Organizers: Giovanni Catino and Fabio Cipriani

**Bozhidar Velichkov**, Université Grenoble Alpes,

*Regularity of optimal sets for spectral functionals*, Wednesday, April 20, 2016, time 11:30 o'clock, Aula seminari 3° piano

**Abstract:****Abstract:**
We consider the variational shape optimization problem of the minimization of the sum of the first $k$ Dirichlet eigenvalues of a set $\Omega$ under a volume constraint $|\Omega|=1$. We prove that the free boundary of the optimal set is $C^{1,\alpha}$ regular up to a set of zero (d-1)-Hausdorff measure. The optimal set is a solution of a free boundary problem of Alt-Caffarelli type involving vector valued functions. We will dedicate most of our attention to the study of the local minimizers for this free boundary problem.
**Matteo Muratori**, Università degli Studi di Pavia,

*Symmetry results in Caffarelli-Kohn-Nirenberg interpolation inequalities*, Wednesday, April 13, 2016, time 16:15 o'clock, Aula seminari 6° piano

**Abstract:****Abstract:**
We investigate the structure of functions that optimize a special family of weighted interpolation inequalities of Caffarelli-Kohn-Nirenberg type (Compositio Math. 1984). The spatial dimension is greater than or equal to 3 and the weight appearing in the L^p norms is an inverse power with exponent between 0 and 2. The non-weighted case, associated with standard Gagliardo-Nirenberg inequalities, has thoroughly been investigated by M. Del Pino and J. Dolbeault in a remarkable paper (JMPA 2002), where they prove that optimal functions coincide with explicit profiles of Aubin-Talenti type. If the exponent ranges strictly between 0 and 2, suitable analogues of Aubin-Talenti-type profiles continue to exist. However, the main issue is related to radial symmetry. Indeed, as soon as optimal functions are radial, then they are necessarily of Aubin-Talenti type. Because of the weight, standard Schwarz symmetrization techniques fail: we have therefore to exploit a completely different method. First of all, by means of a concentration-compactness analysis, we prove that optimal functions converge to the Aubin-Talenti profiles as the exponent of the power of the weight tends to zero. Then we proceed by contradiction with an argument that involves angular derivatives of possibly non-radial optimal functions, which allows us to show that radial symmetry holds at least when the exponent is close to zero. This is a joint work with J. Dolbeault and B. Nazaret.
**Maria Colombo**, University of Zurich,

*Flows of non-smooth vector fields and applications to PDEs*, Wednesday, March 30, 2016, time 16:00 o'clock, Sala Consiglio 7° piano

**Abstract:****Abstract:**
The classical Cauchy-Lipschitz theorem shows existence and uniqueness of the flow of any sufficiently smooth vector field in R^d. In 1989, Di Perna and Lions proved that Sobolev regularity for vector fields, with bounded divergence and a growth assumption, is sufficient to establish existence, uniqueness and stability of a generalized notion of flow, consisting of a suitable selection among the trajectories of the associated ODE. Their theory relies on a growth assumption on the vector field which prevents the trajectories from blowing up in finite time; in particular, it does not apply to fast-growing, smooth vector fields.
In this seminar we give an overview of the topic and we introduce a notion of maximal flow for non-smooth vector fields which allows for finite-time blow up of the trajectories. We show existence and uniqueness under only local assumptions on the vector field and we apply the result to a kinetic equation, the Vlasov-Poisson system, where we describe the solutions as transported by a suitable flow in the phase space. This allows, in turn, to prove existence of weak solutions for general initial data.
This seminar is organized within the PRIN 2012 Research project «Equazioni alle derivate parziali di tipo ellittico e parabolico: aspetti geometrici, disuguaglianze collegate, e applicazioni - Partial Differential Equations and Related Analytic-Geometric Inequalities» Grant Registration number 2012TC7588_003, funded by MIUR – Project coordinator Prof. Filippo Gazzola
**Juan Luis Vazquez**, Universidad Autonoma de Madrid,

*Fractional Diffusion Equations. Theory on bounded domains*, Friday, March 11, 2016, time 11:00 o'clock, Sala Consiglio 7° piano

**Abstract:****Abstract:**
In this talk I will report on some of the progress made by the author and collaborators on the topic of nonlinear diffusion equations involving long distance interactions in the form of fractional Laplacian operators. In recent work with Bonforte and Sire we address the solution of Dirichlet problems in bounded domains, where the correct definition of what is the operator has a number of possible answers leading to different qualititative and quantitative results.
This seminar is organized within the PRIN 2012 Research project «Equazioni alle derivate
parziali di tipo ellittico e parabolico: aspetti geometrici, disuguaglianze collegate, e applicazioni
- Partial Differential Equations and Related Analytic-Geometric Inequalities» Grant Registration
number 2012TC7588_003, funded by MIUR – Project coordinator Prof. Filippo Gazzola
**Tomás Caraballo**, Universidad de Sevilla,

*Pullback attractors for random and non-autonomous dynamical systems: an introduction with applications*, Friday, Febraury 26, 2016, time 10:00 o'clock, Aula seminari III piano

**Abstract:****Abstract:**
The theory of pullback attractors for Random/Non-autonomous Dynamical Systems has been extensively developed over the last three decades, mainly due to the fact that it provides a suitable framework to analyze the global behavior of dynamical systems containing some kind of stochasticity or randomness as well as some time dependent forcing. In this talk we will introduce the basic tools of the theory of pullback attractors for handling both non-autonomous and random dynamical systems. We will show how both problems can be analyzed in a unified formulation thanks to the concept of cocycle. We will also emphasize on the different effects that different kind of noise can produce on the asymptotic behaviour of the solutions. Our results will be illustrated with some basic academic examples as well as some other interesting models appearing in the applied sciences (for example, reaction-diffusion equations and chemostast models).
**Zindine Djadli**, Université Grenoble Alpes,

*Conformal geometry: some old and recent results*, Wednesday, Febraury 24, 2016, time 11:30 o'clock, Aula seminari III piano

**Abstract:****Abstract:**
In this talk I will review some results in conformal geometry. The talk will mainly focus on the applications of conformal geometry in rigidity results. The talk is aimed to be accessible to everyone, especially people not familiar with Riemannian geometry.