Scuola Normale Superiore di Pisa

Born in 1963 in Alba, Italy

EDUCATION

1985-1988 Scuola Normale Superiore
Corso di Perfezionamento (PHD).

1981-1985 Scuola Normale Superiore
Pisa, Italy.

1981-1985 Laurea in Matematica (B.S.)
University of Pisa, Italy. Summa cum laude.
Advisor: Prof. Ennio De Giorgi. Thesis in semicontinuity and relaxation of
integral functionals defined on Sobolev spaces.

PROFESSIONAL EXPERIENCE

1997–present Full Professor
Scuola Normale Superiore, Pisa, Italy.

1995–1997 Full Professor
Department of Mathematics, University of Pavia, Italy.

1994–1995 Full Professor
University of Benevento, Italy.

1992–1994 Associate Professor
University of Pisa, Italy.

1988-1992 Research Assistant
Second University of Rome, Italy

HONORS
1991 “Bortolozzi” Prize of Unione Matematica Italiana.
1996 National Prize for Mathematics and Mechanics of the italian Minister of Education.
1996 Invited speaker at the 2nd European congress of mathematicians in Budapest.
1999 “Caccioppoli” Prize of the Unione Matematica Italiana.
1999 Editor of “Interfaces and Free Boundaries”, Main Editor of the “Journal of the European
Mathematical Society”
2000 Editor of “Archive for Rational Mechanics and Analysis”
2002 Invited speaker at the International Congress of Mathematicians in Beijing, in the
PDE session
Long term visiting scientist in the Max Planck Institute of Leipzig, the Massachussets
Institut of Technology, the ETH of Zurich, the Institute Henri Poincar´e
2003 Awarded with the international Fermat Prize of the University of Toulouse (France)
2005 Managing Editor of “Calculus of Variations and Partial Differential Equations”
2005 Editor of Communications in Partial Differential Equations
2005 Socio Corrispondente Accademia Nazionale dei Lincei, Roma
2006 Socio Corrispondente Istituto Lombardo, Accademia di Scienze e Lettere, Milano
2007 Editor of Communications in Mathematical Sciences

TEACHING ACTIVITY
Has been teaching courses on Γ-convergence, Measure Theory, Elementary Probability, Non Linear
PDE’s, BV functions, Minimal Surfaces and Geometric Measure Theory, Functional Analysis,
Viscosity solutions, Calculus of Variations.

[1] Nuovi risultati sulla semicontinuità inferiore di certi funzionali integrali, Rend. Accad. Naz. dei Lincei, Cl. Sci. Fis. Mat. Natur, 79, 82–89, 1985.

[2] New lower semicontinuity results for integral functionals, Rend. Accad. Naz. Sci. XLMem. Mat. Sci. Fis. Natur., 105, 1–42, 1987.

[3] Integral representation and relaxation for functionals defined on measures (withE.DeGiorgi and G.Buttazzo), Rend. Accad. Naz. dei Lincei, Cl. Sci. Fis. Mat. Natur., 81, 7–13, 1987.

[4] Weak lower semicontinuous envelope of functionals defined on a space of measures (with G.Buttazzo),Ann. Matem. Pura edAppl.,(IV) 110, 311–340, 1988.

[5] Continuous operators of the form Tf(u)= f(x, u, Du)(with G.Buttazzo and A.Leaci), Boll. Un. Mat. Ital., 7 2-B, 99–108, 1988.

[6] A lower closure theorem for autonomous orientor fields, Proceedings of the Royal Society of Edinburgh, 110-A, 249–254, 1988.

[7] Relaxation of autonomousfunctionals withdiscontinuousintegrands,Ann. Univ. diFerrara, 34, 21–47, 1988.

[8] Lipschitz regularity for minimizers of integral functionals with highly discontinuous coefficients (with O.Ascenziand G.Buttazzo), J. Math. Anal. Appl., 142, 301–316, 1989.

[9] Un nuovo tipo di funzionale del Calcolo delle Variazioni (with E.De Giorgi), Rend. Accad. Naz. Lincei, Cl. Sci. Fis. Mat. Natur., 82, 199-210, 1988.

[10] A compactness theorem for a new class of functions of bounded variation, Boll. Un. Mat. It, 3-B, 857–881, 1989.

[11] Existence theory for a new class of variational problems, Archive for Rational Mechanics & Analysis, 111, 291–322, 1990.

