In un incontro a Venezia festeggiamo i 75 anni di Antonio Ambrosetti.
- 9:00-9:45 – Ermanno Lanconelli (Bologna): On the stability of the
Gauss mean value theorem for harmonic functions: slides
Among the various rigidity properties of the Euclidean
balls one of the best known examples is the Gauss mean value formula for harmonic
functions. This property raises the question of its stability. i.e. : if $D$ is an open
set with finite measure and $x_0$ is a point of $D$ such that $u(x_0)$ is close to the
average of $u$ on $D$ for every integrable harmonic functions $u$ in $D$, is it true
that $D$ is close to a ball centered at $x_0$? In this talk we present some positive
answers to this question, obtained in collaboration with Giovanni Cupini, Nicola Fusco
and Xiao Zhong
- 9:45–10:30 – Daniele Bartolucci (Roma Tor Vergata): Mean field
equations and the global bifurcation diagram of the Gel'fand problem: slides
For a large class of two
dimensional domains (which need not be neither simply connected nor symmetric) we
describe the qualitative behavior of the unbounded branch of solutions of the Gel'fand
problem containing the minimal solutions. The proof is based on two main facts: 1)
solutions can be parametrized as a function of the energy of the associated mean field
equation, naturally arising in the Onsager statistical description of two dimensional
turbulence; 2) a kind of inverse maximum principle holds along the non minimal branch of
solutions. At least to our knowledge this is the first result about the monotonicity of
the branch of non-minimal solutions which is not just concerned with radial solutions
and/or with symmetric domains. This is part of a joint research project in collaboration
with A. Jevnikar.
- 10:30–11:00 Coffee break
- 11:00–11:45 – Gianmaria Verzini (Milano Politecnico): Normalized
solutions to semilinear elliptic equations and systems: slides
We study the existence of solutions having prescribed L^2
norm to some semilinear elliptic problems in bounded domains. These kind of solutions
appear in different contexts, such as the study of the Nonlinear Schrödinger equation,
or that of quadratic ergodic Mean Field Games systems. When the nonlinearity is critical
or supercritical with respect to the Gagliardo-Nirenberg inequality, though Sobolev
subcritical, we show that solutions having Morse index bounded from above can exist only
when the mass is sufficiently small. On the other hand, we provide sufficient conditions
for the existence of such solutions, also in the Sobolev critical case. Based on joint
works with Benedetta Noris, Dario Pierotti and Hugo Tavares
- 11:45–12:30 – Emanuele Haus (Roma 3): Strong Sobolev instability of
quasi-periodic solutions of the 2D cubic Schrödinger equation: slides
We consider the defocusing cubic
nonlinear Schrödinger equation (NLS) on the two-dimensional torus. This equation admits
a special family of elliptic invariant quasiperiodic tori called finite-gap solutions.
These solutions are inherited from the integrable 1D model (cubic NLS on the circle) by
considering solutions that depend only on one variable. We study the long-time stability
of such invariant tori for the 2D NLS model and show that, under certain assumptions and
over sufficiently long timescales, they exhibit a strong form of transverse instability
in Sobolev spaces $H^s(\mathbb{T}^2)$ ($0 < s < 1$). More precisely, we construct
solutions of the 2D cubic NLS that start arbitrarily close to such invariant tori in the
$H^s$ topology and whose $H^s$ norm can grow by any given factor. In my talk, I will
also say some words about the ongoing work concerning Sobolev instability of more
general 2D quasi-periodic solutions. The subject of this talk is partly motivated by the
problem of infinite energy cascade for 2D NLS, and it is a joint work with M. Guardia,
Z. Hani, A. Maspero and M. Procesi.
- 12:30–15:00 – pausa pranzo
- 15:00–15:45 – Veronica Felli (Milano Bicocca): On spectral
stability of Aharonov-Bohm operators with moving poles: slides.
In this talk, I will present some results in
collaboration with L. Abatangelo (Milano-Bicocca), L. Hillairet (Orléans), C. Léna
(Stockholm), B. Noris (Amiens), and M. Nys, concerning the behavior of the eigenvalues
of Aharonov-Bohm operators with one moving pole or two colliding poles. In both cases of
poles moving inside the domain and approaching the boundary, the rate of the eigenvalue
variation is estimated in terms of the vanishing order of some limit eigenfunction. An
accurate blow-up analysis for scaled eigenfunctions will be presented too.
