Embedded Files
Ben Seeger (University of North Carolina)
Error estimates for finite dimensional approximations of Hamilton-Jacobi-Bellman equations on the Wasserstein space
6 giugno 2025 alle ore 11:00: in aula 1BC45
We consider finite-dimensional approximations of equations posed on the space of probability measures, which arise in mean field control and game theory. Our main purpose is to develop methods for estimating the error between the L-viscosity solution and its approximation that rely on PDE methods alone, which is a relevant question when there is no useful representation formula for the solution. We also address questions about the optimal quantization of probability measures by a deterministic empirical measure.
---------------
Masato Kimura (Kanazawa University)
Energy-dissipation in irreversible phase field fracture models and their extensions
5 giugno 2025 alle ore 13:30: in aula 1BC45
This talk presents an overview of the Irreversible Phase Field Model for Fracture (Irreversible F-PFM) proposed by the speaker and collaborators, which is based on the concept of irreversible gradient flows. An irreversible gradient flow is a constrained gradient flow in which monotonicity in time is enforced, leading to a natural energy-dissipation identity [Akagi–Kimura, 2019; Kimura–Negri, 2021].
The Irreversible F-PFM applies this framework to the Ambrosio–Tortorelli regularization of the variational fracture model by Francfort and Marigo, ensuring both the irreversibility of crack evolution and monotonic energy decay (i.e., a consistent dissipation structure). This model is compactly formulated using partial differential equations and is readily amenable to standard finite element solvers, enabling crack propagation simulations without a priori knowledge of the crack path.
After reviewing the mathematical structure of the Irreversible F-PFM, the talk will introduce a range of recent extensions and their finite element simulations, each highlighting how the dissipation structure is preserved or adapted. These include: Irreversible F-PFM under unilateral constraints; Thermo-mechanical fracture models; Phase field modeling of fracking in oil and gas reservoirs; Desiccation-induced fracture modeling; Dynamic phase-field fracture models and seismic fault rupture; Through these examples, we aim to highlight the versatility and robustness of the irreversible phase-field approach in simulating complex fracture phenomena across disciplines.
---------------
Alberto Bressan (PennState University)
Uniqueness domains for 2 x 2 conservation laws.
3 giugno 2025 alle ore 11:30: in aula 1BC45
The present well-posedness theory for hyperbolic conservation laws mainly applies to solutions with small total variation.
On the other hand, for a genuinely nonlinear 2 x 2 system of conservation laws, a classical paper by J.Glimm and P.Lax (Memoirs AMS, 1970) established the global existence of weak solutions, for initial data with small L^\infty norm, but possibly unbounded variation. The total variation of these solutions decays like 1/t. Their uniqueness has remained a challenging open problem.
This talk will present some new results in this direction, inspired by the theory of linear analytic semigroups and intermediate domains for sectorial operators. Roughly speaking, we show that solutions whose total variation decays at a faster rate are unique and exhibit Holder continuous dependence.
An auxiliary result identifies initial data with fast decay rate, introducing a suitable class of metric interpolation spaces.
This is a joint research with E.Marconi and G.Vaidya.
---------------
Joe Jackson (University of Chicago)
Quantitative convergence for displacement monotone mean field games of control
30 maggio 2025 alle ore 11:30: in aula 2AB40
Mean field games of controls (MFGC) are mean field games in which players interact with each other through their controls, in addition to their states. MFGC provide a natural modelling framework in economics, where players (households, firms, etc.) often interact with each other through some macroeconomic variable (price, interest rate, etc.) which is determined by the distribution of the players' controls (consumption vs. saving, production rate, etc.). Mathematically, the main novelty is the appearance of a fixed point equation on the space of measures, which is coupled with the usual equations of MFG theory. In this talk, I will discuss an ongoing joint work with Alpár Mészáros, in which we study the convergence problem for MFGC, i.e. the problem of rigorously justifying the connection between MFGC and the corresponding sequence of finite player games. Our main results provide convergence rates for the equilibria of the finite player games towards their MFGC limit as the number of players goes to infinity. Rather than analyzing the master equation (which seems to be quite challenging in the setting of MFGC), we first prove a convergence result for open-loop equilibria via "forward-backward propagation of chaos", and then transfer this estimate to closed-loop equilibria by using some uniform in N estimates on the N-player Nash system.