[12] Functionals defined on partitions in sets of finite perimeter I: integral representation and Γ-convergence (withA.Braides), JournaldeMathematiques Pures etAppliquees,69, 285–306, 1990.

[13] Functionals defined on partitions in sets of finite perimeter II: semicontinuity, relaxation, homogenization (with A.Braides), Journal de Mathematiques Pures et Appliquees, 69, 307– 333, 1990.

[14] A general chain rule for distributional derivatives (with G.Dal Maso), Proc. Amer. Mat. Soc., 108, 691–702, 1990.

[15] Functionals with linear growth defined on vector valued BV functions (with S.Mortola and V.M. Tortorelli), Journal de Mathematiques Pures et Appliquees, 70, 269–323, 1991.

[16] Approximation offunctionalsdepending onjumpsby ellipticfunctionals via Γ-convergence (with V.M.Tortorelli), Communications on Pure and Applied Mathematics, 43, 999–1036, 1990.

[17] Variational problems in SBV, Acta Applicandae Mathematicae, 17, 1–40, 1989.

[18] Metric space valued functions of bounded variation, Annali della Scuola Normale Superiore, 17, 439–478, 1990.

[19] A boundary value problem for nematic liquid crystals with variable degree of orientation (with E.G.Virga), Archive for Rational Mechanics& Analysis, 114, 335–347, 1991.

[20] Existence of minimal energy configurations of nematic liquid crystals with variable degree of orientation, Manuscripta Mathematica, 68, 215–228, 1990.

[21] Regularity of solutions of a degenerate elliptic variational problem, Manuscripta Mathematica, 69, 285–306, 1990.

[22] Hölder continuity of solutions of one dimensional Lagrange problems of the Calculus of Variations (with O.Ascenzi), Ricerche di Matematica, 15, 311–319, 1991.

[23] On the approximation of free discontinuity problems (with V.M.Tortorelli), Boll. Un. Mat. Ital., 6-B, 105–123, 1992.

[24] Singular perturbation problem with a compact support semilinear term (withG. Albertiand G. Buttazzo), Asymptotic Analysis, 8, 247–258, 1994.

[25] Integral representation of relaxed functionals on BV (Ω;Rk)and polyhedral approximation (with D.Pallara), Indiana University Math. J., 42, 295–321, 1993.

[26] Gamma-convergence andtheleast squares method (withP.D’Ancona andS.Mortola),Annali di Matematica Pura e Applicata, 146, 101–127, 1994.

[27] On the representation in BV Ω;Rm of quasi-convex integrals (with G.Dal Maso), Journal of Functional Analysis, 109, 76–97, 1992.

[28] The space SBV (Ω)and free discontinuity problems,Attidel convegno: Variational andfree boundary problems, editors AmbrosioFriedman and J.Spruck, serie IMA volumes in mathe­matics and its applications, Springer 1993.

[29] Free discontinuity problems and the Calculus of Variations, Atti del simposio Italo-Franco­Russo,INRIASophia-Anthipolis,24-27Settembre1991, editoreINRIA,DomainedeVoluceau, Rocquencourt, B.P. 105 78153 Le Chesnay Cedex.

[30] On singularities of convex functions (with G.Albertiand P.Cannarsa), Manuscripta Mathe­matica, 76, 421–436, 1992.

[31] Su alcune proprietà delle funzioni convesse, Rend. Mat. Accad. Naz. Lincei ser. IX, 3, 195–202, 1992.

[32] On the propagation of singularities of semi-convex functions (with P.Cannarsa and H.M.Soner), Ann. Sc. Norm. Sup., 20, 597–616, 1993.

[33] An optimal design problem with perimeter penalization (with G.Buttazzo), Calc. Var., 1, 55–69, 1993.

[34] On thelower semicontinuity ofquasi-convexfunctionalsin SBV, NonlinearAnalysis,Theory Methods and Applications, 23, 405–425, 1994.

[35] Approximation problems for curvature varifolds (with D.Pallara and M.Gobbino), Journal of Geometric Analysis, 8, 1–19, 1998.

[36] Lower semicontinuityproblemsinSobolev spaces with respect to a measure (withG.Buttazzo and I.Fonseca), Journal de Mathematiques Pures et Appliquees 75 (1996), 211–224.

