Invited Speakers
- Gabriele Bonanno, Università di Messina
- Alessandra Lunardi, Università di Parma
- Enrico Miglierina, Università Cattolica del Sacro Cuore, Milano
- Berardino Sciunzi, Università della Calabria
- Vincenzo Vespri, Università di Firenze
- Gianluca Vinti, Università di Perugia
- Fabio Zanolin, Università di Udine
Abstracts
Gabriele Bonanno
The local minimum theorem: complements and applications.
The existence of at least two or three non-zero solutions for nonlinear elliptic eigenvalue problems is established. The basic ingredients are the local minimum theorem obtained in [1] and the classical Ambrosetti-Rabinowitz theorem. Some remarks on the mountain pass theorem and its relationships with the local minima are highlighted; further, a note on parameters for which the above problems admit solutions is done as well as a qualitative property of the obtained local minimum is investigated. By an appropriate combination of previous results, theorems of two and three critical points are obtained and a variant of the three critical points theorem, where the classical compactness condition of Palais-Smale is not assumed, is also emphasized.Alessandra Lunardi
Sobolev and BV functions in infinite dimension.
In Hilbert or Banach spaces $X$ endowed with a good probability measure $\mu$ there are a few "natural" definitions of Sobolev spaces and of spaces of bounded variation functions. The available theory deals mainly with Gaussian measures and Sobolev and BV functions defined in the whole $X$, while the study and Sobolev and BV spaces in domains, and/or with respect to non Gaussian measures, is largely to be developed. As in finite dimension, Sobolev and BV functions are tools for the study of different problems, in particular for PDEs with infinitely many variables, arising in mathematical physics in the modeling of systems with an infinite number of degrees of freedom, and in stochastic PDEs through Kolmogorov equations. In this talk I will describe some of the main features and open problems concerning such function spaces.
Enrico Miglierina
Weak$^*$ -fixed point property in $\ell_1$.
In this talk we study the $w^*$-fixed point property ($w^*$-FPP) for nonexpansive mappings in the space $\ell_1$. First, we provide some sufficient conditions for $w^*$-FPP in $\ell_1$ based on structural properties of the predual $X$ of $\ell_1$. Then, the main result of our paper provides some characterizations of weak-star topologies that fail the fixed point property for nonexpansive mappings in $\ell_1$ space.The key tool of our result is a detailed study of the hyperplanes of the space $c$.
Finally, we deal with the stability properties of $w^*$-FPP in $\ell_1$, by linking it to some geometrical features related to the notion of polyhedral space.
The talk is essentially based on a series of papers written jointly with Emanuele Casini, Łukasz Piasecki, Roxana Popescu and Libor Vesely.