The scientific activity of the Group “Mathematical Analysis” is focused on different subjects in the area of Calculus of Variations, Geometric Measure Theory, Fluid Dynamics, Dispersive equations, Nonlinear Analysis, Ergodic Theory, and Dynamical Systems.
Our research mainly contributes to Pure Mathematics but provides also several applications in the natural and applied sciences, e.g., Physics, Biology, and Mechanics.
The group works in close collaboration with members of the Faculty of Sciences of Scuola Normale Superiore and Centro di Ricerca Matematica Ennio De Giorgi.
The group runs a Seminar Series: further information may be found here.
Research Topics
Calculus of variations and geometric measure theory
The research of a large portion of the group “Mathematical Analysis” is focused on the Calculus of Variations and Geometric Measure Theory. The list of research directions includes shape and spectral optimization, free boundary problems, isoperimetric problems and related stability estimates, local and nonlocal geometric flows, the structure of measures and differentiability, analysis and geometric measure theory in nonEuclidean settings.
A relevant part of the research in this area is devoted to the application of Calculus of Variations and Geometric Measure Theory to problems in Mathematical Physics, Continuum Mechanics, and other areas of Mathematics.
This group has strong connections with Scuola Normale Superiore and Centro De Giorgi.
The repository CVGMT collects preprints in this area, as well as information about seminars and conferences.
Faculty Members Involved:
 Giovanni Alberti
 Giuseppe Buttazzo
 Maria Stella Gelli
 Massimo Gobbino
 Valentino Magnani
 Matteo Novaga
 Emanuele Paolini
 Alessandra Pluda
 Aldo Pratelli
 Vincenzo Maria Tortorelli
 Dario Trevisan
 Bozhidar Velichkov
Postdoctoral Fellows:
 Danka Lucic
 Roberto Ognibene
 Giorgio Tortone
Ph.D. Students:
 Konstantinos Bessas
 Luca Briani Medeiros Dos Santo
 Vincenzo Scattaglia
 Luciano Sciaraffia Rubio
Free boundary problems
The free boundary problems are PDE problems (often elliptic or parabolic) in which the domain of the solution is not a priori prescribed but is part of the variables of the problem. The boundary conditions are overdetermined and might involve the trace of the function, its gradient (or just the normal derivative), and the curvature of the boundary. Solutions to free boundary problems are obtained for instance from the minimization of variational functionals that take into account both the state function (the solution of the PDE) and its domain of definition. Wellknown examples of free boundary problems are the onephase and the twophase Bernoulli problems, the obstacle problem, the thinobstacle problem, the optimal partitions problems, MumfordShah, free transmission problems, and their parabolic counterparts, for instance, the Stefan problem.
The focus is on the free boundary regularity theory, which is the study of the fine local structure of the free boundary and relates to the analysis of the interplay between the geometry of the domain and the behavior of the state function (the solution of the PDE). Of particular interest are theorems that guarantee (under some geometric conditions) that the boundary is a smooth manifold (the socalled epsilonregularity theorems) and the analysis of the free boundaries around singularities. The methods include blowup analysis, monotonicity formulas, density estimates, estimates on the dimension of the singular set, viscosity solutions, analysis of the stable cones, improvementofflatness methods, epiperimetric inequalities.
The free boundary problems are strongly related to problems in Geometric Analysis as the regularity of areaminimizing surfaces and currents; in Elliptic PDEs, for instance to the analysis of the nodal sets to the solutions of PDEs or the behavior at the boundary of the solutions (Boundary Harnack inequalities); and also to some Shape Optimization problems.
Members:
 Roberto Ognibene
 Giorgio Tortone
 Bozhidar Velichkov
Collaborators:
 Guido De Philipps (New York University)
 Dario Mazzoleni (Università di Pavia)
 Luca Spolaor (University of California San Diego)
 Susanna Terracini (Università di Torino)
Geometric flows
The idea of deforming a geometric object by means of a parabolic PDEs to simplify it (e.g. to reduce its topological complexity or to make it more symmetric) goes back to Eells and Sampson who introduced the first geometric evolution equation: the harmonic map flow. Since 1964, geometric flows have been applied to a variety of topological, analytical, and physical problems, giving in some cases very fruitful results.
