Organizers: Giovanni Catino and Fabio Cipriani

**Louis Dupaigne**, Université de Picardie,

*Symmetry and asymptotics in semilinear elliptic problems*, Tuesday, January 26, 2010, time 16:15 o'clock, Aula seminari III piano

**Abstract:****Abstract:**
Consider a domain of Euclidean space having a certain symmetry, say about a point or a hyperplane. Is it true that every solution of a
nonlinear PDE inherits the symmetry property, if the equation allows it? Starting from a classical result of Gidas, Ni and Nirenberg, we
question the optimality of the conditions under which symmetry is preserved/violated, in the following directions: weak solutions, nodal
solutions, non-lipschitz nonlinearities.
I will try to make a case for the asymptotic nature of the problem, by spending the second part of the talk on the case of boundary blow-up
solutions. The proofs rely on classical tools, such as the maximum principle and
the Alexandrov device of moving parallel hyperplanes, as well as a more recent technique developped with O. Costin (Ohio State U),
stemming from exponential asymptotics.
http://web.mate.polimi.it/cdv/
**Susanna Terracini**, Università di Milano-Bicocca, *Problemi matematici nella condensazione di Bose-Einstein*, Wednesday, January 20, 2010, time 16:15, Aula Seminari MOX - VI piano

**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.