Dipartimento di Matematica
Guido Castelnuovo - In dismissione

Lenya Ryzhik (Stanford University)

Diffusion of knowledge and the lottery society

Diffusion of knowledge models in macroeconomics describe the evolution of an interacting system of agents who perform individual Brownian motions (this is internal innovation) but also can jump on top of each other (this is an agent or a company acquiring knowledge from another agent or company). The learning strategy of the individual agents (jump probabilities) are obtained from an additional optimization problem that involves the current configuration of particles and is a solution to a forward-backwards in time mean-field game. We will discuss some results on the basic properties of this system. This is a joint work with H. Berestycki, A. Novikov and J.-M. Roquejoffre.

Il seminario si svolgerà all'interno delle attività del progetto PRIN 2022W58BJ5 ``PDEs and optimal control methods in mean field games, population dynamics and multi-agent models'' finanziato dall'Unione europea - Next Generation EU.

Seminari A.Ma.Ca. (Analisi Matematica al Castelnuovo)

Isabeau Birindelli (Sapienza, Università di Roma)

Test function approach to fully nonlinear equations in thin domains

In this seminar we will illustrate a work in collaboration with Ariela Briani and Hitoshi Ishii that extents the well known result on thin domains of Hale and Raugel. The test function approach of C. Evans is very powerful and gives new results even in the case of the Laplacian.

Emanuele Spadaro (Sapienza, Università di Roma)

The asymptotics of volume preserving geometric flows and a sharp quantitative Alexandrov inequality

I will present a study on the asymptotic behavior of the volume preserving mean curvature and the Mullins-Sekerka flat flows in three dimensional space, for which we need to establish a sharp quantitative version of the Alexandrov inequality for sets with a perimeter bound.

Seminari A.Ma.Ca. (Analisi Matematica al Castelnuovo)

Eugenio Montefusco (Sapienza, Università di Roma)

Only 3-points survive!

We study some qualitative properties of the solutions to a segregation limit problem in planar domains. The main goal is to show that, generically, the limit configuration of N competing populations consists of a partition of the domain whose singular points are (N-2) triple points, meaning that at most three populations meet at any point on the free boundary. To achieve this, we relate the solutions of the problem to a particular class of harmonic maps in singular spaces, which can be seen as the real part of certain holomorphic functions. The genericity result is obtained by tricky perturbation arguments.

Andrea Terracina (Sapienza, Università di Roma)

Radon measure-valued solutions of scalar conservation laws

We present some results for Radon measure-valued solutions of first order scalar conservation laws. In particular we discuss the case in which the singular part of the initial datum is a superposition of Dirac masses. In this case it is necessary to give an entropy formulation for Radon measure solutions. The main point is that the negative and the positive singular part of the measure have support along given characteristics. On the support of the singular part of the measure we have to impose, for the regular part, some compatibility conditions. When the flux is bounded the entropy formulation and the compatibility conditions are enough to characterize a unique solution. In the case of unbounded fluxes the positive and the negative singular part of the measure can intersect in a finite time and the dynamics is not clear. Therefore, we consider a proper approximation of the original problem in order to determine which are the natural conditions to be imposed. In this way it is possible to give a formulation for which the problem is well posed. The results are obtained in collaboration with M. Bertsch, F. Smarrazzo and A. Tesei.

Ugo Bessi (Università Roma Tre)

Dirichlet forms and dynamics on fractals

On several types of fractals, it is possible to build a Dirichlet form in a natural way; it is also possible to define a dynamical system and we shall see that the Dirichlet form has a natural relation with the dynamics. On the dynamical side, we shall use this fact to study the distribution of periodic orbits on the fractal; on the analytical side we shall study the structure of the space of $L^2$ one-forms on the fractal.

Il seminario si svolgerà all'interno delle attività del progetto PRIN 2022W58BJ5 ``PDEs and optimal control methods in mean field games, population dynamics and multi-agent models'' finanziato dall'Unione europea - Next Generation EU.

Angela Pistoia (Sapienza, Università di Roma)

Some properties of Steklov eigenfunctions.

