Dorin Bucur, Univ. Chambery, Isoperimetric inequalities for eigenvalues by direct methods, Friday, Febraury 12, 2010, time 14:15 o'clock, Aula Seminari III piano
Abstract:Abstract:
A possible way to prove isoperimetric inequalities may be: - prove the existence of an optimal shape, without imposing any regularity constraint on the admissible shapes. A relaxation phenomenon may be observed, e.g. the solution is a measure or a quasi-open set. - prove the regularity of the optimal shape. This problem is often very difficult, but sometimes a mild regularity result is enough in order to pass to the next step. - extract optimality conditions and get extra information about the optimal shape. If the regularity is not very high, classical optimality conditions obtained by shape derivative methods can not be written. Other optimality criteria have to be used. This plan will be discussed for the minimization of the the eigenvalues of the Laplacian with Dirichlet and Robin boundary conditions. http://web.mate.polimi.it/cdv/

Monica Conti, Dipartimento di Matematica, Politecnico, Regular attractors for the Cahn-Hilliard equation with memory, Tuesday, Febraury 09, 2010, time 16:15 o'clock, Aula seminari III piano
Abstract:Abstract:
The Cahn-Hilliard equation is a parabolic differential equation of forth order which plays an essential role in material sciences since 1958, when it was introduced by J.W. Cahn and J.E Hilliard. This talk concerns with its memory relaxation, namely an integro-differential version of the original equation which arises as a model for the phenomenological description of phase transition based on the relaxation of the chemical potential. We first deal with the viscous version of the model and present results on the existence of a global attractor of optimal regularity and its stability with respect to the physical sensible parameters involved in the equation. We finally discuss very recent results on the asymptotic behavior of the non-viscous 2D-model; remarkably, in absence of viscosity and instantaneous diffusion effects, even the well-posedness of the 3D-model is an open question.

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.