Among other geometric equations, we mention the mean curvature flow. This flow can be regarded as the gradient flow of the area functional: an ndimensional timedependent surface evolves with normal velocity its mean curvature, at any point and time. In place of the area functional, one may consider generalized perimeters, such as the fractional perimeter, and the corresponding gradient flows. Concerning the study of these flows, we are interested in the longtime existence of (weak and strong) solutions and in the classification of possible singularities. Besides the urge to introduce a nonlocal version of mean curvature flow, one wants also to extend it to the more singular geometric objects.
Motived by the study of grain boundaries, Brakke introduces the first weak versions of the flow and in the years a plethora of weak definitions (level set approach, minimizing movements, thresholding scheme, canonical Brakke flow) has been developed, but the onedimensional version of this problem is also studied in the framework of classical PDEs. In such a situation the flow is called network flow.
We are interested in several aspects of evolution: wellposedness, longtime behavior, analysis of singularities, flow past singularities, and stability. Our main goal is to exploit the known results to a higher dimension.
Members:
 Matteo Novaga
 Alessandra Pluda
Collaborators:
 Antonin Chambolle (CEREMADE, CNRS 7534 – Université ParisDauphine)
 Annalisa Cesaroni (Università di Padova)
 Carlo Mantegazza (Università di Napoli)
 Rafe Mazzeo (Stanford University)
 Massimiliano Morini (Università di Parma)
 Marcello Ponsiglione (Sapienza Università di Roma)
 Marco Pozzetta (Università di Napoli)
 Mariel Saez (Pontificia Universidad Católica de Chile)
Geometric measure theory
The development of Geometric Measure Theory (GMT) began about one hundred years ago, with the first intrinsic notions of dimension for general sets, which could be possibly noninteger, and the corresponding intrinsic notions of measure, where “intrinsic” means that it does not rely on parametrizations.
In the second half of the last century, one of the driving problems was the existence of ddimensional surfaces with minimal area and prescribed boundary (Plateau problem), which led to the development of several “weak” notions of surface, such as finite perimeter set (Caccioppoli, De Giorgi), integral currents (Federer and Fleming), and varifolds (Almgren).
In the past fifty years these notions played a fundamental role in the treatment of many problems of geometric nature besides minimal surfaces, from the asymptotic behavior of singular perturbations problems in continuum mechanics and physics (e.g, scalar and vector models of GinzburgLandau type) to the notions of weak solutions of geometric evolution problems (e.g., the mean curvature flow).
At the same time, many attempts have been made to extend these theories to various nonEuclidean settings (subRiemannian manifolds and metric spaces with bounds on curvature). These attempts have led to several fundamental questions that have not yet been fully answered–indeed in many cases our understanding is still quite limited.
The Italian mathematical community in Calculus of Variations and Geometric Measure Theory, whose origins can be traced back to the work of Ennio De Giorgi here in Pisa, gave essential contributions to the developments mentioned above. Among the topics currently studied in our department, we mention the following:
 GMT in subRiemannian manifolds: area formulas, contact sets, currents;
 Minimal partitions;
 GaussGreen formulas, sets of finite perimeter and rectifiability in stratified Lie groups;
 Weak solutions to geometric evolution problems;
 Structure of measures and differentiability of Lipschitz functions.
Members:
 Giovanni Alberti
 Valentino Magnani
 Matteo Novaga
 Emanuele Paolini
 Vincenzo Maria Tortorelli
Collaborators:
 Marianna Csornyei (University of Chicago)
 Andrea Marchese (Università di Trento)
 Annalisa Massaccesi (Università di Padova)
 David Preiss (Warwick University)
 Eugene Stepanov (Steklov Institute, Saint Petersburg)
Shape optimization
The field of shape optimization had very strong growth in the last decades, mainly due to the several applications in engineering, but also for the mathematical interest related to several problems in spectral geometry. The goal is to find a domain $\Omega$ that minimizes a given cost functional, that may involve geometrical quantities as perimeter or measure, as well as other quantities related to PDEs like for instance the related energies or eigenvalues. Showing the existence of optimal shapes, and their regularity is often a very challenging issue that needs the development of new and deep tools.
