Calculus of variations, geometric theory of measurement and geometric analysis

Several problems of Non Linear Analysis are studied with techniques of Calculus of Variations and Geometric Theory of Measurement. The most recent results of the group in this field include the theory of optimal transport, with applications to the study of irrecgular boundary metric spaces, matching and evolution problems, and geometric equations with applications in Riemannian geometry, as well as the theory of curves and surfaces and  physical-mathematical problems.

Activities carried out in 2019.

(Projects, conferences, workshops, seminars, exhibitions, summer schools, prizes and awards)

Organized conferences

Ambrosio Alessio Figalli, Fields Medallist, 14-17 gennaio Convegno Nazionale di Calcolo delle Variazioni, 4-8 febbraio 2019

Di Marino People in Optimal Transportation and Applications Incontri Indam 2019 Cortona (ITALY), June 24-28, 2019

Malchiodi Variational problems, PDEs and applications, Pisa, January 2019. Recent trends in Geometric analysis and applications, Centro de Giorgi, Pisa, November 2019. Variational methods, with applications to problems in mathematical physics and geometry, Venice, November 2019.


Ambrosio: Balzan Prize

Borrelli: Prix Jeune Chercheur 2019 della Fondation Dauphine.

Projects and research groups

PRIN projects

2015, 2017, GNAMPA, various projects with non-Italian universities.



  • CMI Winter school, Toulouse, 18-22 febbraio, 2019.
  • Optimal Transport and Geometric Analysis, Venezia, 1-5 aprile 2019.
  • Some topics of Geometric Analysis and Geometric Measure Theory, 16-17 aprile, Centro De Giorgi, Pisa


  • Problemas de curvatura prescrita en variedades con borde.
  • Seminar of Young Researchers of the University of Granada. IEMath-GR (University of Granada), Granada (Spain). 28/05/2019
  • Curvature Prescription Problems on Manifolds with Boundary. Séminare Jeunes Chercheurs de la Universidad de Cergy-Pointoise (University of Cergy-Pontoise), Pontoise (France). 12/06/2019

Di Marino 01/19

  • Optimal Transport Methods in Density Functional Theory, Banff 03/19
  • Workshop on optimal transport and applications, Toulouse 04/19 Duality for the Entropic Optimal Transport Problem and Applications, EPFL 07/19
  • Eight or Nine Talks on Contemporary Optimal Transport Problems, Strasbourg 09/19, Second joint IMU-INdAM Conference in Analysis, Naples

Jevnikar 06/2019

  • PDE Workshop, Wuhan Institute of Physics and Mathematics, Chinese Academy ofSciences, China
  • 05/2019 BIRS Workshop – Nonlinear Geometric PDEs, Banff, Canada


  • Conference in Geometry and PDEs, Univ. Granada, January 2019.
  • Nonlinear Meeting in Turin, January 2019.
  • PDE seminar, Rome La Sapienza, February 2019. Workshop on Geometric Functionals, IAS, Princeton, March 2019.
  • Some topics of Geometric Analysis and Geometric Measure Theory, Pisa, April 2019.
  • Nonlinear Geometric PDE's, Banff, May 2019.
  • Analytic and Geometric Aspects of PDEs, Milan, May 2019. Workshop on Geometric Analysis, Toulouse, June 2019.
  • Partial Differential Equations, MFO, Oberwolfach, July 2019. Geometric Analysis, Submanifolds and Geometry of PDE?s, Politecnico di Torino, September, 2019.
  • Second Joint IMU-INDAM Meeting in Analysis, Naples, September 2019.
  • Geometric analysis seminar, Princeton, November 2019.


  • Workshop on Elastic Curves 6.-8. May 2019
  • Mathematisches Institut Albert-Ludwigs-Universität Freiburg 8th May 2019


  • 07/02/2019 Rigidità nella disuguaglianza di Bakry-Èmery per esponente p=1 e insiemi di perimetro finito su spazi RCD(K,N), CDV conference, Levico Terme (Italy).
  • 14/02/2019 New results about the structure theory of RCD(K,N) metric measure spaces, Analysis seminar, Mathematics Institute, University of Warwick, Coventry.
  • 28/02/2019 Polya-Szego Inequality and Dirichlet p-Spectral gap for non-smooth spaces with Ricci curvature bounded below, Analysis seminar, SISSA, Trieste. 24/03/2019
  • Regularity of Lagrangian flows and lower Ricci curvature bounds, AMS meeting, University of Hawaii at Manoa, Honolulu, Hawaii.
  • 01/04/2019 Constancy of the dimension for RCD(K,N) spaces via regularity of Lagrangian flows, Optimal Transport and Geometric Analysis conference, Venezia.
  • 26/06/2019 Quantitative Obata’s theorem via localization, HIM, Bonn.
  • 02/10/2019 Quantitative Obata’s theorem via localization, Analysis seminar, University of Jyväskylä.
  • 24/10/2019 Structure theory of RCD(K,N) spaces up to codimension 1, PDE CDT Lunchtime seminar, University of Oxford.
  • 04/12/2019 A quantitative Obata theorem via localization, Analysis seminar, University of Pisa.
  • 16/12/2019 A quantitative Obata theorem via localization, Séminaire de Géometrié, Institut de Mathématiques de Jussieu-Paris Rive Gauche.


