Matteo Carducci (SNS, Pisa)
An epiperimetric inequality for odd frequencies in the thin obstacle problem
11/06/2025, 14:30, Aula S (Palazzo Campana)
Abstract: In this talk we consider the thin obstacle problem, which consists in minimizing the Dirichlet integral in $B_1$, among functions $u\ge0$ on $B_1\cap\{x_{n+1}=0\}$ and with a prescribed boundary datum on $\partial B_1$. The points of the free boundary of a solution can be classified according to their value of Almgren's frequency function. In particular, we study the points with odd frequency, obtaining a stratification result through the use of an epiperimetric inequality.
These results are obtained in a joint work with B. Velichkov.
Donato Scarcella (Universitat Politècnica de Catalunya)
Breakdown of the homoclinic connection around L4 in the RPC3BP beyond the Routh mass ratio
02/04/2025, 14:30, Aula 2 (Palazzo Campana)
Abstract: we consider the Restricted Planar Circular 3-Body Problem (RPC3BP) close to the Lagrangian critical point L4. The RPC3BP describes the motion of a massless body under the gravitational influence of two massive bodies (the primaries) performing circular orbits, assuming that the massless body moves in the same plane as the primaries. For values of the mass parameter greater than a certain critical threshold (the Routh critical mass ratio), L4 is a complex saddle critical point. We study the two-dimensional stable and unstable manifolds associated with L4 and provide an asymptotic formula for their mutual distance.
This study is a collaborative effort with Inmaculada Baldomá (UPC) and Pau Martín (UPC).
Domenico Vuono (Università della Calabria)
Second and third order estimates for solutions to p-Laplacian equations
26/03/2025, 14:30, Aula Marro (Palazzo Campana)
Abstract: Abstract Vuono
Santiago Barbieri (Universitat Politècnica de Catalunya)
Existence and nonexistence of invariant curves of coin billiards.
19/03/2025, 14:30, Aula 1 (Palazzo Campana)
Abstract: In this talk I will consider the coin billiard introduced by M. Bialy. It is a modification of the classical billiard, obtained as the return map of a nonsmooth geodesic flow on a cylinder that has homeomorphic copies of a classical billiard table on the top and on the bottom (a coin). The return dynamics is described by a map T of the annulus. Together with A. Clarke, we proved the following three main theorems: in two different scenarios (when the height of the coin is small, or when the coin is near-circular) there is a family of KAM curves close to, but not accumulating on, the boundary of the annulus; for any noncircular coin, if the height of the coin is sufficiently large, there is a neighbourhood of the boundary through which there passes no invariant essential curve; and the only coin billiard for which the phase space is foliated by essential invariant curves is the one built on a circular table. These results provide partial answers to questions of Bialy. Finally, I will describe the results of some numerical experiments on the elliptical coin billiard.
Joint work with Andrew Clarke
Marco Caroccia (Politecnico di Milano)
On the singular planar plateau problem
18/02/2025, 14:00, Aula 4 (Palazzo Campana)
Abstract: The classical Plateau problem asks which surface in three-dimensional space spans the least area among all the surfaces with boundary given by an assigned curve S. This problem has many variants and generalizations, along with (partial) answers, and has inspired numerous new ideas and techniques. In this talk, we will briefly introduce the problem in both its classical and modern contexts, and then we will focus on a specific vectorial (planar) type of the Plateau problem. Given a curve S in the plane, we can ask which diffeomorphism T of the disk D maps the boundary of D to S and spans the least area, computed as the integral of the Jacobian of T, among competitors with the same boundary condition. For simply connected curves, the answer is provided by the Riemann map, and the minimal area achieved is the Lebesgue measure of the region enclosed by S. For more complex curves, possibly self-intersecting, new analysis is required. I will present a recent result in this sense, obtained in collaboration with Prof. Riccardo Scala from the University of Siena, where the value of the minimum area is computed with an explicit formula that depends on the topology of S
André Guerra (ETH, Zurigo)
Harmonic maps and the vectorial obstacle problem
23/01/2025, 14:30, Aula Lagrange (Palazzo Campana)
Abstract: I will discuss some recent results obtained in collaboration with A. Figalli, S. Kim and H. Shahgholian. We consider minimizers of the Dirichlet energy among maps constrained to take values outside a smooth domain O in R^m. These minimizers can be thought of either as solutions of a vectorial obstacle problem, or as harmonic maps into the manifold-with-boundary given by the complement of O. I will discuss results concerning the regularity of the minimizers, the location of their singularities, and the structure of the free boundary.
