Talks and abstract

Luca Asselle Ruhr Universität Bochum, Germany

"Morse theory for strongly indefinite functionals"

For strongly indefinite functionals, i.e. functionals all of whose critical points have infinite Morse index and co-index, no classical Morse theory can possibly exist due to the fact that attaching an infinite dimensional cell does not produce any change in the topology of sublevel sets. In this talk I will show how to instead construct a Morse complex for a suitable class of such functionals (including e.g. the Lorentzian energy functional, and the Hamiltonian action in cotangent bundles). The Morse complex is generated by critical points, and the boundary operator counts the number (modulo two) of gradient flow lines connecting pairs of critical points whose (suitably defined) relative Morse indices differ by one. In contrast to Floer theory, such gradient flow lines are obtained as genuine intersection between stable and unstable manifolds, which (despite being infinite dimensional) turn out to have finite dimensional intersection with good compactness properties. Transversality is achieved by generically perturbing the negative gradient vector field within a class of vector fields preserving all good compactness properties. In the particular case of the Hamiltonian action in cotangent bundles, the resulting Morse homology is isomorphic to the Floer homology of T*M as well as to the singular homology of the free loop space of the base manifold M. Nevertheless, the Morse complex approach has potentially several advantages over Floer homology which will be discussed if time permits. This is joint work with Alberto Abbondandolo and Maciej Starostka.

Gabriele Benedetti Vrije Universiteit Amsterdam, Netherlands

"The dynamics of strong magnetic fields on surfaces: periodic orbits and trapping regions"

How does a magnetic field influence the motion of a charged particle on a surface? Are there periodic orbits or trapping regions for the particle? How difficult is to construct a magnetic field for which all orbits are periodic? In this talk we will see that, if the magnetic field is strong, a normal form going back to the Russian school allows us to use Birkhoff theorem and KAM theory to tackle these questions. This is joint work with Luca Asselle.

Renato Colucci Università Politecnica delle Marche - Ancona, Italy

"Special solutions for an equation arising in sand ripple dynamics"

We study a nonlinear fourth order evolution equation arising in the context of sand ripple dynamics. We analyse the set of sta tionary solutions and traveling waves in order to recover the observed phenomenology such as different wave lengths ripples, traveling waves, coarsening and time scales.

Dario Corona, Università degli Studi di Camerino, Italy

"On the existence and multiplicity of the brake orbits"

This talk will show the recent developments on the study of the brake orbits (i.e. pendulum-like periodic orbits that oscillate back and forth between two points) in a potential well of a Hamiltonian system. We see that if the Hamiltonian function is even and strictly convex with respect to the generalized momenta then the brake orbits are in one-to-one correspondence with orthogonal Finsler geodesic chords in a bounded manifold with boundary. Thus, the existence and multiplicity of the brake-orbits can be obtained by appropriate refinements of mini-max methods. The talk is completed with a historical perspective, some applications and further developments of the subject.

Irene De Blasi Università degli Studi di Torino, Italy

"Stability and chaotic properties of a refraction billiard"

Refraction billiards can be considered as a generalization of the classical Birkhoff billiards, and prove themselves to be very useful to describe the motion of a particle under the influence of different potentials separated by an interface. Instead of moving freely inside a planar domain, bouncing off its boundary, our particle's trajectory is free to cross the interface, as soon as a particular refraction Snell's law, justified by variational arguments, is satisfied. The problems coming from the investigation of such models are various, and are connected, for example, to the existence of trajectories with given rotation numbers or the arising of chaotic behaviours. They can be analysed from different points of view, and their study involves techniques coming from calculus of variations, as well as Lyapunov linear stability, KAM and Aubry Mather theory. This seminar analyzes a particular type of refractive billiard, inspired by the study of a Galactic model, where a pure Keplerian potential and a harmonic oscillator one act in complementary regions of the plane. As we will see, its dynamical properties are crucially influenced by the geometry of the refraction interface.

Marco Gallo Università degli Studi di Bari Aldo Moro, Italy

"Even? Not so odd: existence of multiple solutions for some nonlocal problem"

Dealing with Schrödinger-type equations, Berestycki and Lions in [Arch. Ration. Mech. Anal., 1983] succeeded in building infinitely many solutions in the case of general nonlinearities satisfying no Ambrosetti-Rabinowitz condition. On the other hand, the construction relies on two main ingredients: the locality of the source, and its oddness. In this talk I will present a new construction, in order to prove a multiplicity result for a Hartree-type equation, with general (odd or even) nonlocal nonlinearities. To this end, gaining compactness will be quite struggling as well. Finally, I will apply this idea also to the problem of normalized solutions, that is when the L^2-norm is assigned in advance. The talk is based on a joint paper with S. Cingolani and K. Tanaka.

Alessandro Goffi Università degli Studi di Padova, Italy

"New regularity properties for solutions of quasilinear equations with power-growth first-order terms"

In this talk I will discuss some recent advances on the regularity properties of solutions to a large class of quasilinear elliptic and parabolic equations with diffusion modeled over the p-Laplacian having power-like gradient terms and unbounded right-hand side in Lebesgue scales. A special attention will be devoted to the Hölder regularity of distributional semi-solutions and to the so-called problem of maximal L^q-regularity. The proofs rely on different techniques based on integral approaches, refinements of the Bernstein method and/or duality arguments. The results answer to a conjecture raised by P.-L. Lions for the case p=2.