[37] A new proof of the SBV compactness theorem, Calc. Var., 3, 127–137, 1995.

[38] A level set approach for the evolution of surfaces of any codimension (with H.M.Soner), J. Diff.. Geom. 43 (1996), 693–737.

[39] Partial regularity of free discontinuity sets I (with D.Pallara), Ann. Scuola Norm. Sup. Pisa, 24 (1997), 1–38.

[40] Partial regularity offreediscontinuity setsII (with N.Fusco and D.Pallara), Ann. Scuola Norm. Sup. Pisa, 24 (1997), 39–62.

[41] Movimenti minimizzanti, Rend. Accad. Naz. Sci. XL Mem. Mat. Sci. Fis. Natur., 113, 191–246, 1995.

[42] Flow by mean curvature of surfaces of any codimension,(with H.M.Soner) in: Variational Methods for Discontinuous Structures,Como1994,R.SerapionieF.TomarelliEds.,Birkh¨auser, Boston, 1996.

[43] Specialfunctions withbounded variation and with weaklydifferentiabletraces onthejump set (with AmbrosioBraides and AmbrosioGarroni)NoDEA, 5 (1998), 219–243.

[44] Energies in SBV and variational models in fracture mechanics (withA.Braides)Proceedings of theEurHomogenizationcongress,Nizza,GakutoInt. Series,Math. Sci. andAppl., 9, 1–22, 1997.

[45] Ennio De Giorgi, matematico e amante della sapienza Rendiconti Istituto Lombardo, Accademia di Scienze e Lettere, 131 (1997), 337–369.

[46] Free discontinuity problems and special functions with bounded variation. Proceedings ECM2 Budapest 1996, Progress in Mathematics, 168 (1998), 15–35.

[47] Fine properties of functions in BD (with Coscia and G.Dal Maso), Arch. Rat. Mech. Anal. 139 (1997), 201–238.

[48] Curvature and distance function from a manifold. (with C.Mantegazza) J. Geom. Anal., 8 (1998), 719–744.

[49] Higher regularity of solutions of free discontinuity problems. (with N.Fusco and D.Pallara), Differential and Integral Equations, 12 (1999), 499–520.

[50] A geometric approach to monotone functions in Rn . (with G. Alberti) Matematische Zeitschrift, 230 (1999), 259–316.

[51] Partial regularity in free discontinuity problems. (con D. Pallara)Progress in partial differential equation: the Metz surveys 4, M. Chipot and I. Shafrir Eds., Pitman Research Notes in Mathematics 345 (1996), Longman, Harlow, 3–17.

[52] Corso introduttivo alla teoria geometrica della misura ed alle superfici minime. Quaderno della Scuola Normale Superiore di Pisa, Pantograf, Genova, 1997.

[53] Lecture Notes on geometric evolution problems, distance function and viscosity solutions. In “Calculus of Variations and Partial Differential Equations”, G.Buttazzo, A.Marino, M.K.V.Murthy Eds, Springer, 5–93, 1999.

[54] A measure theoretic approach to higher codimension mean curvature flows. (with H.M.Soner) Annali della Scuola Normale Superiore di Pisa, Cl. Sci. IV, 25 (1997), 27–49.

[55] Moto per curvatura mediAmbrosio (conG.Bellettini)VI appendice all’ Enciclopediaitaliana di Scienze, Lettere ed Arti, Treccani Editrice, 1998.

[56] Partial regularityfor thequasi minimizers ofperimeter. (withE.Paolini)RicerchediMatematica, Vol. XLVIII (numero speciale dedicato a Ennio De Giorgi) (1999), 167–186. Errata corrige: (2001), 191-193.

[57] On a volume constrained variationalproblem. (withI.Fonseca, P.Marcellini, L.Tartar)Arch. Rat. Mech. Anal., 149 (1999), 23–47.

[58] Lineenergiesforgradientvector fieldsintheplane. (withC.DeLellis eC.Mantegazza)Calc. Var., 9 (1999), 327–355.

[59] Rectifiable sets in metric and Banach spaces. (with B.Kirchheim)Mathematische Annalen, 318 (2000), 527–555.

[60] Currents in metric spaces. (with B.Kirchheim)Acta Mathematica, 185 (2000), 1–80.

[61] On the aymptotic behaviour of anisotropic energies arising in the cardiac bidomain model. (with P.Colli Franzone and G.Savar`e) Interfaces and Free Boundaries, 2 (2000), 213–266.

