147166.
 Existence,
uniqueness
and regularity of optimal transport maps, SIAM J. Math. Anal., 39
(2007), no. 1, 126137.
 High
action
orbits
for Tonelli Lagrangians and superlinear Hamiltonians on
compact configuration spaces (with A. Abbondandolo), J. Differential Equations,
234 (2007), no. 2, 626653.
 Strong
displacement
convexity on Riemannian manifolds (with C. Villani), Math. Z., 257 (2007), no.
2, 251259.
 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, 497509.
 Existence
and
uniqueness of martingale solutions for SDEs with rough or
degenerate coefficients, J. Funct. Anal., 254 (2008), no. 1, 109153.
 A
simple proof of the MorseSard theorem in Sobolev spaces,
Proc. Amer. Math. Soc., 136
(2008), no. 10, 36753681.
 Geodesics
in
the
space of measurepreserving maps and plans (with L.
Ambrosio), Arch. Ration.
Mech. Anal., 194 (2009), no. 2, 421462.
 Synchronized traffic
plans
and stability of optima
(with M. Bernot), ESAIM
Control Optim. Calc. Var., 14 (2008), no. 4, 864878.
 Optimal
transportation
on noncompact manifolds (with A. Fathi), Israel J. Math., 175
(2010), no. 1, 159.
 Invariant
measures
of Hamiltonian systems with prescribed asymptotical Maslov
index (with A. Abbondandolo), J. Fixed Point Theory Appl., 3 (2008),
no. 1, 95120.
 A
geometric lower bound on Grad's number, ESAIM Control Optim. Calc. Var.,
15 (2009), no. 3, 569575.
 On
the
Hausdorff
Dimension of the Mather quotient (with A. Fathi and L.
Rifford), Comm. Pure Appl.
Math., 62 (2009), no. 4, 445500.
 Absolute
continuity
of Wasserstein geodesics in the Heisenberg group (with N.
Juillet), J. Funct. Anal.,
255 (2008), no. 1, 133141.
 An
approximation
lemma about the cut locus, with applications in optimal
transport theory (with C. Villani), Methods Appl. Anal., 15
(2008), no. 2, 149154.
 On
flows
associated to Sobolev vector fields in Wiener spaces: an
approach à la DiPernaLions (with L. Ambrosio), J. Funct. Anal., 256
(2009), 179214.
 Convergence
to
the viscous porous medium equation and propagation of chaos
(with R. Philipowski), ALEA Lat.
Am.
J. Probab. Math. Stat., 4 (2008), 185203.
 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, 137155.
 A note on
Cheeger sets (with F. Maggi and A. Pratelli), Proc. Amer. Math. Soc., 137
(2009), 20572062.
 The
optimal
partial transport problem, Arch. Ration. Mech. Anal., 195 (2010), no. 2,
533560.
 C^1
regularity
of solutions of the MongeAmpère equation for optimal
transport in dimension two (with G. Loeper), Calc. Var. Partial Differential Equations, 35
(2009), no. 4, 537550.
 Mass
Transportation
on SubRiemannian Manifolds (with L. Rifford), Geom. Funct. Anal., 20
(2010), no. 1, 124159.
 A
note on the regularity of the free boundaries in the
optimal partial transport problem, Rend.
Circ. Mat. Palermo, 58 (2009), no. 2, 283286.
 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, 16701706.
 On
flows
of
H^{3/2}vector fields on the circle, Math. Ann., 347 (2010), no.
1, 4357.

A
refined BrunnMinkowski inequality for convex sets
(with F. Maggi and A. Pratelli), Ann. Inst. H. Poincaré Anal. Non Linéaire,
26 (2009), no. 6, 25112519.
 Some
new
wellposedness
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, 18901920.
 Regularity
properties
of optimal maps between nonconvex domains in the plane, Comm. Partial Differential
Equations, 35 (2010), no. 3, 465479.
 Local
semiconvexity
of Kantorovich potentials on noncompact manifolds (with
N. Gigli), ESAIM Control
Optim. Calc. Var., 17 (2011), no. 3, 648653.
 A
new transportation distance between nonnegative measures,
with applications to gradients flows with Dirichlet boundary
conditions (with N. Gigli), J. Math. Pures Appl., 94 (2010), no. 2,
107130.