---------------
Amandine Aftalion (Université Paris Saclay)
Optimizing running performance
29 maggio 2025 alle ore 11:15: in aula 1BC50
We will describe how math and especially optimal control can
help understand running performance. How can we best distribute our effort and energy to achieve the best time in a race? The human body energy comes from several sources: aerobic energy (breathing) and anaerobic energy. To this, we need to couple muscle control by the brain, and motivation, to get
an economic model of cost-benefit optimization that enables us to
understand how to reach the best time. This depends of course on distance, but also on physiology and motivation. We will show how a system of coupled ODEs can help us understand the best athletes' way of running at world championships.
---------------
Alberto Maione (CRM Barcelona)
H-compactness for nonlocal linear operators in fractional divergence form
20 maggio 2025 alle ore 14:30: in aula 2AB40
In this talk, we present a new result on the compactness of possibly non-symmetric and nonlocal linear operators in fractional divergence form with respect to the H-convergence.
Specifically, we consider cases where the oscillations of the coefficient matrices are prescribed outside the reference domain.
The compactness argument we introduce overcomes the limitations of classical localization techniques, which mismatch with the nonlocal nature of the operators.
In the second part of the presentation, we explore the case of symmetric operators and show an equivalence between the H-convergence of the nonlocal operators and the Gamma-convergence of the corresponding energies.
Finally, we discuss a set of open problems and new research directions arising from this work.
This research is carried out in collaboration with Maicol Caponi (University of L'Aquila) and Alessandro Carbotti (University of Salento).
---------------
Alessandro Carbotti (Universita del Salento)
On the local regularity of very weak s-harmonic functions
19 maggio 2025 alle ore 14:30: in aula 2AB40
The aim of this talk is to give a new proof that any very weak s- harmonic function u in the unit ball B is smooth.
As a first step, we improve the local summability of u. Then, we exploit a suitable version of the Nirenberg difference quotient method tailored to get rid of the singularity of the integral kernel and gain H^s_loc Sobolev regularity on u. Finally, by applying more standard methods, we reach real analyticity of u.
The talk is based on a joint work with S. Cito, D.A. La Manna and D. Pallara.
---------------
Pierre Cardaliaguet (Universite Paris Dauphine)
Traffic flow on junctions: mean field limits and optimization
7 maggio 2025 alle ore 11:30: in aula 1BC50
Traffic flow problems on junctions (or intersections) have been the subject of abundant literature in recent years. The modeling involves scalar conservation laws with discontinuities at the junction points, or, sometimes equivalently, Hamilton-Jacobi equations with discontinuous Hamiltonians. We will present the existence and uniqueness results for these equations, then explain how to derive these continuous models (where traffic is seen as a fluid) from discrete models (describing in detail the individual behavior of vehicles). We will also explain how to optimize the traffic in the case of a simple junction. The talk is based on joint works with N. Forcadel, R. Monneau and P. Souganidis.
---------------
Rossano Sannipoli (Università di Padova)
On the optimal sets in Pólya type inequalities
30 aprile 2025 alle ore 12:00: in aula 2AB40
In this talk, we examine two scaling invariant shape functionals introduced by Pólya, involving the Lebesgue measure, the perimeter and, respectively, the torsional rigidity and the first Dirichlet-Laplacian eigenvalue for open, bounded convex sets of R^n. It is known that these functionals are bounded and some of their bounds are achieved by suitable optimizing sequences. We establish new quantitative estimates, which give us key properties and information on the behavior of such optimizing sequences. In particular, we consider two remainder terms that provide details about the geometrical structure of these sequences: the first one gives us information about their thickness; the second one is another shape functional that allows us to fully characterize them in a purely geometrical way.