I present some results concerning the number of critical points and the number of nodal domains of Steklov eigenfunctions. The results have been obtained in collaboration with Luca Battaglia (Roma 3), Alberto Enciso (ICMAT Madrid) and Luigi Provenzano (Sapienza Roma).

Il seminario si svolgerà all'interno delle attività del progetto PRIN 2022W58BJ5 ``PDEs and optimal control methods in mean field games, population dynamics and multi-agent models'' finanziato dall'Unione europea - Next Generation EU.

Emanuele Haus (Università Roma Tre)

Normal form and dynamics of the Kirchhoff equation

In this talk I will present some recent results on the Kirchhoff equation of nonlinear elasticity, describing transversal oscillations of strings and plates, with periodic boundary conditions. We are interested in the long-time existence and long-time dynamics of small amplitude solutions; to investigate such questions, we compute the normal form of the equation close to the elliptic equilibrium corresponding to the null solution. At the first step of normal form, one is able to erase from the equation all the cubic terms giving a nonzero contribution to the time evolution of the Sobolev norm of solutions; thus we deduce that, for initial data of size ϵ in Sobolev class, the time of existence of the solution is at least of order ϵ−4 (which improves the lower bound ϵ−2 coming from the linear theory). After the second step of normal form, there remain some resonant terms (which cannot be erased) of degree five that give a non-trivial contribution to the time evolution of the Sobolev norm of solutions. Nonetheless, we show that small initial data satisfying a suitable nonresonance condition produce solutions that exist over a time of order at least ϵ−6. On the other hand, we use the effective terms of degree five to construct some special solutions exhibiting a chaotic-like behavior. These results were obtained in collaboration with P. Baldi, F. Giuliani, M. Guardia.

This seminar is part of the activities of the Excellence Department Project CUP B83C23001390001.

Antonia Diana (Sapienza, Università di Roma)

The Surface Diffusion Flow: long time behavior and stability

In a recent paper, we study the long-time behavior and the stability of the surface diffusion flow of smooth hypersurfaces in the flat torus T^n. According to this flow, smooth hypersurfaces move with the outer normal velocity given by the Laplacian of their mean curvature. A first local-in-time existence (and uniqueness) theorem was presented by Escher, Mayer and Simonett, then long-time existence results, in dimensions two and three, were shown by Elliot and Garcke, Wheeler, Acerbi, Fusco and Morini, etc. Even if the three-dimensional case is the most relevant from the physical point of view, since it describes the evolution in time of interfaces between solid phases of a system, driven by the surface diffusion of atoms under the action of a chemical potential, we aim to generalize these results to arbitrary dimensions. More precisely, we show that if the initial set is sufficiently close to a strictly stable critical set for the volume-constrained Area functional and it has ``small energy'', then the flow actually exists for all times and asymptotically converges in a suitable sense to a ``translated'' of the critical set. This is a joint work with Nicola Fusco and Carlo Mantegazza (Scuola Superiore Meridionale and Università degli Studi di Napoli Federico II).

Paolo De Donato (Sapienza, Università di Roma)

The Stepanov theorem for multiple valued functions

The classical Stepanov theorem strengthens the Rademacher theorem by establishing almost-everywhere differentiability for pointwise Lipschitz functions into Euclidean spaces. In this seminar, I will discuss my recent extension of the Stepanov theorem to a broader setting: pointwise Lipschitz functions between general metric spaces with a generalized notion of differentiability that unifies traditional differentiability in Euclidean spaces with Almgren's framework for multiple valued functions.

Emanuela Radici (Università de L'Aquila)

Entropy solutions of scalar nonlocal conservation laws with congestion

In this talk we consider a class of scalar nonlinear models describing crowd dynamics. The congestion term appears in the transport equation in the form of a compactly supported nonlinear mobility function, thus making standard weak-type compactness arguments and uniqueness of weak solutions fail. We introduce two different approaches to the problem and discuss their connections with the wellposedness of entropy solutions of the target pde in the sense of Kruzkov. A deterministic particle ap- proach relying on suitable generalisations of the Follow-the-leader scheme, which can be interpreted as the Lagrangian discretisations of the problem; and a variational approach in the spirit of a minimising movement scheme exploiting the gradient flow structure of the evolution in a suitable metric framework. This seminar is part of the activities of the Excellence Department Project CUP B83C23001390001 and it is funded by the European Union – Next Generation EU.

