{"id":55,"date":"2022-03-16T22:31:48","date_gmt":"2022-03-16T22:31:48","guid":{"rendered":"https:\/\/www.dm.unipi.it\/research\/mat-05-analysis\/"},"modified":"2026-01-23T10:10:30","modified_gmt":"2026-01-23T09:10:30","slug":"mathematical-analysis","status":"publish","type":"page","link":"https:\/\/www.dm.unipi.it\/research\/areas\/mathematical-analysis\/","title":{"rendered":"Mathematical Analysis"},"content":{"rendered":"\n

The scientific activity of the Group \u201cMathematical Analysis\u201d 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<\/a> and  Centro di Ricerca Matematica Ennio De Giorgi<\/a>.<\/p>\n\n\n\n

The group runs a Seminar Series: further information may be found here<\/a>.<\/p>\n\n\n\n

<\/div>\n\n\n\n

Research Topics<\/h4>\n\n\n\n
Calculus of variations and geometric measure theory <\/h5>
\n

The research of a large portion of the group \u201cMathematical Analysis\u201d is focused on the Calculus of Variations and Geometric Measure Theory. The list of research directions includes shape and spectral optimization, compensated compactness and variational integrals, free boundary problems, isoperimetric problems and related stability estimates, local and non-local geometric flows, the structure of of PDE-constrained measures and differentiability, analysis and geometric measure theory in both Euclidean and non-Euclidean 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.<\/p>\n\n\n\n

This group has strong connections with Scuola Normale Superiore and Centro De Giorgi.
The repository CVGMT<\/a> collects preprints in this area, as well as information about seminars and conferences.<\/p>\n\n\n\n

<\/div>\n\n\n\n
Members<\/h6>
\n\n\n
\n
\n
\n<\/i><\/a>\n\nGiovanni Alberti\n\n<\/span>
\n<\/i>giovanni.alberti@unipi.it<\/a>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
\n<\/i><\/a>\n\nAdolfo Arroyo Rabasa\n\n<\/span>
\n\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
\n<\/i><\/a>\n\nGiuseppe Buttazzo\n\n<\/span>
\n<\/i>giuseppe.buttazzo@unipi.it<\/a>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
\n<\/i><\/a>\n\nMaria Stella Gelli\n\n<\/span>
\n<\/i>maria.stella.gelli@unipi.it<\/a>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
\n<\/i><\/a>\n\nMassimo Gobbino\n\n<\/span>
\n<\/i>massimo.gobbino@unipi.it<\/a>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
\n<\/i><\/a>\n\nIlaria Lucardesi\n\n<\/span>
\n<\/i>ilaria.lucardesi@unipi.it<\/a>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
\n<\/i><\/a>\n\nValentino Magnani\n\n<\/span>
\n<\/i>valentino.magnani@unipi.it<\/a>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
\n<\/i><\/a>\n\nCyrill Muratov\n\n<\/span>
\n<\/i>cyrill.muratov@unipi.it<\/a>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
\n<\/i><\/a>\n\nFrancesco Nobili\n\n<\/span>
\n<\/i>francesco.nobili@dm.unipi.it<\/a>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
\n<\/i><\/a>\n\nMatteo Novaga\n\n<\/span>
\n<\/i>matteo.novaga@unipi.it<\/a>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
\n<\/i><\/a>\n\nFilippo Paiano\n\n<\/span>
\n<\/i>filippo.paiano@phd.unipi.it<\/a>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
\n<\/i><\/a>\n\nEmanuele Paolini\n\n<\/span>
\n<\/i>emanuele.paolini@unipi.it<\/a>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
\n<\/i><\/a>\n\nNicola Picenni\n\n<\/span>
\n<\/i>nicola.picenni@unipi.it<\/a>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
\n<\/i><\/a>\n\nAlessandra Pluda\n\n<\/span>
\n<\/i>alessandra.pluda@unipi.it<\/a>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
\n<\/i><\/a>\n\nAldo Pratelli\n\n<\/span>
\n<\/i>aldo.pratelli@unipi.it<\/a>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
\n<\/i><\/a>\n\nDario Trevisan\n\n<\/span>
\n<\/i>dario.trevisan@unipi.it<\/a>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
\n<\/i><\/a>\n\nBozhidar Velichkov\n\n<\/span>
\n<\/i>bozhidar.velichkov@unipi.it<\/a>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div><\/div>\n\n\n\n
<\/div>\n\n\n\n

