Invited Speakers

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.

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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.

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Berardino Sciunzi

On the moving plane method for semilinear and quasilinear elliptic problems.
Starting from the seminal paper of J. Serrin and driven by the celebrated result of B. Gidas, W. M. Ni and L. Nirenberg, the PDE community exploited the Moving Plane Method in various issues in order to prove symmetry and monotonicity properties of solutions to elliptic PDE. The method can be carried on in bounded or in unbounded domains. We will consider the leading examples given by the case of bounded domains, the case of the whole space and the case of the half space.

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Vincenzo Vespri

Pointwise estimates for nonnegative solutions to degenerate quasilinear parabolic equations.
We consider the Cauchy problem associated to a class of nonlinear degenerate/singular parabolic equations, whose prototype is the parabolic p-Laplacian ($\frac {2N} {N+1} < p < \infty$). In his seminal paper, after stating the Harnack estimates, Moser proved almost optimal estimates for the parabolic kernel by using the so called ”Harnack chain” method. In the linear case sharp estimates come by using Nash’s approach. Fabes and Stroock proved that Gaussian estimates are equivalent to a parabolic Harnack inequality. In several papers in collaboration with Bögelein, Calahorrano, Piro-Vernier and Ragnedda, by using the DiBenedetto-DeGiorgi approach we prove optimal kernel estimates for quasilinear parabolic equations. Lastly we use these results to prove existence and sharp pointwise estimates from above and from below for the fundamental solutions.

Gianluca Vinti

Approximation problems and Applications to Digital Images
We discuss some approximation problems by means of sampling type operators for the reconstruction and enhancement of digital images together with some applications to concrete problems.

Fabio Zanolin

Remarks on the Ambrosetti-Prodi periodic problem.
We present recent results on the Ambrosetti-Prodi problem for nonlinear equations with periodic boundary conditions. In particular, we improve some classical assumptions of uniform coercivity to nonlocal ones. We also discuss the presence of subharmonic solutions and complex dynamics.