 On
the
MaTrudingerWang curvature on surfaces (with L. Rifford
and C. Villani), Calc. Var.
Partial Differential Equations, 39 (2010), no. 34,
307332.
 Almost
everywhere
wellposedness of continuity equations with measure initial
data (with L. Ambrosio), C. R. Math. Acad. Sci. Paris, 348 (2010), no.
56, 249252.
 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, 207252.
 Partial
regularity
of Brenier solutions of the MongeAmpère equation (with
Y.H. Kim), Discrete Contin.
Dyn. Syst., 28 (2010), no. 2, 559565.
 Global
in
time measurevalued solutions and finitetime aggregation for
nonlocal interaction equations (with J. A. Carrillo, M. Di
Francesco, T. Laurent, and D. Slepcev), Duke Math. J., 156
(2011), no. 2, 229271.
 A
mass transportation approach to quantitative isoperimetric
inequalities (with F. Maggi and A. Pratelli), Invent. Math., 182 (2010),
no. 1, 167211.
 Fine
properties
of minimizers of mechanical Lagrangians with Sobolev
potentials (with V. Mandorino), Discrete
Contin. Dyn. Syst., 31 (2011), no. 4, 13251346.
 On
the
shape of liquid drops and crystals in the small mass regime
(with F. Maggi), Arch.
Ration. Mech. Anal., 201 (2011), no. 1, 143207.
 Surface
measures
and
convergence of the OrnsteinUhlenbeck semigroup in Wiener
spaces (with L. Ambrosio), Ann. Fac. Sci. Toulouse
Math. (6), 20 (2011), no. 2, 407438.
 When
is
multidimensional
screening a convex program? (with Y.H. Kim and R. J.
McCann), J. Econom. Theory,
146 (2011), 454478.
 Nearly
round
spheres
look convex (with L. Rifford and C. Villani), Amer. J. Math., 134
(2012), no. 1, 109139.
 Tangent
cut
loci on surfaces (with L. Rifford and C. Villani), Differential Geom. Appl.,
29 (2011), no. 2, 154159.
 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, 11991242.
 NonLocal
TugofWar
and the Infinity Fractional Laplacian (with C. Bjorland
and L. Caffarelli), Comm.
Pure Appl. Math., 65 (2012), no. 3,
337380.
 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, 855876.
 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, 11311166.
 Isoperimetrictype
inequalities
on constant curvature manifolds (with Y. Ge), Adv. Calc. Var., 5 (2012),
no. 3, 251284.
 Confinement
in
nonlocal
interaction equations (with J. A. Carrillo, M. Di
Francesco, T. Laurent, and D. Slepcev), Nonlinear Anal., 75
(2012), no. 2, 550558.
 A
sharp stability result for the relative isoperimetric
inequality inside convex cones (with E. Indrei), J. Geom. Anal., 23 (2013),
no. 2, 938969.
 Semiclassical
limit
for
mixed states with singular and rough potentials (with M. Ligabò and T.
Paul), Indiana Univ. Math. J.,
61 (2012), no. 1, 193222.
 Regularity
of
solutions to the parabolic fractional obstacle problem
(with L. Caffarelli), J.
Reine Angew. Math., 680 (2013), 191233.
 Total
Variation
Flow and Signed Fast Diffusion in one dimension (with M.
Bonforte), J. Differential
Equations, 252 (2012), no. 8, 44554480.
 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, 114.
 Existence
of
Eulerian solutions to the semigeostrophic equations in
physical space: the 2dimensional periodic case (with L.
Ambrosio, M. Colombo, and G. De Philippis), Comm. Partial Differential
Equations, 37 (2012), no. 12, 22092227.
 NonLocal
Gradient
Dependent Operators (with C. Bjorland and L. Caffarelli),
Adv. Math., 230 (2012),
no. 46, 18591894.
 W^{2,1}
regularity
for solutions of the MongeAmpère equation (with G. De
Philippis), Invent. Math.,
192 (2013), no. 1, 5569.
 Sharp
stability theorems for the anisotropic Sobolev and logSobolev
inequalities on functions of bounded variation (with F.
Maggi and A. Pratelli), Adv.
Math., 242 (2013), 80101.
 Stability
for a GNS inequality and the LogHLS inequality, with
application to the critical mass KellerSegel equation
(with E. Carlen), Duke Math.