- 15:45–16:30 – Alberto Maspero (SISSA): Traveling quasi-periodic
water waves with constant vorticity.
We prove the existence and
the linear stability of small amplitude time quasi-periodic traveling waves solutions of
the gravity-capillary water waves equations with constant vorticity, for a
bi-dimensional fluid over a flat bottom delimited by a space-periodic free interface.
These quasi-periodic solutions exist for all the values of depth, gravity and vorticity,
and restricting the surface tension to a Borel set of asymptotically full Lebesgue
measure. This is a joint work with M. Berti and L. Franzoi.
- 16:30–17:00 Coffee break
- 17:00–17:45 – Margherita Nolasco (L'Aquila): Ground state for the
relativistic one electron atom: slides
We study stationary solutions for the Dirac-Maxwell system coupled with an
external Coulomb potential. In particular we give a minimax characterization for the
“ground state” of the system.
- 17:45–18:30 – Alessandro Portaluri (Torino Università): Keplerian
orbits through the Conley-Zehnder index: slides
It was discovered by William Gordon (1977) that Keplerian ellipses in
the plane are minimizers of the Lagrangian action and spectrally stable as periodic
points of the associated Hamiltonian flow. The aim of this talk is to give a homotopy
theoretical proof of these results through a self-contained, explicit and simple
computation of the Conley-Zehnder index. The techniques developed in this talk can be
used to investigate the higher dimensional case of Keplerian ellipses, where the
classical variational proof no longer applies.
- 9:00-9:45 – Patrizia Pucci (Perugia): Unità locale di Perugia:
alcuni risultati e prospettive di ricerca: slides
Presentiamo brevemente alcuni risultati ottenuti durante il periodo
inerente al progetto PRIN, insieme a vari problemi rimasti ancora aperti.
- 9:45–10:30 – Giusi Vaira (Università della Campania "Luigi
Vanvitelli"): Maximal solution of the Liouville equation in doubly connected
domains: slides
In this talk we
consider the Liouville equation $\Delta u +\lambda^2 e^{\,u}=0$ with Dirichlet boundary
conditions in a two dimensional, doubly connected domain $\Omega$. We show that there
exists a simple, closed curve $\gamma\in \Omega$ such that for a sequence $\lambda_n\to
0$ there exist a sequence of solutions $u_{n}$ such that
$\frac{u_{n}}{\log\frac{1}{\lambda_n}}\to H$, where $H$ is a harmonic function in
$\Omega\setminus\gamma$ and $\frac{\lambda_n^2}{\log\frac{1}{\lambda_n}}\int_\Omega
e^{\,u_n}\,dx\to \ell_\gamma$, where $\ell_\gamma$ is a weighted length of $\gamma$.
- 10:30–11:00 Coffee break
- 11:00–11:45 – Vieri Benci (Università di Pisa): The idea of
infinity in Mathematics between Science and Philosophy: slides
According to Bertrand Russell the great problems of
mathematical philosophy are three and all connected to the notion of infinity:
- the problem of infinite numbers;
- the problem of the continuous;
- the problem of infinitesimals.
According to our philosopher, these problems have all been solved: the first was
solved by Cantor with the introduction of cardinal numbers, the second was solved by
Dedekind (and Cantor) identifying the Euclidean line with the real line and the third
was solved by Weierstrass who eliminated the infinitesimal numbers from the realm of
mathematics. But are we sure that this view is correct? In reality I do not at all agree
and I will try to bring arguments in favor of a new vision of the infinite.
- 11:45–12:30 – David Arcoya (Granada, Spain): Lorentz Force Equation
and Mountain Pass Theorem: slides
In Special Relativity, the motion of a slowly accelerated electron under the influence
of an electric field $E$ and a magnetic field $B$ is modeled by the Lorentz force
equation \begin{equation*} \left(\frac{q'}{\sqrt{1-|q'|^2}}\right)'=E(t,q) + q'\times
B(t,q). \end{equation*} We review the main results of a joint work with Cristian
Bereanu (University of Bucharest) and Pedro J. Torres
(Universidad de Granada). Specifically, we provide a rigourous critical point theory by
showing that the solutions are the critical points in the Szulkin's sense of the
corresponding Poincaré non-smooth Lagrangian action. Moreover, by using a Mountain Pass
minimax principle without compactness, we prove a variety of existence and multiplicity
of solutions of the Lorentz force equation with periodic or Dirichlet boundary
conditions.