Emanuele Pastorino (Politecnico di Milano)
Unpredictable behavior of a partially damped system of PDEs modeling suspension bridges
12/12/2024, 14:30, Aula 1 (Palazzo Campana)
Abstract: We consider a nonlinear nonlocal coupled system of beam-wave equations, governing the dynamics of a degenerate plate modeling the behavior of supspension bridges. In line with the physical observations, the beam equation is both damped and forced, whereas the wave equation is "isolated". Since the resulting overall damping is degenerate, the full dynamical system is only partially dissipative. This leads to an unpredictable behavior for the solutions of the system, which prevents to forecast the general behavior of bridges, encompassing the instability of the unique stationary solution and the richness of the ω-limit set, which contains infinitely many periodic solutions. We also discuss some striking differences between the sole beam equation and the coupled system, which may be the cause of some not fully understood paradoxes and erroneous conclusions. The talk is based on joint work with Maurizio Garrione and Filippo Gazzola (Politecnico di Milano).
Erica Scantamburlo (Politecnico di Torino)
Effects of multiple periselene passages on satellites attitude dynamics in near-rectilinear halo orbits
19/11/2024, 14:30. Aula Lodi (Palazzo Campana)
Abstract: With the announcement of the NASA Artemis program ‘Lunar Gateway’, a lot of attention has been dedicated to the near-rectilinear halo orbits (NRHOs) originating at the Earth–Moon L2 point.
In this talk, we discuss the phenomena characterizing the rotational dynamics of an axisymmetric satellite along the NRHOs. For this purpose, we consider a model where only the gravity gradient due to the Earth and the Moon is considered as an external moment. We prove that the gravity gradient along the NRHOs can be approximated as an impulse acting at the periselene, and we investigate the effects of multiple periselene passages on the attitude.
Farid Bozorgnia (Technical University of Lisbon)
Graph-Based Semi-supervised Learning Using Spatial Segregation Theory
12/11/2024, 14:30. Aula 5 (Palazzo Campana)
Abstract: In this talk, we briefly explain various models of Reaction-Diffusion
Systems characterized by high competition rates. We investigate the existence and uniqueness of solutions for each model and the numerical approximation of their singular limit. Next, I address graph-based semisupervised learning that leverages the theory of these competitive-type systems of PDEs to classify data when only a few labels are available.
Gabriele Fioravanti (Università di Torino)
Regularity for elliptic PDEs degenerating on lower dimensional manifolds
06/11/2024, 14:30. Aula 5 (Palazzo Campana)
Abstract: link
Marco Gallo (Università Cattolica del Sacro Cuore, Brescia)
Limit is not unique? Concavity for a logarithmic equation
16/10/2024, 14:30. Aula Lagrange (Palazzo Campana)
Abstract: In this talk we will discuss the concavity properties of positive solutions to Dirichlet problems set in convex domains. In this framework, the best one can hope for is that a certain composition of the solution with a suitable concave function (e.g. a logarithmic, a square root, etc.) is, indeed, concave.
We will briefly review some classical results, trying to understand that the precise shape of the nonlinearity plays a crucial role, and thus general theorems are not available. In particular, we will eventually discuss the case of the logarithmic equation, and try to figure out why this is a tough guy and how to tackle it. This talk is based on a joint work with Sunra Mosconi and Marco Squassina.
Yuying Liang (Beihang University, Beijing, PR China)
Artificial Intelligence for Dynamical System: More than Application
9/10/2024, 14:30. Aula Lagrange (Palazzo Campana)
Understanding the internal structure of a small body, especially an irregular-shaped asteroid, is not only significant for close-proximity operations of exploration missions, but also provides scientific insights into its origin, current state, and evolution. Unfortunately, the current technology is not advanced enough to scan through asteroids to provide direct measurements of their internal structures (and seems to remain impossible in the near future). An efficient and indirect approach that is widely accepted and has been applied to missions is to inverse the density distribution from the full gravity field. The optical measurements also provide information on the spacecraft’s location in the asteroid-centred body frame. This technical fact enlightens us with the question of whether it is applicable to “read” the density distribution of asteroids directly from the spacecraft’s flight trajectory. As a remarkably challenging underfit inverse problem, the classic method, e.g., the least-square approach, does not show reliable capacity due to the highly unmatched dimension and then a kind of invertible neural networks are introduced. Compared with Dawn team's results, we will present how this data-drive approach is able to accelerate our understanding of a dynamical system and space science.
Valerio Assenza (IMPA, Rio de Janeiro, Brasile)
Magnetic curvature and existence of a closed magnetic geodesic on low energy levels
27/09/2024, 14:30. Aula Lagrange (Palazzo Campana)
Magnetic systems are the natural toy model for the motion of a charged particle moving on a Riemannian manifold under the influence of a (static) magnetic force. In this talk we introduce a curvature operator called magnetic curvature which encodes the information of the classical Riemannian curvature together with terms of perturbation due to the magnetic interaction. We use this new notion of curvature to approach the problem of finding closed trajectories. In a variational setting, with techniques a la Bonnet-Myers, we prove the existence of a closed trajectory for small energies positively curved in this new magnetic sense.