  • A distributional approach to fractional Sobolev spaces and fractional variation: asymptotic analysis, \textit{Analysis seminar}, Padova, 11 December 2019
  • A distributional approach to fractional Sobolev spaces and fractional variation, \textit{International Conference on Elliptic and Parabolic Problems -- Mini-symposium ``Regularity in Nonlinear Elasticity''}, Gaeta, 20--24 May 2019
  • A distributional approach to fractional Sobolev spaces and fractional variation, Basel, 24 April 2019
  • Carnot groups from an RCD point of view: heat, entropy and their properties, \textit{Seminario Dottorandi SNS}, Pisa, 21 January 2019


  • 23/10/2019 - "Some results on codimension one rectifiability for subsets in Carnot groups" (Seminar, Sala Seminari, Dipartimento di Matematica, Università di Pisa)
  • 26/09/2019 - Presentation of the poster "On a smooth hypersurface in a Carnot group which is not Lipschitz parametrizable by countably many tangents" in the conference "Geometry and Analysis: celebrating the mathematics of Pierre Pansu" (Poster, Oxford)
  • 14/03/2019 - "How can harmonic functions split your manifold?" (Seminar, Sala Seminari, Dipartimento di Matematica, Università di Pisa)


  • 03/06/2019 - Scuola Normale Superiore, Pisa
  • 20/06/2019 - 3 Days on Evolution PDEs (invited short talk), Agropoli.


  • Propagation of regularity for solutions of the incompressible continuity equation with non Lipschitz velocity field. Seminar at SISSA (Trieste), February 2019.
  • Rigidity of the $1$-Bakry-\'Emery inequality and applications. 1147 AMS meeting in Honolulu. Special Session on Analysis of Nonlinear Geometric Equations, I. Organized by Aaron Naber and Richard Bamler, March 2019.
  • Rigidity of the 1-Bakry-Émery inequality and sets of finite perimeter on RCD spaces. Conference ``Optimal Transport and Geometric Analysis'' (Venice), April 2019.
  • Structure theory for sets of finite perimeter in RCD spaces. Seminar at Hausdorff center for mathematics (Bonn), June 2019.
  • Propagation of regularity for solutions of the transport equation with non-Lipschitz velocity field. Conference ``GMT and PDEs in Basel -- A young researchers meeting'' (Basel), July 2019.
  • Structure theory for sets of finite perimeter on RCD spaces. Seminar at department of mathematics and statistics university of jyv\"askyl\"a, September 2019.
  • Constancy of the dimension for non-smooth spaces with Ricci bounded below, via regularity of Lagrangian flows. Colloquium talk at Shanghaitech University (Shanghai), November 2019.
  • Constancy of the dimension for RCD spaces via regularity of Lagrangia flows. Seminar at Oberwolfach conference ``Heat Kernels, Stochastic Processes and Functional Inequalities'' (invited participation), November 2019.
  • Propagation of regularity for transport equations with non-Lipschitz velocity fields. Seminar at ``Seminari d'analisi de Barcelona'', December 2019. Comi XXIX Convegno Nazionale di Calcolo delle Variazioni, Levico Terme, Italy, 4-8 February 2019.
  • Talk: “A distributional approach to fractional Sobolev spaces and fractional variation: existence of blow-up”.


  • Aug. 2019: "Yang-Mills-Higgs-Dirac", Talk in Workshop on geometric analysis, AMSS, CAS, Beijing, China.
  • Aug. 2019 : "Blowup analysis and energy identities for a nonlinear sigma model", Talk in Conference on Sharp geometric inequalities and applications to geometry and PDEs, TSIMF, Sanya, China. Main External Collaborators