J., 162 (2013), no. 3, 579625.
 On
the isoperimetric problem for radial logconvex densities
(with F. Maggi), Calc. Var.
Partial Differential Equations, 48 (2013), no. 34,
447489.
 Asymptotics
of the sperimeter as s > 0 (with S. Dipierro, G.
Palatucci, and E. Valdinoci), Discrete Contin. Dyn. Syst.,
33 (2013), no. 7, 27772790.
 Bootstrap
regularity for integrodifferential operators, and its
application to nonlocal minimal surfaces (with B. Barrios
Barrera and E. Valdinoci), Ann.
Scuola
Norm. Sup. Pisa Cl. Sci., to appear.
 A
note on interior W^{2,1+\epsilon} estimates for the
MongeAmpère equation (with G. De Philippis and O. Savin),
Math. Ann., 357 (2013), no. 1, 1122.
 Second
order stability for the MongeAmpère equation and strong
Sobolev convergence of optimal transport maps (with G. De
Philippis), Anal. PDE, 6 (2013), no. 4, 9931000.
 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,
12511268.
 On
sets of finite perimeter in Wiener spaces: reduced boundary
and convergence to halfspaces (with L. Ambrosio and E.
Runa), Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur.
Rend. Lincei (9) Mat. Appl., 24 (2013), no. 1, 111122.
 WKB
analysis of Bohmian dynamics (with C. Klein, P. Markowich,
and C. Sparber), Comm. Pure
Appl. Math., 67 (2014), no. 4, 581620.
 Hölder
continuity and injectivity of optimal maps (with Y.H. Kim
and R. J. McCann), Arch.
Ration. Mech. Anal., 209 (2013), no. 3, 747795.
 Regularity
results for very degenerate elliptic equations (with M.
Colombo), J. Math. Pures
Appl. (9),
101 (2014), no. 1, 94117.
 Sobolev
regularity for MongeAmpère type
equations (with G. De Philippis), SIAM J. Math. Anal., 45
(2013), no. 3, 18121824.
 How
to recognize convexity of a set from its marginals (with
D. Jerison), J. Funct. Anal.,
266 (2014), no. 3, 16851701.
 On
supporting hyperplanes to convex bodies (with Y.H. Kim
and R. J. McCann), Methods
Appl. Anal., 20 (2013), no. 3, 261271.
 Closing Aubry
sets I (with L. Rifford), Comm. Pure Appl. Math., 68 (2015), no. 2,
210285.
 Closing Aubry
sets II (with L. Rifford), Comm. Pure Appl. Math., 68 (2015), no. 3,
345412.
 Optimal
regularity of the convex envelope (with G. De Philippis),
Trans. Amer. Math. Soc.,
367 (2015), no. 6, 44074422.
 An
excessdecay result for a class of degenerate elliptic
equations (with M. Colombo), Discrete Contin. Dyn. Syst. Ser. S, 7 (2014),
no. 4, 631652.
 Higher
integrability for minimizers of the MumfordShah functional
(with G. De Philippis), Arch.
Ration. Mech. Anal., 213 (2014), no. 2, 491502.
 A
general class of free boundary problems for fully nonlinear
elliptic equations (with H. Shahgholian), Arch. Ration. Mech. Anal.,
213 (2014), no. 1, 269286.
 A
general class of free boundary problems for fully nonlinear
parabolic equations (with H. Shahgholian), Ann. Mat. Pura Appl. (4),
194 (2015), no. 4, 11231134.
 Strongly
nonlocal dislocation dynamics in crystals (with S.
Dipierro and E. Valdinoci), Comm. Partial Differential
Equations, 39 (2014), no. 12, 23512387.
 Quantitative
stability for sumsets in R^n (with D. Jerison), J.
Eur. Math. Soc. (JEMS), 17 (2015), no. 5, 10791106.
 Generic
hyperbolicity of Aubry sets on surfaces (with G. Contreras
and L. Rifford), Invent. Math., 200 (2015), no. 1,
201261.
 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, 441507.
 Partial
regularity for optimal transport maps (with G. De
Philippis), Publ. Math. Inst. Hautes Études Sci., 121
(2015), 81112.