Members:
 Luca Briani Medeiros Dos Santo
 Giuseppe Buttazzo
 Danka Lucic
 Aldo Pratelli
Collaborators:
 Michiel van den Berg (University of Bristol)
 Guy Bouchitté (Université de Toulon)
 Lorenzo Brasco (Università di Ferrara)
 Juan Casado Diaz(Universidad de Sevilla)
 Thierry Champion (Université de Toulon)
 Luigi De Pascale (Università di Firenze)
 Faustino Maestre (Universidad de Sevilla)
 Francesca Agnese Prinari (Department of Agricultural, Environmental and Food Sciences, Università di Pisa)
PDEs, ODEs, and dynamical systems
Faculty Members Involved:
 Jacopo Bellazzini
 Luigi Carlo Berselli
 Carlo Carminati
 Elisabetta Chiodaroli
 Stefano Galatolo
 Marco Gipo Ghimenti
 Marina Ghisi
 Massimo Gobbino
 Carlo Romano Grisanti
 Francesco Grotto
 Vladimir Simeonov Gueorguiev
 Pietro Majer
 Claudio Saccon
 Nicola Visciglia
Ph.D. Students:
 Daniele Barbera
 Mario Rastrelli
Dynamical systems and ergodic theory
Our research focuses on the statistical properties of deterministic and random dynamical systems.
These studies have a wide range of applications, from a number of theoretical and geometrical questions to the understanding of the behavior of models of realworld phenomena like the dynamics of fluids, the evolution of climate and its extreme events, the dynamics of extended systems in a meanfield coupling and other complex systems. The methods used in this research are often related to probability, functional analysis, transport, and geometric measure theory.
Our research also focused on reliable computational tools for the study of the statistical properties of dynamics, computeraided proofs, and their relation to theoretical computer science.
Members:
 Carlo Carminati
 Stefano Galatolo
 Francesco Grotto
 Leonardo Roveri
Fluid dynamics equations
Our research focuses on the pure and applied aspects of mathematical fluid dynamics of Newtonian and nonNewtonian fluids.
The research interests span from the most theoretical questions of existence, uniqueness, and regularity of weak solutions towards applications to the numerical analysis of such flows (with finite element/differences methods) and to the study of turbulent flows, with applications to geophysical problems. The methods we use stem from those of classical analysis of PDEs in Sobolev spaces, but for the problems we consider we are using also: convex integration, analysis in Orlicz, weighted spaces, and approximate methods like reducedorder ones. More specifically for the Equations of classical fluid mechanics: incompressible Euler and NavierStokes equations. Existence, uniqueness, properties, and regularity of weak and suitable solutions.
Concerning nonNewtonian fluids, we are interested in the existence, regularity, numerical approximation, and longtime behavior of classes of pfluids or electrorheological ones.
For the turbulent flows, we study the mathematical properties and the mathematical modeling of such methods, having in mind the application to the study of boundary layers and airsea interaction, together with other multiphase problems arising in the study of volcanic eruptions.
Collaborators:
 Camillo De Lellis (Princeton)
 Eduard Feireisl (Prague)
 Ondrej Kreml (Prague)
 Roger Lewandowski (Rennes)
 Paolo Maremonti (Caserta)
 Francesca Crispo (Caserta)
 Michael Ruzicka (Freiburg)
 Emil Wiedemann (Ulm)
Hyperbolic Equations, dispersive equations, and Kirchhoff equations
Members:
 Daniele Barbera
 Jacopo Bellazzini
 Marina Ghisi
 Vladimir Simeonov Gueorguiev
 Mario Rastrelli
 Nicola Visciglia
Collaborators:
 Nicolas Burq (Université de ParisSaclay)
 Tohru Ozawa (Waseda University)
 Fabrice Planchon (Université de Paris Sorbonne)
 David Ruiz (Granada University)
 Nikolay Tzvetkov (ENS Lyon)
Nonlinear analysis
Nonlinear analysis group focuses on partial differential equations of elliptic and hyperbolic types in which the presence of the nonlinear term is crucial for the existence and qualitative properties of solutions.