[62] Connected components of sets of finite perimeter and applications to image processing. Journal of the European Mathematical Society, 3 (2001), 39–92.

[63] Semilinear Elliptic equations in R3 and a conjecture of De Giorgi. (with X.Cabr´e) Journal of the AMS, 13 (2000), 725–739.

[64] Variational approximation of a second order free discontinuity problem in computer vision. (with L.Faina and R.March)SIAM J. Math. Anal., 32 (2001), 1171–1197.

[65] Some fine properties of sets of finite perimeter in Ahlfors regular metric measure spaces. Advances in Mathematics, 159 (2001), 51–67.

[66] Functions of Bounded Variation and Free Discontinuity Problems. (with N.Fusco and D.Pallara) Oxford University Press, 2000.

[67] Some regularity resultsfor minimal crystals. (withM.Novaga andE.Paolini)ESAIMCOCV, 8 (2002), 69–103.

[68] On a long standing conjecture of De Giorgi: old and recent results. (with G.Alberti and X.Cabré) Acta Applicandae Mathematicae 65 (2001), special volume dedicated to A.Avantaggiati, 9–33.

[69] Lecture Notes on Optimal Transport Problems. Lecture Notes in Mathematics “Mathematical aspects of evolving interfaces” (CIMESeries,Madeira(PT),2000)1812,P.Colli andJ.F. Rodrigues Eds., 1–52, 2003.

[70] Selected topicson “AnalysisinmetricSpaces”. (withP.Tilli)QuadernodellaScuolaNormale Superiore, 2000 and Oxford Lecture Series in Mathematics 25, 2003.

[71] Higher integrability of the gradient and dimension of the singular set for minimisers of the Mumford–Shah functional. (with N.Fusco and J.Hutchinson)Calc. Var., 16 (2003), 187–215.

[72] Fine properties of sets of finite perimeter in doubling metric measure spaces. Set Valued Analysis, 10 (2002), 111-128.

[73] A viscosity property of minimizing micromagnetic configurations. (with M.Lucumberryand T.Riviere) Communications on Pure and Applied Mathematics, 56 (2003), 681–688.

[74] Weak differentiability of BV functions on sub-Riemanniangroups. (withV.Magnani)Math. Z., 245 (2003), 123–153.

[75] Existence and stability results in the L1-theory of optimal transportation. (with A.Pratelli) LectureNotesinMathematics “Optimal transportationand applications” (CIMESeries,Mar­tina Franca, 2001) 1813, L.A. Caffarelli and S. Salsa Eds., 123–160, 2003.

[76] On the rectifiability of defect measures arising in a micromagnetics model. (with B.Kirchheim,M.LucumberryandT.Riviere)NonlinearProblemsinMathematical Physics and related topics II, in honor of O.A. Ladyzhenskaya. M.S. Birman, S. Hildebrandt, V.A. Solon­nikov andN.UraltsevaEds.,InternationalMathematicalSeries,Kluwer/Plenum,2002(pages 29–60).

[77] Adirect variational approach toaproblemarisinginimagereconstruction. (withS.Masnou) Interfaces and Free Boundaries, 5 (2003), 63–82.

[78] On a variational problem arising in image reconstruction. (with S.Masnou) International Series ofNumericalMathematics(P.Colli,C.Verdi,A.VisintinEds.), 147 (2003), 17–26.

[79] Optimal mass transportation in the Heisenberg group. (with S.Rigot)Journal of Functional Analysis, 208 (2004), 261–301.

[80] Transport equation and Cauchy probem for BV vector fields. Inventiones Mathematicae, 158 (2004), 227–260.

[81] Existence of solutions for a class of hyperbolic systems of conservation laws in several space dimensions. (with C.De Lellis)IMRN, 41 (2003), 2205–2220.

[82] Existence of optimal transport maps for crystalline norms. (with B.Kirchheim and A.Pratelli) Duke Math. J., 125 (2004), 207–241.

[83] Well posedness for a class of hyperbolic systems of conservation laws in several space dimen­sions. (with C.De Lellis and F.Bouchut)Comm. PDE, 29 (2004), 1635–1651.

[84] Gradient flows in metric spaces and in the space of Probability measures. (with N.Gigliand G.Savar`e) Lectures in Mathematics ETH Zurich, Birkh¨auser, 2005.