 On
the convexity of injectivity domains on nonfocal manifolds
(with T. Gallouët and L. Rifford), SIAM J. Math. Anal., 47
(2015), no. 2, 9691000.
 Boundary
εregularity in optimal transportation (with S. Chen), Adv.
Math., 273 (2015), 540567.
 Regularity
and Bernsteintype results for nonlocal minimal surfaces
(with E. Valdinoci), J. Reine Angew. Math., to appear.
 A
note on the dimension of the singular set in free interface
problems (with G. De Philippis), Differential Integral
Equations, 28 (2015), 523536.
 BMOtype
norms related to the perimeter of sets (with L. Ambrosio,
J. Bourgain, and H. Brezis), Comm. Pure Appl. Math., to
appear.
 Transport
maps for βmatrix models and universality (with F.
Bekerman and A. Guionnet), Comm. Math. Phys., 338
(2015), no. 2, 589619.
 Existence
and uniqueness of maximal regular flows for nonsmooth vector
fields (with L. Ambrosio and M. Colombo), Arch. Ration. Mech. Anal.,
218 (2015), no. 2, 10431081.
 On
the density function on moduli spaces of toric 4manifolds
(with Á. Pelayo), Adv. Geom., to appear.
 Rigidity
and sharp stability estimates for hypersurfaces with constant
and almostconstant nonlocal mean curvature (with G.
Ciraolo, F. Maggi, and M. Novaga), J. Reine Angew. Math.,
to appear.
 Nonlinear
bounds in Hölder spaces for the MongeAmpère equation
(with Y. Jhaveri and C. Mooney), J. Funct. Anal., 270 (2016), no. 10, 38083827.
 Stability
results on the smoothness of optimal transport maps with
general costs (with S. Chen), J. Math. Pures Appl. (9), 106 (2016), no. 2,
280295.
 Infinite
speed of propagation and regularity of solutions to the
fractional porous medium equation in general domains (with
M. Bonforte and X. RosOton), Comm. Pure Appl. Math., to
appear.
 Regularity
of codimension1 minimizing currents under minimal assumptions
on the integrand, J. Diff. Geom., to appear.
 Characterization
of isoperimetric sets inside almostconvex cones (with E.
Baer), Discrete Contin. Dyn. Syst., to appear.
 Weak
KAM Theory for a Weakly Coupled System of HamiltonJacobi
Equations (with D. Gomes and D. Marcon), Calc. Var.
Partial Differential Equations, 55 (2016), no. 4, 5579.
 Quantitative
stability of the BrunnMinkowski inequality for sets of equal
volume (with D. Jerison), Chin. Ann. Math., to
appear.
 Universality
in severalmatrix models via approximate transport maps
(with A. Guionnet), Acta Math., to appear.
 Quantitative stability for the BrunnMinkowski inequality
(with D. Jerison)
 On the Lagrangian structure of transport equations: the
VlasovPoisson system (with L. Ambrosio and M. Colombo)
 Global regularity for the free boundary in the obstacle
problem for the fractional Laplacian (with B. Barrios and X.
RosOton)
 Global wellposedness of the spatially homogeneous
KolmogorovVicsek model as a gradient flow (with M.J. Kang
and J. Morales)
 Gradient stability for the Sobolev inequality: the case p≥2
(with R. Neumayer)
 Lipschitz changes of variables between perturbations of
logconcave measures (with M. Colombo and Y. Jhaveri)
 Symplectic Gcapacities and integrable systems (with J.
Palmer and Á. Pelayo)
 The sharp quantitative Euclidean concentration inequality
(with F. Maggi and C. Mooney)
 A quantitative analysis of metrics on R^n with almost
constant positive scalar curvature, with applications to
Yamabe and fast diffusion flows (with G. Ciraolo and F. Maggi)
 Free boundary regularity in the parabolic fractional
obstacle problem (with B. Barrios and X. RosOton)
 Rigidity and stability of
Caffarelli's
logconcave perturbation theorem (with G. De Philippis)
 Partial W^{2,p}
regularity for optimal transport maps (with
S. Chen)
 Geometry of minimizers for the interaction energy with mildly
repulsive potentials (with J. A. Carrillo and F. S.
Patacchini)
 Lecture notes and
reviews:
 Optimal
Transport
and Curvature (with C. Villani), Nonlinear
PDE's
and applications, 171217, Lecture Notes in Math., 2028, Springer,
Heidelberg, 2011.
