Geodesic flows on a closed Riemannian manifold (M,g) typically exhibit a rich collection of periodic orbits, or closed geodesics. This abundance raises the natural question of whether one can recover information about the manifold (M,g) from data associated with these closed geodesics — such as their lengths, homotopy classes, and other invariants. In this talk, I will survey both classical and recent results related to this theme.
The existence of solutions $(\lambda, u) \in \mathbb{R} \times H^1_0(\Omega)$ of nonlinear Schrödinger equations like \[ −\Delta u + \lambda u = f (u) \qquad \text{in} \quad \Omega \subset \mathbb{R}^N \] with prescribed $L^2$-norm \[ \int_{\Omega} u^2 \, dx = \alpha \] has found considerable interest in the last decade. We present a new approach to this problem based on a nonlocal heat flow. As an application, if $\Omega$ is a ball or $\Omega = \mathbb{R}^N$ we prove the existence of radial solutions with a prescribed number of nodal domains under suitable assumptions on $f$.
The talk is based on work with Shijie Qi and Wenming Zou.
In this talk we consider the existence of positive solutions to the following nonlinear Schrödinger equations: $$\tag{*} \begin{cases} -\Delta u+\mu u=g(u) &\text{in}\ \mathbb{R}^N, \\ \frac12\int_{\mathbb{R}^N} u^2 \, dx =m, \end{cases} $$ where $ N \geq 2$, $g(s)\in C(\mathbb{R},\mathbb{R})$, $m>0$ are given and $(\mu,u)\in (0,\infty)\times H^1_r(\mathbb{R}^N)$ is unknown. We consider the situation $g(s) \sim |s|^{p-1}p$. The power $p$ plays important roles: when $p\in (1,1+ \frac{4}{N})$ (resp. $p \in (1+\frac{4}{N}, \frac{N+2}{N-2})$), $(*)$ are called $L^2$-subcritical (resp. $L^2$-supercritical) problems. These cases are extensively studied by many researchers. In this talk, we give a focus on the boundary case: $p=1+\frac{4}{N}$, i.e., $L^2$-critical case. We take a Lagrangian approach to this problem.
We also give a mention to a related problem in Hamiltonian systems: $$\tag{**} \frac{d^2q}{dt^2} + \nabla V(q) = 0, \quad \frac12\left| \frac{dq}{dt}\right|^2 + V(q) = E, $$ where $V(q) \sim - \frac{1}{|q|^\alpha}$, $\alpha > 0$ and $E \in \mathbb{R}$.
This talk is based on joint works with S. Cingolani and M. Gallo.
We will present some asymptotical results concerning the minimization of the positive principal eigenvalue associated with a weighted Neumann problem settled in a bounded regular domain.
These results provide, in the asymptotical regime, detailed answers to some long-standing questions in this framework.
Joint works with Lorenzo Ferreri (Scuola Normale Superiore di Pisa), Dario Mazzoleni (Università di Pavia), Gianmaria Verzini (Politecnico di Milano).
Conformally symplectic diffeomorphisms transform a symplectic form on a manifold into a multiple (the conformal factor) of itself. They appear naturally in applications.
In this talk, we study the relations between symplectic properties and dynamical properties. We consider Normally hyperbolic manifolds and their (un)stable manifolds. We prove that a NHIM is symplectic if and only if the rates satisfy certain pairing rules and if and only if the rates and the conformal factor satisfy certain (natural) inequalities. We also study the properties of scattering maps associated to homoclinic intersections. We prove that the scattering maps are symplectic even if the dynamics is dissipative. We also show that if the symplectic form is exact, then the scattering maps are exact, even if the dynamics is not exact.
This is a joint work with M. Gidea and R. de la LLave
We focus on the problem of existence and nonexistence of positive solutions for the Sobolev-subcritical Lane–Emden equation on certain Riemannian manifolds with negative curvature, which, from the viewpoint of the volume growth of geodesic balls, can be regarded as intermediate cases between the Euclidean and the hyperbolic spaces.