[85] Special functions of bounded variation in doubling metric measure spaces. (withM.Miranda and D.Pallara). In “Calculus of Variations: topics from the mathematical heritage of E. De Giorgi”,Quaderni di Matematica 14 (2004),IIUniversitàdiNapoli(D.Pallara ed.),1–45.

[86] Traces and fine properties of a BD class of vector fields and applications. (with G.Crippa and S.Maniglia). Ann. Sci. Toulouse, XIV (4)(2005), pp. 527–561.

[87] A note on admissible solutions of 1d scalar conservation laws and 2d Hamilton–Jacobi equa­tions. (withC.DeLellis). Journal o fDifferentialHyperbolicEquations, 1 (4)(2004), 813–826.

[88] Stability of flows associated to gradient vector fields and convergence of iterated transport maps. (with G.Savar´e and S.Lisini) Manuscripta Mathematica, 121 (2006), 1–50.

[89] Problemadi trasportoeequazionediCauchypercampi vettoriali a variazionelimitata. Atti XVII Convegno UMI, Milano 8-13 Settembre 2003, Edito da Monograf, 2004.

[90] Gradient flowswith metric and differentiablestructures,and applicationstotheWasserstein space. (with N.Gigliand G.Savar´e) Rend. Mat. Acc. Lincei, 15 (9) (2004), 327–343.

[91] Lipschitz regularity and approximate differentiability of the DiPerna-Lions flow. (with M.Lecumberry and S.Maniglia) Rendiconti del Seminario Fisico Matematico di Padova, 114 (2005), 29–50.

[92] Multiscale Young measures in almost periodic homogenization and applications. (with H.Frid) Preprint, 2005

[93] Very weak notions of differentiability. (with J.Mal´y) Proceedings of the Royal Society of Edinburgh, 137 A, 447–455, 2007.

[94] On the chain rule for the divergence of BV like vector fields: applications, partial results, open problems. (with C. De Lellis and J.Mal´y) In “Perspectives in Nonlinear Partial Differential Equations: in honour of Haim Brezis”, Contemporary Mathematics, 446 (2007), 31–67, AMS.

[95] Intrinsic regular hypersurfaces in the Heisenberg group. (with F.Serra Cassano and D.Vittone). Journal of Geometric Analysis, 16(2)(2006), 187–232.

[96] The flow associated to weakly differentiable vector fields: recent results and open problems. Boll. UMI,(8) 10-B (2007), 25–41.

[97] Hamiltonian ODE’s in the Wasserstein space of probability measures. (with W.Gangbo) Communications in Pure and Applied Mathematics, LXI (2008), 18–53.

[98] Necessary optimality conditions for geodesics in weighted Wasserstein spaces. (with F.Santambrogio) Rendiconti Lincei, Series IX, 43 23–37, 2007.

[99] Gradient Flows of Probability Measures. (with G.Savar´e) In Volume 3, Evolutionary equa­tions, of Handbook of Differential Equations, C.M. Dafermos and E. Feireisl Eds., Elsevier, 2007.

[100] Approximation of geometric densities of random closed sets. (with F.Capasso and E.Villa)

[101] Geodesics in the space of measure-preserving maps and plans. (with A.Figalli)

[102] A gradient flow approach to an evolution problem arising in superconductivity. (with S.Serfaty)

[103] Existence and stability results for Fokker-Planck equations with log-concave reference mea­sures. (with G.Savarè, L. Zambotti)

[104] Leaf superposition property for integer rectifiable currents. (with G. Crippa, P. Le Floch)

[105] On the regularity of the pressure field of Brenier’s weak solutions to incompressible Euler equations. (with A. Figalli)

[106] Rectifiability of setsof finiteperimeterinCarnotgroups: existenceof atangenthyperplane. (with E.Le Donne, B.Kleiner)

[107] One-sided Minkowski content for some classes of closed sets and applications to stochastic geometry. (with A. Colesanti, E. Villa)

[108] DiPerna–Lions theory in Wiener spaces. (with A.Figalli)

[109] Transport equation and Cauchy problem for non-smooth vector fields. Lecture Notes in Mathematics “Calculus of Variationsand Non-Linear Partial Differential Equations” (CIME Series, Cetraro, 2005) 1927, B. Dacorogna, P. Marcellini eds., 2–41, 2008.