The research interests span from nonlinear Schroedinger equations and GrossPitaevskij in a Euclidean setting to Yamabe problems on compact manifolds. The main topics of the research concerning elliptic equations are the existence and multiplicity of concentrated solutions, qualitative properties, blowup analysis, and compactness of solutions. On the timedependent side, we focus on the existence of solitons, stability and instability issues, and longtime dynamics for equations of physical interest. Another field of research is related to functional inequalities in presence of nonlocal terms: the existence of maximizers and the role of symmetry.
Members:
 Jacopo Bellazzini
 Marco Gipo Ghimenti
 Pietro Majer
Collaborators:
 Luigi Forcella
 Enno Lenzmann
 Vitali Moroz
 David Ruiz
 Louis Jeanjean
 Anna Maria Micheletti
 Angela Pistoia
 Pietro d’Avenia
 Simone Dovetta
 Monica Clapp
 Jean Van Schaftingen
Members
Staff
Name  Surname  Links  Personal Card 

Giovanni  Alberti  [Google Scholar] [Mathscinet]  
Jacopo  Bellazzini  [Google Scholar] [Mathscinet] [Orcid]  
Luigi Carlo  Berselli  [Google Scholar] [Mathscinet] [Orcid]  
Giuseppe  Buttazzo  [Google Scholar] [Mathscinet] [Orcid]  
Carlo  Carminati  [arXiv] [Mathscinet] [Orcid]  
Elisabetta  Chiodaroli  [Google Scholar] [Mathscinet] [Orcid]  
Stefano  Galatolo  [Google Scholar] [Mathscinet] [Orcid]  
Maria Stella  Gelli  [Mathscinet]  
Marco Gipo  Ghimenti  [Google Scholar] [Mathscinet] [Orcid]  
Marina  Ghisi  [Mathscinet] [Orcid]  
Massimo  Gobbino  [Mathscinet] [Orcid]  
Carlo Romano  Grisanti  [Google Scholar] [Mathscinet]  
Francesco  Grotto  [Google Scholar] [Mathscinet]  
Vladimir Simeonov  Gueorguiev  [Mathscinet] [Orcid]  
Placido  Longo  [Mathscinet]  
Valentino  Magnani  [Mathscinet]  
Pietro  Majer  [Mathscinet]  
Matteo  Novaga  [Google Scholar] [Mathscinet] [Orcid]  
Emanuele  Paolini  [Google Scholar] [Mathscinet] [Orcid]  
Alessandra  Pluda  [Google Scholar] [Mathscinet] [Orcid]  
Aldo  Pratelli  [Mathscinet] [Orcid]  
Giuseppe  Puglisi  [Mathscinet]  
Claudio  Saccon  [Mathscinet] [Orcid]  
Vincenzo Maria  Tortorelli  [Mathscinet]  
Bozhidar  Velichkov  [Google Scholar] [Mathscinet] [Orcid]  
Nicola  Visciglia  [Mathscinet] 
Postdoctoral Fellows
Name  Surname  Links  Personal Card 

Danka  Lucic  [Google Scholar] [Mathscinet]  
Roberto  Ognibene  [Mathscinet] [Orcid]  
Giorgio  Tortone  [Google Scholar] [Mathscinet] [Orcid] 
Ph.D. Students
Name  Surname  Links  Personal Card 

Daniele  Barbera  
Konstantinos  Bessas  
Luca  Briani Medeiros Dos Santo  [Mathscinet]  
Mario  Rastrelli  
Vincenzo  Scattaglia  
Luciano  Sciaraffia Rubio 
Past Ph.D. Students
2021
 Obinna Kennedy Idu (Università di Pisa), “HigherOrder Rectifiability Criteria And a Model for Soap Films”, supervised by Giovanni Alberti.
 Kosuke Kita (Università id Pisa and Waseda University, Tokyo, Japan), “A study on the qualitative theory of solutions for some parabolic equations with nonlinear boundary conditions”, supervised by Vladimir S. Gueorguiev and Tohru Ozawa.