A number of interesting phenomena arise: the subcritical regime naturally divides into three further regimes, characterized respectively by existence phenomena, nonexistence phenomena, and by a mixed behavior where existence depends on additional assumptions on the manifold, thereby revealing substantial differences with respect to both the Euclidean and the hyperbolic settings.
This is a joint work with Alessandra De Luca and Matteo Muratori.
The behavior of eigenvalues of Aharonov-Bohm operators and their stability with respect to changes in the position of the poles are discussed. In particular, the problem of determining the exact asymptotic behavior of the eigenvalue variation, under small perturbations of the poles’ configuration, is addressed.
Both the case of multiple colliding poles with circulation 1/2 and the case of any circulations are considered. In the case of half-integer circulations, a gauge transformation makes the problem equivalent to an eigenvalue problem for the Laplacian in a domain with straight cracks, lying along the directions of motion of the poles. For this problem, an asymptotic expansion for simple eigenvalues shows, as the dominant term, the minimum of an energy functional associated with the configuration of poles and defined on a space of functions suitably jumping through the cracks. In the case of operators with non-half-integer circulations, the problem is reformulated as a system of two equations (with unknowns given by the real and imaginary parts of the gauged eigenfunction) coupled through prescribed jumps of the unknown functions and their normal derivatives on some segments identified by the configuration of the poles.
In the case of a single moving pole, double eigenvalues are also considered: it is observed that they split into two branches of simple eigenvalues as the pole moves along certain directions.
The results presented in the talk have been obtained in a series of papers in collaboration with L. Abatangelo, B. Noris, R. Ognibene, and G. Siclari.
A major component of the Hilbert sixth problem concerns the derivation of macroscopic equations of motion, and the associated kinetic equations, from microscopic first principles. In the classical setting of Boltzmann's kinetic theory, this corresponds to the derivation of the Boltzmann equation from particle systems governed by Newtonian dynamics; in the theory of wave turbulence, this corresponds to the derivation of the wave kinetic equation from nonlinear dispersive equations.
In this talk we present recent joint works with Zaher Hani and Xiao Ma, where we consider the hard sphere model in the particle setting, and the cubic nonlinear Schrödinger equation in the wave setting. In both cases we derive the corresponding kinetic equation up to arbitrarily long times, as long as the solution to this kinetic equation exists. This is a key step towards the resolution of the Hilbert sixth problem.
Restriction theory studies functions whose Fourier transforms are supported on some curved manifold in $\mathbb{R}^n$ (for example, solutions to the linear Schrodinger equation or to the wave equation). Projection theorems study the Hausdorff dimension of fractal sets under orthogonal projections from $\mathbb{R}^n$ to its subspaces.
We will survey some recent works in both fields and discuss their interactions
Starting with the pioneering computations of Stokes in 1847, the search of traveling waves in fluid mechanics has always been a fundamental topic, since they can be seen as building blocks to determine the long time dynamics.
In this talk I shall discuss some recent results concerning the constructioin of quasi-periodic waves in Fluid Mechanics (for Euler, Navier-Stokes and Water Waves equations) by means of Nash-Moser type bifurcation techniques, Normal Form and spectral methods, pseudo-differential operators techniques. I will then focus on the construction of Quasi-periodic 3D Stokes waves for a free surface fluid in finite depth. These solutions are not stationary in any reference frame and they are approximately given by finite sums (arbitrarily big) of traveling waves, whose velocity speeds are rationally independent. This is a very hard small divisors problem due to the fact that one deals with a dispersive quasi-linear PDE in dimension greater than two, implying that one has to deal with very strong perturbative effects of the quasi-linear nonlinearity and very strong resonance phenomena.
Partecipation is free, but you must register by sending an email to vittorio.cotizelati at unina.it
Scientific Committee: Massimiliano Berti, Vittorio Coti Zelati, Andrea Malchiodi, Paul H. Rabinowitz, Susanna Terracini