 Regularity
of
optimal
transport maps [after MaTrudingerWang and Loeper],
Séminaire Bourbaki. Vol. 2008/2009. Exposés 9971011. Astérisque, 332 (2010), Exp. No.
1009, ix, 341368.
 Cédric
Villani
reçoit
un prix de la Société Mathématique Européenne
(French) [Cédric Villani, 2008 EMS Prize] (with L.
Desvillettes), Gaz. Math.,
no. 120 (2009), 7681.
 Optimal
Transport.
Old
and New. [book review], Bull. Amer. Math. Soc. (N.S.), 47 (2010), no. 4, 723727.
 Almost
everywhere wellposedness of continuity equations with
measure initial data (with L. Ambrosio), C. R. Math.
Acad. Sci. Paris 348 (2010), no. 56, 249252.
 Lecture
notes
on variational models for incompressible Euler equations
(with L. Ambrosio), Optimal transportation, 5871, London
Math. Soc. Lecture Note Ser., 413, Cambridge Univ.
Press, Cambridge, 2014.
 Quantitative
isoperimetric
inequalities, with applications to the stability of liquid
drops and crystals, Concentration, functional
inequalities and isoperimetry, 7787, Contemp. Math., 545, Amer. Math. Soc.,
Providence, RI, 2011.
 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,
195204, IMA Vol. Math.
Appl., 153, Springer, New York, 2011.
 Variational
models
for the incompressible Euler equations (with S. Daneri),
HCDTE lecture notes. Part II. Nonlinear hyperbolic PDEs,
dispersive and transport equations, 51 pp., AIMS Ser.
Appl. Math., 7, Am. Inst. Math. Sci. (AIMS),
Springfield, MO, 2013.
 Stability
in geometric and functional inequalities, Proceedings of
the 6th European Congress of Mathematics.
 Sobolev
regularity for the MongeAmpère 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,
103118, 242; translation in J. Math. Sci. (N. Y.) 196
(2014), no. 2, 175183.
 Aubry
sets, HamiltonJacobi equations, and the Mañé Conjecture
(with L. Rifford), Geometric analysis, mathematical
relativity, and nonlinear partial differential equations,
83104, Contemp. Math., 599, Amer. Math. Soc.,
Providence, RI, 2013.
 The
MongeAmpère equation and its link to optimal transportation
(with G. De Philippis), Bull. Amer. Math. Soc. (N.S.), 51
(2014), no. 4, 527580.
 Partial
regularity results in optimal transportation (with G. De
Philippis), Trends in Contemporary Mathematics, Springer
INdAM Series, Volume 8, (2014), 293307
 Quantitative
stability results for the BrunnMinkowski inequality,
Proceedings of the ICM 2014.
 Stability
results for the BrunnMinkowski inequality, Colloquium
De Giorgi 2013 and 2014, 119127, Colloquia, 5, Ed.
Norm., Pisa, 2014.
 Perimeter
of sets and BMOtype norms (with L. Ambrosio, J.
Bourgain, and H. Brezis), C. R. Math. Acad. Sci. Paris, 352
(2014), no. 9, 697698.
 An
overview of unconstrained free boundary problems (with
H. Shahgholian), Philos. Trans. A, 373 (2015), no.
2050, 20140281, 11 pp.
 Global
existence for the semigeostrophic equations via Sobolev
estimates for MongeAmpère, CIME Lecture Notes,
Springer, to appear.
 A
transportation approach to universality in random matrix
theory, Boll. Unione Mat. Ital., to appear.
 Optimal
transportation
and actionminimizing measures, Thesis, Scuola Normale Superiore, Pisa,
2007. Tesi. Scuola Normale Superiore di Pisa (Nuova Series)
[Theses of Scuola Normale Superiore di Pisa (New Series)], 8.
Edizioni della Normale, Pisa, 2008. xx+254 pp.
 Autour
des inégalités isopérimétriques. (French) [On isoperimetric
inequalities] (with
W. Bench, C. de Franchis, L. Deproit, S. Gilles, B. Oh, A.
Tenne, K. Webster), Edited and with a preface by Figalli.
Éditions de l'École Polytechnique, Palaiseau, 2011. 124 pp.