Organizers: Giovanni Catino and Fabio Cipriani

**Susanna Terracini**, Università di Milano-Bicocca, *Problemi matematici nella condensazione di Bose-Einstein*, Thursday, January 07, 2010, time 17:00, Sala Consiglio - VII piano

**Elvise Berchio**, Politecnico di Milano,

*Hardy-Rellich type inequalities with boundary terms and applications.*, Tuesday, December 15, 2009, time 16:15 o'clock, aula seminari III piano

**Abstract:****Abstract:**
We present a family of Hardy-Rellich type inequalities having boundary terms and where the optimal constants are not necessarily the classical Hardy-Rellich ones. When the domain is the unit ball the exact values of the constants are performed. We exploit this fact to study the regularity of the “extremal solution” to some semilinear elliptic problems under Steklov boundary conditions.
**Gabriele Grillo**, Politecnico di Torino,

*On the asymptotic behaviour of some solutions of the fast diffusion equation.*, Tuesday, December 01, 2009, time 16:15 o'clock, Aula seminari terzo piano

**Abstract:****Abstract:**
The so-called Barenblatt profiles are known to be explicit solution to the fast diffusion equation. Such solutions play, in the study of such equation, a role similar to the one played by the Gaussian solutions when dealing with the heat equation. In this talk we shall in fact show that, in suitable senses, certain classes of solutions to the fast diffusion equation converge to the Barenblatt profiles, giving explicit rates of convergence. Such rates are related to the best constant, explicitly determined, in a suitable Hardy-Poincaré inequality. A particular case, in which such constant vanishes, shows a polynomial decay instead of an exponential one. Such behaviour is proved using a geometric interpretation of the linearized evolution, the Li-Yau theory on the heat kernel on manifolds with nonnegative Ricci curvature, some weighted Nash inequalities and, finally, an appropriate use of the parabolic Harnack inequality.
This is a report of joint works with M. Bonforte, J.L. Vazquez and, in part, A. Blanchet e J. Dolbeault.
**Kazuhiro Ishige**, Tohoku University (Sendai, Giappone),

*Blow-up for a semilinear parabolic equation with large diffusion on R^N*, Tuesday, November 24, 2009, time 16:15 o'clock, aula seminari III piano

**Abstract:****Abstract:**
We study the blow-up time and the location of the blow-up set of
the solution for a semilinear heat equation with large diffusive coefficient.
In particular, we prove that, if the diffusive coefficient is sufficiently lar
ge, then the location of the blow-up set depends on the large time behavior of
the hot spots of the solutions for the heat equation. This is a joint
work with Yohei Fujishima.
**Wu Hao**, Fudan University Shanghai e WIAS Berlin,

*On a semiconductor drift-diffusion-Poisson model in R3
*, Wednesday, May 20, 2009, time 16:15, Sala Seminari Fausto Saleri , VI piano

**Abstract:****Abstract:**
We study a time-dependent as well as a stationary
drift-diffusion-Poisson system for semiconductors. Global existence
and uniqueness of weak solution of the time-dependent problem are
proven and we also prove the existence and uniqueness of the steady
state. Finally, we discuss the large time asymptotics of the
time-dependent problem. This is a joint work with Prof. P. Markowich
and Prof. S. Zheng.
**Alexander Ioffe**, Department of Mathematics, The Technion, Haifa,

*Tame Functions and Variational Analysis
*, Wednesday, May 20, 2009, time 17:15, Aula seminari Fausto Saleri, 6 piano

**Abstract:****Abstract:**
All non-trivial theorems of local variational analysis applied to generic
non-differential functions (e.g. optimization problems with generic
Lipschitz data) produce not very informative, often just trivial
results. Fortunately, functions that usually appear in applications
have some special structures (e.g. polyhedral, linear-quadratic, spline
etc.). Typically such structures are particular cases of semi-algebraic
(or more generally, tame) structures. The latter turn out to be perfectly
compatible with basic constructions of local variational analysis which
excludes any possibility for the mentioned unpleasant phenomena to happen.
Moreover, in this case a number of powerful results can be proved that
are not otherwise valid.
The latter statements will be clarified in the talk, both in general terms
and for some important classes of problems, including standard problems of
mathematical programming, gradient dynamical systems and optimal control
of state-linear systems.