Daniele Castorina (Università di Napoli Federico II)

Mean-Field sparse optimal control of systems with additive white noise

We analyze the problem of controlling a multiagent system with additive white noise through parsimonious interventions on a selected subset of the agents (leaders). For such a controlled system with an SDE constraint, we introduce a rigorous limit process toward an infinite dimensional optimal control problem constrained by the coupling of a system of ODEs for the leaders with a McKean--Vlasov type of SDE, governing the dynamics of the prototypical follower. The latter is, under some assumptions on the distribution of the initial data, equivalent with a (nonlinear parabolic) PDE-ODE system. The derivation of the limit mean-field optimal control problem is achieved by linking the mean-field limit of the governing equations together with the Gamma-limit of the cost functionals for the finite-dimensional problems. This is a joint research project with Francesca Anceschi (Ancona), Giacomo Ascione (SSM Napoli) and Francesco Solombrino (Napoli Federico II).

This seminar is part of the activities of the Excellence Department Project CUP B83C23001390001 and it is funded by the European Union – Next Generation EU.

Marco Cirant (Università di Padova)

Some results on Nash systems with several equations

Nash systems are strongly coupled systems of semilinar parabolic equations that describe closed-loop Nash equilibria in stochastic differential games. Despite existence, uniqueness and regularity for this class of systems is now classical, new questions regarding the regularity of solutions as the number of players increases have been posed within the study of large population games. It is indeed desirable to justify limiting models for these games as the number of players goes to infinity; from a PDE perspective, this amounts to obtain Lipschitz estimates (and further rather precise information on higher order derivatives) as the number of equations goes to infinity. The talk will be devoted to the presentation of some results in this direction, obtained in collaboration with D. F. Redaelli (University of Rome Tor Vergata).

This seminar is part of the activities of the Excellence Department Project CUP B83C23001390001 and it is funded by the European Union – Next Generation EU.

Annalisa Cesaroni (Università di Padova)

Self-organizing equilibria in a Kuramoto mean field game

I will discuss a mean field game model on the synchronization of coupled oscillators, initially proposed by Yin, Mehta, Meyn, and Shanbhag, and recently examined by Carmona, Cormier, and Soner. This model aims to study phase transitions in non-cooperative dynamic games involving a large number of agents. I will present a recent result obtained on this model in collaboration with Marco Cirant (University of Padova). Our work focuses on characterizing synchronized equilibria and their local dynamic stability.

This seminar is part of the activities of the Excellence Department Project CUP B83C23001390001 and it is funded by the European Union – Next Generation EU.

Davide Barilari (Università di Padova)

Towards unified synthetic curvature bounds for Riemannian and sub-Riemannian geometry

The Lott-Sturm-Villani theory of CD(K, N) metric measure spaces satisfying Ricci curvature lower bounds in a synthetic sense via optimal transport, though extremely successful, has been shown not to directly apply to sub-Riemannian geometries. Nonetheless, still using optimal transport tools, some entropy inequalities have been proved to hold in the case of the Heisenberg group and more in general in sub-Riemannian manifolds. In this talk we survey the known results and motivate a new approach we propose aiming to unify Riemannian and sub-Riemannian synthetic Ricci lower bounds, introducing suitable curvature dimension conditions. (joint with Andrea Mondino (Oxford) and Luca Rizzi (SISSA))

This seminar is part of the activities of the Excellence Department Project CUP B83C23001390001 and it is funded by the European Union – Next Generation EU.