 Luigi Marangio (Université de Bourgogne FrancheComté), “Rigorous computational methods for understanding behavior of random dynamical systems”, supervised by Stefano Galatolo and Christophe Guyeux.
2020
 François Générau (Université Grenoble Alpes), “On a stable variational approximation of the cut locus, and a nonlocal isoperimetric problem”, supervised by Bozhidar Velichkov and Édouard Oudet.
 Andrea Merlo (Scuola Normale Superiore di Pisa), “Geometry of 1codimensional measures in Heisenberg groups”, supervised by Giovanni Alberti and Roberto Monti.
 Dinh Duong Nguyen (Université Rennes), “Some results on Turbulent models”, supervised by Roger Lewandowski and Luigi Berselli.
 Marco Pozzetta (Università di Pisa), “Willmoretype Energies of Curves and Surfaces”, supervised by Matteo Novaga.
 Baptiste Trey (Université Grenoble Alpes), “Existence and regularity of optimal shapes for some spectral optimization problems”, supervised by Bozhidar Velichkov and Emmanuel Russ.
2019
 Francesco Bartaloni (Università di Pisa), “Infinite horizon optimal control problems with noncompact control space. Existence results and dynamic programming” supervised by Paolo Acquistavate and Fausto Gozzi.
 Giacomo Del Nin (Università di Pisa), “A Some asymptotic results on the global shape of planar clusters”, supervised by Giovanni Alberti.
 Valerio Pagliari (Università di Pisa), “Aymptotic behaviour of rescaled nonlocal functionals and evolutions”, supervised by Matteo Novaga.
2018
 Ricardo Bioni Liberalquino (Univ. Fed. Rio de Janeiro) “Computation of stationary densities of systems with additive noise”, supervised by Maria Jose Pacifico.
 Dayana Pagliardini (Scuola Normale Superiore di Pisa), “Fractional minimal surfaces and AllenCahn equations”, supervised by Andrea Malchiodi and Matteo Novaga.
 Harish Shrivastava (Università di Pisa), “Shape optimisation problems for integral functionals and regularity properties of optimal domains”, supervised by Giuseppe Buttazzo.
2017
 Luigi Forcella (Scuola Normale Superiore di Pisa), “Asymptotic problems for some classes of dispersive PDEs”, supervised by Nicola Visciglia.
 Anna Rita Giammetta (Università di Pisa), “Perturbed Hamiltonians in one dimension: analysis for linear and nonlinear Schrödinger problems”, supervised by Vladimir Gueorguiev.
2016
 Matteo Cerminara (Scuola Normale Superiore di Pisa), “Modeling dispersed gas–particle turbulence in volcanic ash plumes”, supervised by Luigi Berselli and Tomaso Esposti Ongaro.
 Francois Dayrens (Université de Lyon), “Minimisations sous contraintes et flots du périmètre et de l’énergie de Willmore”, supervised by Simon Masnou and Matteo Novaga.
 Augusto Gerolin (Università di Pisa), “Multimarginal optimal transport and potential optimization problems for Schrödinger operators”, supervised by Giuseppe Buttazzo.
 Anna Rita Giammetta (Università di Pisa) , “Perturbed Hamiltonians in one dimension: analysis for linear and nonlinear Schrödinger problems”, supervised by Vladimir S. Gueorguiev.
 Alessandra Pluda (Università di Pisa), “Minimal and Evolving Networks”, supervised by Matteo Novaga.
 Emanuela Radici (FAU ErlangenNürnberg ), “Diffeomorphic approximation of planar elastic deformations”, supervised by Aldo Pratelli.
 Andrea Tamagnini (Università di Firenze) “Planar Clusters”, supervised by Emanuele Paolini
 Florian Zeisler (FAU ErlangenNürnberg ), “On the optimal transport problem with relativistic cost”, supervised by Aldo Pratelli.
2015
 Marco Caroccia (Università di Pisa), “On the isoperimetric properties of planar $N$clusters”, supervised by Giovanni Alberti.
 Rafael Lucena (Univ. Fed. Rio de Janeiro), “Spectral Gap for Contracting Fiber Systems and Applications”, supervised by Stefano Galatolo and Maria Jose Pacifico.