---------------
Filomena De Filippis (Università di Parma)
μ-ellipticity and nonautonomous integrals
28 aprile 2025 alle ore 14:30: in aula 2AB40
µ-ellipticity describes certain degenerate forms of ellipticity typical of convex integrals at linear or nearly linear growth, such as the area integral or the iterated logarithmic model. The regularity of solutions to autonomous or totally differentiable problems is classical after Bombieri, De Giorgi and Miranda, Ladyzhenskaya and Ural’tseva and Frehse and Seregin. The anisotropic case is a later achievement of Bildhauer, Fuchs and Mingione, Beck and Schmidt and Gmeineder and Kristensen. However, all the approaches developed so far break down in presence of nondifferentiable ingredients. In particular, Schauder theory for certain significant anisotropic, nonautonomous functionals with Hölder continuous coefficients was only recently obtained by C. De Filippis and Mingione. We will see the validity of Schauder theory for anisotropic problems whose growth is arbitrarily close to linear within the maximal nonuniformity range, and discuss sharp results and insights on more general nonautonomous area type integrals. From recent, joint work with Cristiana De Filippis (Parma) and Mirco Piccinini (Pisa).
---------------
Alberto Roncoroni (Politecnico di Milano)
Liouville theorems for critical equations
3 aprile 2025 alle ore 14:30: in aula 2BC30
In this talk I will present some classification results for positive solutions to some classical critical elliptic equations in the Euclidean and Riemannian settings.
This is a joint work with G. Catino and D.D. Monticelli (Politecnico di Milano).
Moreover, if time permits, I will present some classification results in the sub-Riemannian setting.
These are joint works with G. Catino, D.D. Monticelli (Politecnico di Milano), Y.Y. Li (Rutgers University) and X. Wang (Michigan State University).
---------------
Ivan Gudoshnikov (Czech Academy of Sciences, Prague)
State-dependent sweeping process approach to modeling of elastoplastic networks with softening plasticity
2 aprile 2025 alle ore 14:30: in aula 2AB40
Softening plasticity and fracture mechanics lead to ill-posed mathematical problems due to the loss of monotonicity. Multiple co-existing solutions are possible when softening elements are coupled together, and solutions cannot be continued beyond the point of complete degradation of elastic range. Moreover, spatially continuous models with softening suffer from localization of strains and stresses to measure-zero submanifolds. We formulate a problem of quasistatic evolution of elasto-plastic spring networks (Lattice Spring Models) with a plastic flow rule which describes linear hardening, linear softening and perfectly plastic springs in a uniform manner. The fundamental kinematic and static characteristics of the network are described by the rigidity theory and structural mechanics.
To solve the evolution problem we convert it to a type of a differential quasi-variational inequality known as the state-dependent sweeping process. The existence of solution to the associated time-stepping problem (implicit catch-up algorithm) can be proved, and the related estimates obtain imply the existence of a solution to the (time-continuous) sweeping process. Using numerical simulations of regular grid-shaped networks with softening we demonstrate the development of non-symmetric shear bands.
At the same time, in toy examples it is easy to analytically derive multiple co-existing solutions, appearing in a bifurcation which happens when the parameters of the networks continuously change from hardening through perfect plasticity to softening.
---------------
Debora Amadori (Università dell'Aquila)
Conservation laws with discontinuous gradient-dependent flux
26 marzo 2025 alle ore 14:30: in aula 2AB40
This talk concerns a scalar conservation law with discontinuous gradient-dependent flux, in which the discontinuity depends on the sign of the derivative u_x of the solution; namely, the flux is given by two different smooth functions f(u) or g(u), when u_x is positive or negative, respectively.