Daniele Bartolucci (Rome Tor Vergata), Raffaele Carlone (Università Federico II di Napoli), Jih-Hsin Cheng (Acad. Sinica, Taipei), Maria Colombo (EPFL), Massimo Fornasier (TUM, Monaco), Nicola Gigli (SISSA), Matthew Gursky (Notre Dame Univ.), Philipp Harms (Freiburg), Juergen Jost (Max Planck, Leipzig), Enno Kessler (Harvard University),Tobias Lamm (KIT, Karlsruhe), Mathieu Lewin (CNRS, Paris Dauphine), Danka Lučić (Jyvaskyla University), Valentino Magnani (Univ. Pisa), Luca Martinazzi (Univ. Padova), Luciano Mari (Univ. Torino), Mario Micallef (Uniuv. Warwick), Andrea Mondino (Univ. Oxford), Matteo Novaga (Pisa), Mark A. Peletier (Eindhoven), David Ruiz (Univ. Granada), Filippo Santambrogio (Université Lyon 1), Giuseppe Savare' (Pavia), Lorenzo Tentarelli (Politecnico di Torino). Paul Yang (Princeton Univ.), Miaomiao Zhu (Shanghai Jiao Tong University)

Publications 2019

  • Ambrosio, Brue, Semola: Rigidity of the $1$-Bakry-\'Emery inequality and sets of finite perimeter in $RCD$ spaces, GAFA.
  • Ambrosio, Glaudo: Finer estimates on the matching problem on 2-dimensional manifolds. Journal of the Ecole Polytechnique.
  • Ambrosio, Glaudo, Trevisan: On the optimal map in the $2$-dimensional random matching problem. DCDS.
  • Antonelli, Bruè, Semola: Volume bounds for the quantitative singular strata of non collapsed RCD metric measure spaces. Anal. Geom. Metr. Spaces.
  • Bartolucci, Gui, Jevnikar, Moradifam: A singular Sphere Cover-ing Inequality: uniqueness and symmetry of solutions to singular Liouville-typeequations,Math. Ann.
  • Bartolucci, Jevnikar, Lee, Yang: Uniqueness of bubbling solutions ofmean field equations,J. Math. Pures Appl.
  • Bartolucci, Jevnikar, Lin: Non-degeneracy and uniqueness of solutionsto singular mean field equations on bounded domains, J. Diff. Eq.
  • Bartolucci, Jevnikar, Lee, Yang: Local uniqueness ofm-bubblingsequences for the Gel’fand equation,Comm. PDEs
  • Bartolucci, Gui, Hu, Jevnikar, Yang: Mean field equation ontorus: existence and uniqueness of evenly symmetric blow-up solutions, DiscreteContin. Dyn. Syst.
  • Benamou, Carlier, Di Marino, Nenna: An entropy minimization approach to second-order variational mean-field games. Math. Models Methods Appl.
  • Borrelli, Carlone, Tentarelli: An overview on the standing waves of nonlinear Schrödinger and Dirac equations on metric graphs with localized nonlinearity, Symmetry.
  • Borrelli, Carlone, Tentarelli: Nonlinear Dirac equation on graphs with localized nonlinearities: bound states and nonrelativistic limit, SIAM J. Math. Anal.
  • Bruè, Nguyen: On the Sobolev space of functions with derivative of logarithmic order. Adv. Nonlinear Anal.
  • Bruè, Nguyen, Stefani: A maximal function characterization of absolutely continuous measures and Sobolev functions. Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl.
  • Colombo, Di Marino, Stra: Continuity of multimarginal optimal transport with repulsive cost. SIAM J. Math. Anal.
  • Comi, Stefani, A distributional approach to fractional bounded variation: existence of blow-up, Journal of Functional Analysis.
  • Chen, Comi, Torres, Cauchy fluxes and Gauss-Green formulas for divergence-measure fields over general open sets, Archive for Rational Mechanics and Analysis.
  • Di Marino,Squassina, New characterizations of Sobolev metric spaces. J. Funct. Anal.
  • Franceschi, Stefani: Symmetric double bubbles in the Grushin plane. ESAIM Control Optim. Calc. Var.
  • Jevnikar, Yang: A mean field equation involving positively supported prob-ability measures: blow-up phenomena and variational aspects, Proc. Royal Soc.Edinburgh
  • Jost, Wu, Zhu: Energy quantization for a nonlinear sigma model with critical gravitinos. Trans.AMS
  • Jost, Kessler, Tolksdorf, Wu, Zhu: From harmonic maps to the nonlinear supersymmetric sigma model of quantum field theory: at the interface of theoretical physics, Riemannian geometry, and nonlinear analysis, Vietnam Journal of Mathematics
  • Malchiodi: On conformal metrics with constant Q-curvature, Anal. Theory Appl. Mennucci: On Perimeters and Volumes of Fattened Sets, International Journal of Mathematics and Mathematical Sciences
  • Mennucci: Designing metrics; the delta metric for curves, ESAIM, COCV.
  • Mondino, Semola: Polya-Szego inequality and Dirichlet spectral gap for non-smooth spaces with Ricci curvature bounded below, Journal de Mathématiques Pures et Appliquées.