Nonlinear Analysis Seminar 

Abstract: We introduce a new type of eigenvalues whose corresponding eigenfunctions satisfy an orthogonality constraint with respect to a given function. Then we study a shape optimization problem arising in this setting.

Based on a joint work with D. Zucco. 

Abstract: Magnetic flows are generalizations of geodesic flows that describe the motion of a charged particle in a magnetic field. While every closed Riemannian manifold admits at least one closed geodesic, the analogous problem for magnetic orbits (also known as magnetic geodesics) is significantly more challenging and has received considerable attention in recent decades. I will present a result establishing that every low energy level of any magnetic flow admits at least one contractible closed orbit, assuming only that the magnetic strength is not identically zero, has a compact strict local maximum K, and that the cohomology class of the magnetic field is spherically rational. Moreover, this magnetic geodesic can be localized within an arbitrarily small neighborhood of K. This is joint work with Valerio Assenza and Gabriele Benedetti.

Abstract:  In this talk we deal with some nonlocal and nonlinear PDEs driven

by the fractional Laplacian. In particular we discuss the existence of nontrivial

solutions for critical equations, also in presence of jumping nonlinearities.

The results obtained generalize to the nonlocal setting what is known in the

classical setting of elliptic PDEs.

This is a joint paper with Giovanni Molica Bisci, Kanishka Perera and Caterina

Sportelli.

Abstract: In this talk we deal with some nonlocal and nonlinear PDEs drivenby the fractional Laplacian. In particular we discuss the existence of nontrivial solutions for critical equations, also in presence of jumping nonlinearities.
The results obtained generalize to the nonlocal setting what is known in the classical setting of elliptic PDEs. This is a joint paper with Giovanni Molica Bisci, Kanishka Perera and Caterina Sportelli.

Abstract: By using mountain pass arguments and a novel group-theoretical approach, in this talk we discuss the existence of multiple sequences of nodal solutions with prescribed different symmetries for a wide class of critical elliptic problems settled on the unit sphere of dimension $d\geq 5$, whose simple prototype is given by the celebrated Yamabe equation on the sphere.

Abstract:  This talk explores how the intersection theory of curves of Lagrangian subspaces in a symplectic space serves as a powerful and flexible tool for studying the Morse index in singular variational problems.

We begin by revisiting the classical interpretation of the Morse index as the number of negative eigenvalues of a Sturm–Liouville boundary value problem. This viewpoint can be reframed in terms of intersection numbers between Lagrangian curves and hypersurfaces determined by boundary conditions, all within the Gelfand–Robbin quotient framework.

We then present a general spectral flow formula expressed as the Maslov index of a path of Cauchy data spaces. Building on this, we extend the theory to cover one- and two-sided singular Sturm–Liouville operators by employing symplectic reduction techniques. This generalization captures and unifies existing abstract results underlying the index theory for asymptotic and doubly asymptotic motions—such as collisions or parabolic trajectories—in Celestial Mechanics.

Our results synthesize analytic and symplectic approaches, providing robust tools for computing spectral flows in singular variational settings. Applications span a range of fields, including celestial mechanics (e.g., perturbations in planetary systems), time-dependent oscillatory systems, wave propagation in inhomogeneous media, and quantum mechanics—particularly for Schrödinger operators with periodic or singular potentials.

Abstract: This talk will focus on two-dimensional vector fields with values intothe circle S¹, and divergence bounded in L². This is motivated by liquid crystal models in a regime of degenerate ellipticity. It turns out that the divergence bound implies a bound on their "half-gradient", in some fractional Sobolev/Besov sense. Moreover, this half-order of differentiability is essentially optimal. I will describe some ideas behind these facts, based on tools from hyperbolic

conservation laws, and discuss several open questions. Based on joint work

with L. Bronsard, D. Golovaty and P.Sternberg.

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.

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).

Abstract: Abstract Vuono  

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

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

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.

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).

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.

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.