Below is the list of the specific topics studied in this research area, each of them with a detailed description and the lists of members and collaborators.<\/p>\n\n\n\n

Free boundary problems <\/h5>
\n

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. Well-known examples of free boundary problems are the one-phase and the two-phase Bernoulli problems, the obstacle problem, the thin-obstacle problem, the optimal partitions problems, Mumford-Shah, free transmission problems, and their parabolic counterparts, for instance, the Stefan problem.<\/p>\n\n\n\n

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 so-called epsilon-regularity theorems) and the analysis of the free boundaries around singularities. The methods include blow-up analysis, monotonicity formulas, density estimates, estimates on the dimension of the singular set, viscosity solutions, analysis of the stable cones, improvement-of-flatness methods, epiperimetric inequalities.<\/p>\n\n\n\n

The free boundary problems are strongly related to problems in Geometric Analysis as the regularity of area-minimizing 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.<\/p>\n\n\n\n

Members<\/h6>
\n\n\n
\n
\n
\n<\/i><\/a>\n\nFederico Lai\n\n<\/span>
\n<\/i>federico.lai@phd.unipi.it<\/a>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
\n<\/i><\/a>\n\nFilippo Paiano\n\n<\/span>
\n<\/i>filippo.paiano@phd.unipi.it<\/a>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
\n<\/i><\/a>\n\nBozhidar Velichkov\n\n<\/span>
\n<\/i>bozhidar.velichkov@unipi.it<\/a>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div><\/div>\n\n\n\n
Collaborators and Ph.D. students at other institutions<\/h6>
\n\n\n
\n
\n
\n<\/i><\/a>\n\nGiulia Bevilacqua (Universit\u00e0 Ca' Foscari)\n\n<\/span>
\n<\/i>giulia.bevilacqua@unive.it<\/a>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
\n<\/i>\n\nMatteo Carducci (SNS, Pisa)\n\n<\/span>
\n<\/i>matteo.carducci@sns.it<\/a>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
\n<\/i><\/a>\n\nNicola De Nitti (Universit\u00e0 di Pisa)\n\n<\/span>
\n<\/i>nicola.denitti@unipi.it<\/a>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
\n<\/i>\n\nGuido De Philippis (Courant Institute of Mathematics)\n\n<\/span>
\n<\/i>guido@cims.nyu.edu<\/a>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
\n<\/i>\n\nLorenzo Ferreri (SNS, Pisa)\n\n<\/span>
\n<\/i>lorenzo.ferreri@sns.it<\/a>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
\n<\/i>\n\nDario Mazzoleni (Universit\u00e0 degli Studi di Pavia)\n\n<\/span>
\n<\/i>dario.mazzoleni@unipv.it<\/a>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
\n<\/i>\n\nRoberto Ognibene (Universit\u00e0 degli Studi di Milano-Bicocca)\n\n<\/span>
\n<\/i>roberto.ognibene@unimib.it<\/a>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
\n<\/i>\n\nMaria Antonietta Palladino (SNS, Pisa)\n\n<\/span>
\n<\/i>mariaantonietta.palladino@sns.it<\/a>\n<\/div>\n<\/div>\n<\/div>\n
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\n<\/i>\n\nLuca Spolaor (University of California San Diego)\n\n<\/span>
\n<\/i>lspolaor@ucsd.edu<\/a>\n<\/div>\n<\/div>\n<\/div>\n
\n
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\n<\/i>\n\nSusanna Terracini (Universit\u00e0 di Torino)\n\n<\/span>
\n\n<\/div>\n<\/div>\n<\/div>\n
\n
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\n<\/i>\n\nGiorgio Tortone (Universit\u00e0 di Torino)\n\n<\/span>
\n<\/i>giorgio.tortone@unito.it<\/a>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div><\/div>\n<\/div><\/div>\n\n\n\n
Geometric flows<\/h5>
\n

