SISSA – International School for Advanced Studies

Functional Analysis and Applications

• People
• Research topics
• Ph.D. courses
• Activities
  • Seminars
  • Workshops
  • Events
  • Open Day
• Publications
• Grants
• Join Us
• Links
• mathLab

Seminars 2006-2007

Existence and multiplicity of positive solutions for a singular elliptic equation, involving Hardy potentials and critical growth.

Synthesis and modeling of electri-field responsive nematic gels.

Computational methods for quantum control problems.

A variational approach to the ground states of the XY model.

The discrete structure of the ground states of a spin system is often neglected by introducing a mesoscopic scale that captures the main features of the problem while simplifying its analysis. In general this procedure is not rigorous and not even well understood. In this seminar we will show that in the case of the ground states of the XY (N dimensional possibly anysotropic) spin type models it is indeed possible to justify, from a variational point of view, the coarse graining procedure which leads to a Ginzburg-Landau type energy on a mesoscopic scale. Topological type phase transitions will be highlighted by showing the presence of vortice-like ground states. In the 2-dimesional case we will show how it is possible to address the problem in the case of long range interactions. Finally we will discuss several possible developments and present some open problems.

3-manifolds under an integral pinching assumption are space form.

we will show that a 3-manifold with positive scalar curvature and satisfying an integral pinching assumption beetwen the $L^2$-norm of the Ricci curvature and the $L^2$-norm of the scalar curvature is a spherical space form (ie admits an Einstein metric with positive curvature). This will be done by solving a fully non linear differential equation.

Sharp Moser-Trudinger inequalities on the CR sphere.

La disuguaglianza di Sobolev in forma quantitativa.

Nel 1976 T.Aubin e G.Talenti determinarono (indipendentemente) una classe di funzioni $\cal T$ per cui la disuguaglianza di Sobolev, per $p>1$, si riduce ad un'uguaglianza. Solo nel 2004 D.Cordero-Erausquin, B.Nazaret e C.Villani hanno provato che le funzioni di $\cal T$ sono le uniche per cui vale l'uguaglianza.

Nel seminario verrà presentato un risultato recente, ottenuto in collaborazione con A.Cianchi, F.Maggi e A.Pratelli, che fornisce una versione della disugaglianza di Sobolev in cui il norma del gradiente in $L^p$ di una funzione $u$ controlla, sempre con la stessa costante di Sobolev, non solo la sua norma in $L^{p^*}$, ma anche la sua distanza in $L^{p^*}$ da $\cal T$.

Transitions of Shocks and Singularities of Minimum Functions.

The shock discontinuities, generically present in inviscid solutions of the forced Burgers equation, and their bifurcations happening in the course of time (perestroikas) are classified in two and three dimensions -- the one-dimensional case is well known. This classification is a result of selecting among all the perestroikas occurring for minimum functions depending generically on time, the ones permitted by the convexity of the Hamiltonian of the Burgers dynamics. Topological restrictions on the admissible perestroikas of shocks are obtained. The resulting classification can be extended to the so-called viscosity solutions of a Hamilton--Jacobi equation, provided the Hamiltonian is convex.

Sub-Riemannian contact structures in the functional architecture of the primary visual cortex.

A difficult problem is to understand from within how the connectivity of the visual cortex provides the well behaved geometric structures of visual perception. It can be shown that the horizontal cortico-cortical connections of V1 (the first visual area) implements the contact structure of the fi bration with base the visual plane and fiber the projective line of orientations in this plane. This explains many striking phenomena of integration of local sensory data into global perceptual structures.

Teoremi di comparison in geometria riemanniana.

Recent developments in the theory of optimal mass transportation.

In this talk I will describe some developments, occurred in the last 10-15 years, of the theory of optimal mass transportation, that finds its origin in a problem raised by G. Monge in 1781. I will give a skectchy picture of the broad range of applications of this theory, which covers variational formulations of evolution equations, functional and geometric inequalities, (nonsmooth) Riemannian geometry, and other fields.

A vanishing viscosity approach to fracture growth in a cohesive zone model with prescribed crack path.

On the optimal synthesis for infinite horizon variational problems.

Periodic solutions of forced Kirchhoff equations.

Existence and removable singularity results for some fully nonlinear equations in geometry.

Some linear and nonlinear equations involving differential forms.

Convergence of the Schroedinger dynamics to the Hartree equations as the number of particles tends to infinity, with uniform estimates with respect to the Planck constant.

After a short review on why the increasing number of degrees of freedom makes possible to approximate many-body quantum mechanics, an intrinsically linear evolution, with a nonlinear single-body equation, the proof is sketched of a new method for deriving the nonlinear Hartree equation form the mean field limit $N\to\infty$ for a system of $N$ bosons which allows us to obtain convergence estimates uniform with respect to the Planck constant.

On the Cauchy Problem of the Boltzmann and Landau equations with soft potentials

In this talk, we will consider the existence and the time decay of global classical solutions to the Boltzmann and Landau equations near Maxwellian under the soft potentials.

Course: Dissipative mechanisms in systems of conservation laws

• Various dissipative mechanisms: viscosity, relaxation, coupling with elliptic equations, BGK models, ... Dissipative limit vs time asymptotics.
• Dissipation and entropy. A priori estimate.
• Linear theory: Well-posedness for the Cauchy problem, time asymptotics, relaxation limit. Kawashima's condition ; link with control theory. Various instances of the sub-characteristic property.
• Discontinuities. How shocks are approached in dissipative structures: shock profiles. Can discontinuities happen in dissipative structures?
• Nonlinear setting. Entropy formalism. Characteristic fields in hyperbolic-parabolic or hyperbolic-elliptic systems. Nonlinear diffusion waves, asymptotic decay. The scalar case; L^1 stability of travelling waves.

Lecture notes (accessible only from SISSA intranet)

Singularities in relaxation oscillations and geometric control theory from the common viewpoint

In this talk we have deal with the singularities occur in the slow motion of slow-fast systems and in the affine-control systems with scalar input. These singularities can be considered from the common viewpoint. A proper construction allows to present singularities in these problems as singular points of special vector fields, which have a common property: among the germs of their components there exist some independent germs, which generate the local ideal containing all others components. Some theorems about such vector fields will be presented, and after that we will consider application of these theorems to the affine-control systems with scalar input.

Last modified: 30 October 2007

Webmaster: mauro.bardelloni at sissa dot it