2014
 Serena Guarino Lo Bianco (Università di Pisa), “Some optimization problems in mass transport theory”, supervised by Giuseppe Buttazzo.
 Annalisa Massaccesi (Scuola Normale Superiore di Pisa), “Currents with coefficients in groups, applications, and other problems in Geometric Measure Theory”, supervised by Giovanni Alberti.
 Dario Mazzoleni, (FAU ErlangenNürnberg and Università di Pavia), “Existence and regularity results for solutions of spectral problems”, supervised by Aldo Pratelli.
2013
 Xin Yang Lu (Scuola Normale Superiore di Pisa), “Geometric and regularity properties of solutions of some problems related to the average distance functional”, supervised by Giuseppe Buttazzo.
 Andrea Marchese (Università di Pisa), “Two applications of the theory of currents”, supervised by Giovanni Alberti.
 Berardo Ruffini (Scuola Normale Superiore di Pisa), “Optimization problems for solution of elliptic equations and stability issues”, supervised by Giuseppe Buttazzo.
 Matteo Scienza (Università di Pisa), “Differentiability properties and characterization of Hconvex functions”, supervised by Valentino Magnani.
 Bozhidar Velichkov (Scuola Normale Superiore di Pisa), “Existence and regularity results for some shape optimization problems”, supervised by Giuseppe Buttazzo.
2012
 Minh Nguyet Mach (University of Pisa), “Weak solutions to rateindependent systems: existence and regularity”, supervised by Giovanni Alberti.
 Stefano Spirito (Univesità del L’Aquila), “The artificial compressibility approximation and the inviscid limit for the incompressible NavierStokes equations”, supervised by Pierangelo Marcati and Luigi Berselli.
2011
 Alhassem Nayam (Università di Pisa), “Shape optimization problems of higher codimension”, supervised by Giuseppe Buttazzo.
2010
 Lorenzo Brasco (Università di Pisa), “Geodesics and PDE methods in transport models”, supervised by Giuseppe Buttazzo, Guillaume Carlier, Filippo Santambrogio.
 Jimmy Alfonso Mauro (Università di Pisa), “Some analytic questions in mathematical physic problems”, supervised by Vladimir S. Gueorguiev.
2008
 Davide Catania (Università di Pisa), “Linear and nonlinear wave equations”, supervised by Vladimir S. Gueorguiev.
 Mathieu Hoyrup (ENS Paris and Universite Paris DiderotParis VII), “Computability, Randomness and Ergodic Theory on Metric Spaces”, supervised by Stefano Galatolo and G. Longo.
 Cristobal Rojas (ENS Paris and École Polytechnique), “Randomness and Ergodic Theory: an Algorithmic point of view”, supervised by Stefano Galatolo and G. Longo.
2006
 Alessio Brancolini (Scuola Normale Superiore di Pisa), “Optimization problems for transportation networks”, supervised by Giuseppe Buttazzo.
 Angel Ivanov (Università di Pisa), “Nonlinear evolution equations in mathematical physics”, supervised by Vladimir S. Gueorguiev.
 Filippo Santambrogio (Scuola Normale Superiore di Pisa), “Variational problems in transport theory with mass concentration”, supervised by Giuseppe Buttazzo.
 Mirko Tarulli Di Giallonardo (Università di Pisa), “SmoothingStrichartz estimates for dispersive equations perturbed by a firstorder differential operator”, supervised by Vladimir S. Gueorguiev.
2005
 Niccolò Desenzani (Università di Milano), “Variational convergence of GinzburgLandau functionals with supercritical growth”, supervised by Giovanni Alberti and Ilaria Fragalà.
 Luca Granieri (Università di Pisa), “Mass transportation problems and minimal measures”, supervised by Giuseppe Buttazzo.
 Stefano Zappacosta (Università de L’Aquila), “Resolvent estimates and dispersive estimates for the wave equation with potential”, supervised by Vladimir S. Gueorguiev.
2004
 Andrea Davini (Università di Pisa), “Finsler metrics in optimization problems and HamiltonJacobi equations”, supervised by Giuseppe Buttazzo.
