Nonlinear Analysis Seminar 

Abstract:
TBA

Abstract:
In this talk I will discuss some recent rigidity and classification results for positive Einstein metrics in dimension four.  I will begin with a short survey of this rich subject, then talk about recent work with Andrea Malchodi where we prove estimates for the Green's function of an operator associated to an Einstein metric. Assuming the sectional curvature is positive, I will explain how these estimates give a classification result under certain topological conditions. 

Abstract:
The Born-Infeld theory replaces the quadratic Maxwell Lagrangian with a nonlinear model designed to render the self-energy of a point charge finite. From the PDE viewpoint, this comes at the price of a nonuniformly elliptic structure, making the regularity of weak solutions a very delicate question. In this talk, I will discuss novel advances on the (ir)regularity of solutions to the electrostatic Born-Infeld equation, with emphasis on the structure of their singular set (the set of light rays).
From recent, joint work with Giuseppe Mingione (Parma).

Abstract:
Many phenomena in materials science can be described through nonlocal variational models in which long-range particle interactions compete with a multi-well potential. Classical examples include phase transitions governed by Allen–Cahn type equations and crystal dislocations arising from the Peierls–Nabarro model.

A central challenge in these models is to understand the structure of the transition layers connecting different equilibrium states, particularly when the energy wells exhibit highly degenerate or oscillatory behavior.

In this talk, I will discuss recent results on the existence, uniqueness, and sharp asymptotic decay of such non-standard heteroclinic profiles. I will also address the inverse problem of reconstructing and characterising double-well potentials from prescribed power-type layers.

Abstract:
We deal with weighted Hardy–Sobolev type inequalities for functions on R^d, d\ge 2.

The weights involved are anisotropic, given by products of powers of the distance to the origin and to a nontrivial subspace.

We establish necessary and sufficient conditions for validity of these inequalities, and investigate the existence/nonexistence of extremal functions.

This is a joint research with Gabriele Cora and Alexander I. Nazarov.

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:
Magnetic dynamics was formalized by Arnold in the early 1980s and has since been investigated by several mathematical communities up to the present day. In this seminar, accessible to a broad audience, I will try to motivate the following phenomenon: if the magnetic force acts in every direction, then, as its intensity increases, the magnetic action tends to regularize the motion of the charged particle. More precisely, we will prove the following result: if the magnetic form is symplectic, then the Lyapunov exponents converge uniformly to zero as the norm of the magnetic force tends to infinity. In the opposite direction, I will present a large class of examples in which the magnetic form is degenerate and, for every sufficiently large magnetic intensity, one can find orbits whose Lyapunov exponents are uniformly bounded away from zero. This is a joint work with Leonardo Macarini and James Marshall.  

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:  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.