By introducing a suitable vanishing viscosity approximation, the mutual positions of the graphs of f and g identify two different situations, resulting in a well-posed parabolic problem when f(u)<g(u) for all u, or an ill-posed problem if f(u)>g(u) for all u.
We refer to these problems as the stable/unstable case, respectively.
In the stable case, for periodic solutions, we prove that semigroup trajectories obtained as a limit of a suitable wave-front tracking algorithm coincide with the unique limits of an appropriate class of vanishing viscosity approximations.
In the unstable case, examples show that infinitely many solutions can occur. For initial data that are piecewise monotone, i.e., increasing or decreasing on a finite number of intervals, a global solution of the Cauchy problem is obtained. The solution is unique under the requirement that the number of interfaces, where the flux switches between f and g, remains as small as possible.
This is a joint work with Alberto Bressan and Wen Shen (Penn State University).
---------------
Elisa Continelli (Università di Padova)
First and second-order Cucker-Smale models involving time delays
12 marzo 2025 alle ore 11:30: in aula 2AB40
Recently, multiagent systems have caught the attention of many reserachers in various scientific fields. Among them, there are the Hegselmann-Krause opinion formation model and its second-order version, the Cucker-Smale model, introduced for the description of flocking phenomena. Typically, for solutions of such systems, their asymptotic behavior, consensus for the first-order model and exhibition of flocking for the second-order model, is investigated. In these models, it is natural to introduce time delay effects due to reaction times or to the propagation of information. In this talk, we will focus on some first and second-order Cucker-Smale type models involving time-dependent time delays, providing exponential consensus estimates for their solutions without assuming any smallness conditions on the delay size. Non-universal interactions and possible lack of interaction among the system's agents will be also considered.
Joint works with Chiara Cicolani and Cristina Pignotti
---------------
Felix Otto (Max Planck Institute - Leipzig)
Convection-enhanced diffusion and intermittency
6 marzo 2025 alle ore 13:30: in aula 1BC45
Convection in a divergence-free and time-independent drift enhances the
diffusivity. If the drift is chosen from an ensemble that scales like white
noise on large scales, two space dimensions are critical: a particle spreads
super-diffusively by the square root of a logarithm.
We use scale-by-scale stochastic homogenization to relate the Lagrangian
particle coordinates, in terms of their increments or rather their differential,
to a canonical diffusion on the Lie group SL(2).
Since this diffusion on SL(2) can be compared to a stochastic exponential,
we learn that the increments of the particle positions acquire an intermittency
over large time scales.
This is joint work with Morfe and Wagner.
---------------
Daniele Castorina (Università di Napoli Federico II)
Optimal control problems driven by nonlinear degenerate Fokker-Planck equations
13 febbraio 2025 alle ore 14:30: in aula 2BC30
The well-posedness of a class of optimal control problems is analysed, where
the state equation couples a nonlinear degenerate Fokker-Planck equation with a system
of Ordinary Differential Equations (ODEs). Such problems naturally arise as meanfield
limits of Stochastic Differential models for multipopulation dynamics, where a large
number of agents (followers) is steered through parsimonious intervention on a selected
class of leaders. The proposed approach combines stability estimates for measure solutions
of nonlinear degenerate Fokker-Planck equations with a general framework of assumptions
on the cost functional, ensuring compactness and lower semicontinuity properties.
The Lie structure of the state equations allows one for considering non-Lipschitz
nonlinearities, provided some suitable dissipativity assumptions are considered in addition
to non-Euclidean H¨older and sublinearity conditions. Joint work with F. Anceschi (Ancona), G. Ascione (Napoli SSM) e F. Solombrino (Lecce).
---------------
Daniele Castorina (Università di Napoli Federico II)
Mean-field sparse optimal control of systems with additive white noise
13 febbraio 2025 alle ore 11:30: in aula 7B1
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 infinitedimensional
optimal control problem constrained by the coupling of a system of ODE 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. Joint work with G. Ascione (Napoli SSM) e F. Solombrino (Lecce).