Filippo Giuliani Politecnico di Milano, Italy

"Transfers of energy in infinite dimensional dynamical systems"

In recent years, a great effort has been made to understand transfers of energy phenomena in dynamical systems on lattices and PDEs on compact manifolds. This problem is naturally related to the study of instability and chaotic phenomena in dynamical systems. The most prominent example of these studies is the Arnold diffusion theory, which still presents several important open questions. In this talk we briefly discuss some results on transfers of energy phenomena for nonlinear Hamiltonian PDEs under periodic boundary conditions and we present a recent result of existence of transfers of energy orbits in an infinite pendulum-lattice system. The latter is, to the best of our knowledge, the first example of Arnold diffusion phenomena for a Hamiltonian system with infinitely many degrees of freedom. We develop geometric techniques (in the spirit of the seminal work by Arnold) in infinite dimensional settings to construct unstable solutions in either spaces of bounded sequences or sequences with decay. This is a joint work with M. Guardia.

Teresa Isernia Università Politecnica delle Marche - Ancona, Italy

"Partial regularity fo parabolic systems with Orlicz growth"

In this talk I will speak about a partial regularity result for weak solutions to the following homogeneous parabolic system u_t-\nabla a(Du)=0 in \Omega_T= \Omega \times (-T, 0) where a: R^{Nn}\to R^{Nn} is a C^1-vector field satisfying ellipticity and growth conditions in terms of Orlicz functions. The main result is obtained by using a new A-caloric approximation lemma compatible with an Orlicz setting. This talk is based on a joint work with M. Foss, C. Leone and A.Verde.

Beatrice Langella SISSA - Trieste, Italy

Growth of Sobolev norms in quasi integrable quantum systems

In this talk I will analyze an abstract linear time dependent Schrödinger equation of the form i∂_t ψ = (H + V (t)) ψ , with H a pseudo-differential operator of order d > 1 and V (t) a time dependent family of pseudo-differential operators of order strictly less than d. I will introduce abstract assumptions on H, namely steepness and global quantum integrability, under which we can prove a |t|^ϵ upper bound on the growth of Sobolev norms of all the solutions of the above equation. The result I will present applies to several models, as perturbations of the quantum anharmonic oscillator in dimension 2, and perturbations of the Laplacian on a manifold with integrable geodesic flow, and in particular: flat tori, Zoll manifolds, rotation invariant surfaces and Lie groups. The case of several particles on a Zoll manifold, a torus or a Lie group is also covered. The proof is based a on quantum version of the proof of the classical Nekhoroshev theorem. This is a joint work with Dario Bambusi.

Alberto Maspero SISSA - Trieste, Italy

"Growth of Sobolev norms on linear Schrodinger equations as a dispersive phenomenon"

We consider linear, time dependent Schrödinger equations of the form i∂tψ=H+V(t), where H is a strictly positive selfadjoint operator with discrete spectrum and constant spectral gaps, and V(t) a time periodic potential. We give sufficient conditions on the potential ensuring that the Hamiltonian generates unbounded orbits. The main condition is that the resonant average of V(t) has a nonempty absolutely continuous spectrum and fulfills a Mourre estimate. These conditions are stable under perturbations. The proof combines pseudodifferential normal form with dispersive estimates in the form of local energy decay.

Federico Murgante SISSA - Trieste, Italy

"Well posedness of the Euler-Korteweg system with periodic boundary conditions"

The Euler-Korteweg system is a system of conservation laws governing the motion of liquid-vapor mixtures, which takes into account the surface tension of interfaces by means of a capillarity coefficient. I will focus on periodic and irrotational solutions of the Euler-Korteweg system. Allowing the capillarity coefficient K to depend (in a smooth way) on the density ρ, the system becomes quasilinear and therefore more difficult to apprehend than in the special case K ≡ constant. I will present a local well–posedness result for the system with a general smooth capillarity coefficient K = K(ρ) (joint work with Berti M., Maspero A.). It is remarkable that in a different physical framework, namely in Quantum Hydrodynamics (QHD), the very same system of PDEs arises with precisely K proportional to 1/ρ, in this case I will present a long–time stability result for small solutions (joint work with Feola R., Iandoli F.).

Paolo Ventura SISSA - Trieste, Italy

"Modulational instability in Water Waves"

A problem of fundamental importance in fluid mechanics regards stability/instability of Stokes waves, namely steady solutions of the water-waves system, with respect to longitudinal space periodic long-wave perturbations. After producing a global picture of the problem at its linear level, I will describe, both in the finite and infinite-depth cases, the behavior of the four eigenvalues close to zero of the linearized equations at the Stokes wave, as the Floquet exponent is turned on. In particular it will be justified the conjecture that a pair of non-purely imaginary eigenvalues depicts a closed figure “8”, parameterized by the Floquet exponent, in full agreement with numerical simulations.