Antonio Esposito (DISIM, Università dell'Aquila)

Recent developments on evolution equations on graph

The seminar concerns the study of evolution equations on graphs, motivated by applications in data science and opinion dynamics. We will discuss graph analogues of the continuum nonlocal-interaction equation and interpret them as gradient flows with respect to a graph Wasserstein distance, using Benamou--Brenier formulation. The underlying geometry of the problem leads to a Finslerian gradient flow structure, rather than Riemannian, since the resulting distance on graphs is actually a quasi-metric. We will address the existence of suitably defined solutions, as well as their asymptotic behaviour when the number of vertices converges to infinity and the graph structure localises. The two limits lead to different dynamics. From a slightly different perspective, by means of a classical fixed-point argument, we can show the existence and uniqueness of solutions to a larger class of nonlocal continuity equations on graphs. In this context, we consider general interpolation functions of the mass on the edges, which give rise to a variety of different dynamics. Our analysis reveals structural differences with the more standard Euclidean space, as some analogous properties rely on the interpolation chosen. The latter study can be extended to equations on co-evolving graphs. The talk is based on works in collaboration with G. Heinze (Wias Berlin), L. Mikolas (Oxford), F. S. Patacchini (IFP Energies Nouvelles), A. Schlichting (University of Ulm), and D. Slepcev (Carnegie Mellon University).

This seminar is part of the activities of the Excellence Department Project CUP B83C23001390001 and it is funded by the European Union – Next Generation EU.

Mariapia Palombaro (Università de L’Aquila)

Differential inclusions and polycrystals

We discuss a differential inclusion arising in the context of bounding effective conductiv- ities of polycrystalline composites. The datum is a set of three positive numbers identified with a positive definite diagonal matrix S. The aim is to find suitable solutions to the inclusion DU ∈ K := {λR^t S R : λ ∈ R, R ∈ SO(3)}. We will show how to construct a class of so-called approximate solutions via infinite-rank laminations. The resulting average fields provide an inner bound for the quasi-convex hull of K, which is an improvement of the bounds that were previously established by Avellaneda et al. [1] and Milton & Nesi [2]. We will also discuss some open problems related to the lack of outer bounds and the existence of exact solutions to the differential inclusion. This is joint work with N. Albin and V. Nesi.

References

[1] M. Avellaneda, A. V. Cherkaev, K. A. Lurie, G. Milton. On the effective conductivity of polycrystals and a three-dimensional phase-interchange inequality. J. Appl. Phys. 63, 4989–5003, 1988.

[2] V. Nesi, G. Milton. Polycrystalline configurations that maximize electrical resistivity. J. Mech. Phys. Solids no. 4, 525–542, 1991.

Fabio Cavalletti (Università di Milano)

Timelike Ricci bounds in the non-smooth setting: an optimal transport approach

Optimal transport tools have been extremely powerful to study Ricci curvature, in particular Ricci lower bounds in the non-smooth setting of metric measure spaces (which can be been as a non-smooth extension of Riemannian manifolds). Since the geometric framework of general relativity is the one of Lorentzian manifolds (or space-times), and the Ricci curvature plays a prominent role in Einstein’s theory of gravity, a natural question is whether optimal transport tools can be useful also in this setting. The goal of the talk is to introduce the topic and to report on recent progress. More precisely: After recalling some basics of optimal transport, we will define "timelike Ricci curvature and dimension bounds" for a possibly non-smooth Lorentzian space in terms of displacement convexity of suitable entropy functions and discuss applications, including the extension of classical singularity theorems to settings of low regularity, and prove a new isoperimetric-type inequality. Based on joint works with A. Mondino.

This seminar is part of the activities of the Excellence Department Project CUP B83C23001390001 and it is funded by the European Union – Next Generation EU.

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This seminar is part of the activities of the Excellence Department Project CUP B83C23001390001 and it is funded by the European Union – Next Generation EU.

Salvatore Stuvard (Università di Milano)

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This seminar is part of the activities of the Excellence Department Project CUP B83C23001390001 and it is funded by the European Union – Next Generation EU.

Camilla Polvara (Sapienza Università di Roma)

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This seminar is part of the activities of the Excellence Department Project CUP B83C23001390001 and it is funded by the European Union – Next Generation EU.