 Michele Gori (Università di Pisa), “Lower semicontinuity and relaxation for integral and supremal functionals”, supervised by Giuseppe Buttazzo.
 Nicola Visciglia (Scuola Normale Superiore di Pisa), “a priori estimates for the linear and semilinear perturbed wave equation”, supervised by Vladimir S. Gueorguiev.
2003
 Francesca Prinari (Università di Pisa), “Calculus of variations for supremal functionals”, supervised by Giuseppe Buttazzo.
2001
 Luigi De Pascale (Università di Pisa), “MorseSard theorem in Sobolev spaces, transport problems and applications”, supervised by Giuseppe Buttazzo and Giovanni Alberti.
2000
 Ilaria Fragalà (Università di Pisa), “Tangential calculus and variational integrals with respect to a measure”, supervised by Giuseppe Buttazzo.
1999
 Ariela Briani (Università di Pisa), “HamiltonJacobiBellman equations and $\Gamma$convergence for optimal control problems”, supervised by Giuseppe Buttazzo.
1998
 Paola Trebeschi (Università di Pisa), “Esistenza di soluzioni in problemi di ottimizzazione di forma e ostacoli”, supervised by Giuseppe Buttazzo.
1996
 Marino Belloni (Università di Pisa), “Rilassamento di problemi variazionali con fenomeno di Lavrentiev”, supervised by Giuseppe Buttazzo.
1994
 Loris Faina (SISSA – Trieste), “Some nonconvex variational problems in BV spaces”, supervised by Giuseppe Buttazzo.
 Lorenzo Freddi (Università di Pisa), “Rilassamento e convergenza di problemi di controllo ottimo”, supervised by Giuseppe Buttazzo.
External Collaborators
Name  Surname  Links  Personal Card 

Paolo  Acquistapace  [Mathscinet]  
Hugo  Beirao da Veiga  [Mathscinet] [Orcid]  
Ferruccio  Colombini  [Mathscinet]  
Antonio  Marino  [Mathscinet]  
Anna Maria  Micheletti  [Mathscinet]  
Sergio  Spagnolo  [Mathscinet]  
Antonio  Tarsia  [Google Scholar] [Mathscinet] 
Grants
Current Grants

Variational approach to the regularity of the free boundaries (H2020EU.1.1.  Excellent Science  European Research Council (erc))
Principal Investigator: Bozhidar Velichkov

Gradient flows, Optimal Transport and Metric Measure Structures (Prin 2017)
Principal Investigator: Luigi Ambrosio (SNS, Pisa)  Coordinator of the Research Unit: Giuseppe Buttazzo

Variational methods for stationary and evolution problems with singularities and interfaces (Prin 2017)
Principal Investigator: Gianni Dal Maso (SISSA, Trieste)  Coordinator of the Research Unit: Giovanni Alberti
Members of the Research Unit: De Luca (CNR, Roma), Maria Stella Gelli, Matteo Novaga

Hamiltonian and dispersive PDEs (Prin 2020)
Principal Investigator: Massimiliano Berti (SISSA, Trieste)  Coordinator of the Research Unit: Nicola Visciglia
Members of the Research Unit: Jacopo Bellazzini, Vladimir S. Gueorguiev, Marina Ghisi, Scipio Cuccagna, Daniele Barbera, Kosuke Kita

Nonlinear evolution PDEs, fluid dynamics and transport equations: theoretical foundations and applications (Prin 2020)
Principal Investigator: Stefano Bianchini (SISSA, Trieste)  Coordinator of the Research Unit: Luigi Carlo Berselli

Nuovi sviluppi per minimi dell’energia in problemi di ottimizzazione di forma (Progetti di Ateneo)
Coordinator of the Research Unit: Aldo Pratelli
Members of the Research Unit: Giovanni Alberti, Konstantinos Bessas, Giuseppe Buttazzo, Maria Stella Gelli, Matteo Novaga, Alessandra Pluda, Vincenzo Scattaglia
Past Grants

Problemi di ottimizzazione e di evoluzione in ambito variazionale (Progetti di Ateneo)
Coordinator of the Research Unit: Matteo Novaga