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.<\/p>\n\n\n\n

Among other geometric equations, we mention the mean curvature flow. This flow can be regarded as the gradient flow of the area functional: an n-dimensional time-dependent 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 long-time existence of (weak and strong) solutions and in the classification of possible singularities. Besides the urge to introduce a non-local version of mean curvature flow, one wants also to extend it to the more singular geometric objects.<\/p>\n\n\n\n

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 one-dimensional version of this problem is also studied in the framework of classical PDEs. In such a situation the flow is called network flow.<\/p>\n\n\n\n

We are interested in several aspects of evolution: well-posedness, long-time behavior, analysis of singularities, flow past singularities, and stability. Our main goal is to exploit the known results to a higher dimension.<\/p>\n\n\n\n

Members<\/h6>
\n\n\n
\n
\n
\n<\/i><\/a>\n\nMatteo Novaga\n\n<\/span>
\n<\/i>matteo.novaga@unipi.it<\/a>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
\n<\/i><\/a>\n\nAlessandra Pluda\n\n<\/span>
\n<\/i>alessandra.pluda@unipi.it<\/a>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div><\/div>\n\n\n\n
Collaborators<\/h6>
\n\n\n
\n
\n
\n<\/i>\n\nAnnalisa Cesaroni (Universit\u00e0 di Padova)\n\n<\/span>
\n\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
\n<\/i>\n\nAntonin Chambolle (Universit\u00e9 Paris Dauphine)\n\n<\/span>
\n\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
\n<\/i><\/a>\n\nNicola De Nitti (Universit\u00e0 di Pisa)\n\n<\/span>
\n<\/i>nicola.denitti@unipi.it<\/a>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
\n<\/i>\n\nCarlo Mantegazza (Universit\u00e0 degli Studi di Napoli Federico II)\n\n<\/span>
\n\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
\n<\/i>\n\nRafe Mazzeo (Stanford University)\n\n<\/span>
\n\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
\n<\/i>\n\nMassimiliano Morini (Universit\u00e0 di Parma)\n\n<\/span>
\n<\/i>massimiliano.morini@unipr.it<\/a>\n<\/div>\n<\/div>\n<\/div>\n
\n
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\n<\/i>\n\nMarcello Ponsiglione (Sapienza Universit\u00e0 di Roma)\n\n<\/span>
\n\n<\/div>\n<\/div>\n<\/div>\n
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\n<\/i>\n\nMarco Pozzetta (Politecnico di Milano)\n\n<\/span>
\n<\/i>marco.pozzetta@gmail.com<\/a>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
\n<\/i>\n\nMariel Saez (Pontificia Universidad Cat\u00f3lica de Chile)\n\n<\/span>
\n\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div><\/div>\n<\/div><\/div>\n\n\n\n
Geometric measure theory   <\/h5>
\n

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 non-integer, 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 d-dimensional 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).<\/p>\n\n\n\n

In the past fifty years these notions played a fundamental role in the treatment of many problems of geometric and variational nature besides minimal surfaces, from the asymptotic behavior of singular perturbations problems in continuum mechanics and physics (e.g, scalar and vector models of Ginzburg-Landau type) to the notions of weak solutions of geometric evolution problems (e.g., the mean curvature flow), and even the notion of generalized surfaces (Young measures) as a means to solve all types of variational and non variational problems.<\/p>\n\n\n\n

At the same time, many attempts have been made to extend these theories to various non-Euclidean settings (sub-Riemannian 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.<\/p>\n\n\n\n

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:<\/p>\n\n\n\n