---------------
Alessandra Buratto (Università di Padova)
Controlling an Evolutionary Game to limit clientelism
6 febbraio 2025 alle ore 15:30: in aula 2AB45
We analyze the challenge of countering electoral clientelism in a given population of voters.
We model the problem as a Stag Hunt dilemma within evolutionary game theory, and adopt an infinite-time horizon optimal control approach, where an authority minimizes the social cost of clientelism through costly institutional reforms.
We incorporate a control function into the replicator dynamics of the game and derive the optimal steady-state controls using Pontryagin's Maximum Principle.
We analyze the new "controlled" dynamics, examining the optimal solution for different discount rate levels that define the authority's preferences.
Several complexities may arise, including the proof of sufficiency conditions (since the dynamics are neither concave/convex nor semiconcave/semiconvex) and the existence of multiple equilibria (Skiba points, Stalling equilibria, critical mass...).
Joint work with Laura Caravenna, Rudy Cesaretto, Oto Murer Montagner
The introduction of the problem will be given by the Visiting Scientist Oto Murer Montagner (School of International Relations, Fundação Getulio Vargas, Brazil https://ri.fgv.br/en/pessoas/oto-murer-kull-montagner)
---------------
Bahman Gharesifard (Queen's University, Kingston)
Structural output controllability of ensembles
23 gennaio 2025 alle ore 12:30: in aula 2BC60
The core idea in ensemble control, which originate from quantum spin systems, is that instead of controlling a single large system, we control a large population of small, independent ones using a single control input. A major technical challenge in ensemble control stems from the requirement that the control input be generated irrespective of the parameters of the individual systems. Roughly speaking, the larger the parameterization space is, the more individual systems are contained in the ensemble and hence, the more difficult it is to simultaneously control all of them. An important result among the recent advances on this topic states that real-analytic linear ensemble systems are not $\mathrm{L}^p$-controllable, for $2 \le p \le \infty$, when the underlying parameterization spaces are multidimensional. Therefore, one has to relax the notion of controllability and seek more relaxed controllable characteristics of the ensemble system. In this talk, I consider continuum ensembles of linear time-invariant control systems with single inputs, featuring a sparsity pattern structure, and study structural output controllability, in particular, structural average controllability, as a relaxation of ensemble controllability. I then provide a necessary and sufficient condition for a sparsity pattern to be structurally averaged controllable.
---------------
Giuseppe Negro (Istituto Superior Tecnico, Lisboa)
On sharp Fourier extension from spheres in arbitrary dimensions
18 dicembre 2024 alle ore 12:30: in aula 2BC30
As a corollary of the classical Stein--Tomas inequality, the Fourier extension operator from the (d-1)-dimensional sphere into R^d is bounded from L^2(S^(d-1)) into L^4(R^d) for all d>2. Since the seminal 2013 work of D.Foschi, it has been conjectured that this operator should attain its norm on functions that are constant on the sphere, as naturally suggested by rotational symmetry. This conjecture is still open for d>7. We will discuss recent progress towards it.
---------------
Luca Benatti (Università di Pisa)
A Unified Perspective on Monotonicity Formulas and its Impact on the Penrose Inequality
17 dicembre 2024 alle ore 12:30: in aula 1BC45
In this talk, we provide an overview of a broad family of monotonicity formulas developed over the past two decades. Our goal is to bring under one roof the most recent formulas in nonlinear potential theory and Geroch's monotonicity along the Inverse Mean Curvature Flow (IMCF), established in the groundbreaking 2001 paper by Huisken and Ilmanen. We will highlight the impact of this unified approach on the proof of geometric inequalities, with particular emphasis on the Riemannian Penrose inequality. This talk is based on joint work with Mattia Fogagnolo, Lorenzo Mazzieri, Alessandra Pluda, and Marco Pozzetta.
---------------
Enrico Pasqualetto (Università di Jyväskylä)
Vector calculus on extended metric-topological measure spaces
11 dicembre 2024 alle ore 11:00: in aula 7B1
With the aim of providing a framework for optimal transport problems and nonsmooth analysis in infinite-dimensional spaces, Ambrosio, Erbar and Savaré introduced the concept of extended metric-topological measure space (e.m.t.m. space, for short).
The class of e.m.t.m. spaces includes, inter alia, all `standard' metric measure spaces, abstract Wiener spaces, configuration spaces and extended Wasserstein spaces. In this setting, Savaré developed several notions of first-order Sobolev space and studied their relation. In this talk, I will introduce a new definition of Sobolev space via an integration-by-parts formula and prove its equivalence with the other approaches. Furthermore, I will present two notions of measurable vector fields, which are encoded into the theory of Lipschitz derivations. The differential calculus on e.m.t.m. spaces gives new insights also into classical Sobolev spaces, due to the fact that the category of e.m.t.m. spaces is closed under two useful operations: `taking the compactification' and `passing to the length-conformal distance'.
Based on joint works with Luigi Ambrosio, Toni Ikonen, Danka Lučić and Janne Taipalus.
---------------
Andrea Davini (Università La Sapienza di Roma)
Stochastic Homogenization of viscous HJ equations in 1d
25 ottobre 2024 alle ore 10:30: in aula 2AB45
In this talk I will present some new results I have recently obtained about homogenization of viscous Hamilton-Jacobi equations in dimension one in stationary ergodic environments with nonconvex Hamiltonians. In the non-degenerate case, i.e., when the diffusion coefficient is strictly positive, homogenization is established for superlinear Hamiltonians of fairly general type. This closes a long standing question. When, on the other hand, the diffusion coefficient degenerates, meaning that it is zero at some points or on some regions of the real line, homogenization is proved for Hamiltonians that are additionally assumed quasiconvex in the momentum variable. Furthermore, the effective Hamiltonian is shown to be quasiconvex. This latter result is new even in the periodic setting, despite homogenization being known for quite some time.
---------------
Gianmarco Giovannardi (Università di Firenze)
Schauder estimates up to the boundary on H-type groups: an approach via the double layer potential
16 ottobre 2024 alle ore 11:30: in aula 2AB45
We will show how to obtain the Schauder estimates at the boundary away from the characteristic points for the Dirichlet problem by means of the double layer potential in a Heisenberg-type group G.
Despite its singularity we manage to invert the double layer potential restricted to the boundary thanks to a reflection technique for an approximate operator in G.
This is the first instance where a reflection-type argument appears to be useful in the sub-Riemannian setting. This is a joint work with G. Citti and Y. Sire.
---------------
Steven Flynn (Università di Padova)
Harmonic Analysis and Sub-Riemannian Geometry
24 settembre 2024 alle ore 14:30: in aula 1BC45
The Heisenberg group has the simplest nontrivial Lie algebra, making it an ideal setting for gener- alizing Fourier analysis beyond Euclidean space. It is also the simplest example of a sub-Riemannian manifold, equipped with a natural sub-Laplacian. As observed by Strichartz, harmonic analysis on the Heisenberg group can be understood in two equivalent ways: through its representation theory or as the joint spectral analysis of the sub-Laplacian and the Reeb vector field. We introduce the necessary elements and demonstrate how the joint spectral decomposition of a function on the Heisenberg group corresponds to its decomposition into irreducible unitary representations of the group of Heisenberg isometries. Time permitting, we will present applications of this decomposition.
---------------
Mario Santilli (Università dell'Aquila)
Rigidity with (almost) constant higher order mean curvatures
18 settembre 2024 alle ore 11:00: in aula 1BC50
Page updated
Google Sites
Report abuse