Matematica alla SISSA:

il contributo degli ex allievi

24-27 novembre 2008

SISSA, Trieste

Roberto Alicandro (Università di Cassino)**Variational limits of stochastic discrete systems.**

Debora Amadori (Università dell'Aquila)**A hyperbolic model of granular flow.**

In this talk we present a model for the evolution of granular matter, such as avalanches, proposed by Hadeler and Kuttler. The model describes the evolution of a moving layer flowing downhill and of a standing layer, in one space dimension. From the analytical point of view, it consists of a 2x2 system of conservation laws with a source term, and can be successfully analyzed within the framework of BV solutions. We will discuss the existence of solutions for this problem and their limiting behavior as the thickness of the flowing down material vanishes.

This is a joint work with Wen Shen (Penn State University).

Giovanni Bellettini (Università di Roma "Tor Vergata")**Reconstruction of a three-dimensional shape from the visible part of its apparent contour.**

Luca Biasco (Università di Roma Tre)**Periodic orbits approaching invariant manifolds of finite and infinite dimensional hamiltonian systems**

Abstract: I present some recent results obtained in collaboration with M. Berti, L. Di Gregorio and E. Valdinoci about the existence of infinitely many periodic solutions accumulating on invariant manifolds of hamiltonian systems. I show some applications to Celestial Mechanics (n-body problem) and hamiltonian PDEs (nonlinear wave equation, nonlinear Schroedinger equation).

Paolo Caldiroli (Università di Torino)**Blowing bubbles**

Giuseppe Maria Coclite (Università di Bari)**Conservation Laws with Singular Nonlocal Sources**

Abstract: In this lecture we consider a one-dimensional hyperbolic conservation law with integral source that contains a singular nonlinear term in the origin.

In several gas-dynamics and traffic models we have fluxes or sources depending on the reciprocal of the density: our equation is an integral regularization of such models. The sharp assumptions on the integral kernel are satisfied by several Green's functions of elliptic problems,

in these cases our equation is equivalent to an hyperbolic-elliptic system similar to the ones associated to the Camassa-Holm, Degasperis-Procesi and radiating gases models.

We work on the initial-boundary value problem with homogenous Dirichlet boundary conditions and prove the existence of weak solutions that are almost everywhere positive.

The results were obtained in collaboration with Professor Mario M. Coclite.

Giovanni Colombo (Università di Padova)**Sets with positive reach: regularity properties and applications to control theory.**

Abstract: Sets with positive reach were defined and thoroughly studied by Federer (T.A.M.S. 1959) as a common generalization of convex sets and sets with a smooth boundary. Some new properties of such sets will be presented, including a comparison of De Giorgi and Clarke normals and improved versions of the isoperimetric inequality. Next, functions whose epigraph has positive reach will be discussed. Some regularity properties will be presented, including twice a.e. differentiability and a representation formula for the generalized gradient. Finally, as a motivating application, it will be shown that the epigraph of the minimum time function of some linear control problems has positive reach. Infinite dimensional generalizations and open problems will also be discussed.

Rinaldo Colombo (Università di Brescia)**On the Continuum Modeling of Crowds.**

Abstract: Defining evacuation strategies from buildings or stadiums, designing airports or commercial centers, optimizing corridors in subways or stations. These activities need the ability to effectively describe the motion of a crowd. Various analytical techniques are of help in this task. In turn, crowd management poses questions that lead to new analytical problems.

This presentation will overview some results in this area. Starting from some phenomenological observations, recent analytical as well as experimental results will be presented.

Vittorio Coti Zelati (Università di Napoli "Federico II")**Solutions of Hamiltonian systems homoclinic to invariant tori.**

Veronica Felli (Università di Milano Bicocca)**On the behavior of solutions to Schrödinger equations near an isolated singularity of the electromagnetic potential.**

Abstract: We will present the results obtained in collaboration with Alberto Ferrero and Susanna Terracini about the asymptotics of solutions to Schrödinger equations with singular magnetic and electric potentials. By using a Almgren type monotonicity formula, separation of variables, and an iterative Brezis-Kato type procedure, we describe the exact behavior near the singularity of solutions to linear and semilinear (critical and subcritical) elliptic equations with an inverse square electric potential and a singular magnetic potential with a homogeneity of order -1.

Adriana Garroni (Università di Roma "La Sapienza")**Variational models for phase transitions: a multi-phase non local problem**

We consider a variational model for the study of an important class of defects in crystals, the dislocations. Dislocations are line defect that can be seen as discontinuities of a phase field that represents the plastic slip. We formulate a multi-phase model that exhibits the competition between two terms: a multiple well potential that forces the phase to take values in a discrete set, enforcing the compatibility of the slip with the crystalline structure, and a regularizing term that accounts for the non local elastic distortion due to the slip. After a proper scaling of the energy we study the asymptotic behaviour in terms of $\Gamma$-convergence. In the limit the competition between the two terms in resolved by an anisotropic line tension energy. We will recall the main classical results related with variational model for phase transitions and stress the main differences with the case under consideration. The non local term in the energy produces a logarithmic behaviour, so that many scales play a role in the characterization of the line tension energy density, a multiscale analysis and a relaxation process are required.

Alessandro Giacomini (Università di Brescia)**Crack initiation in brittle materials**

Stefano Luzzatto (Imperial College, London)**Invariant measures for maps of the interval**

Abstract: I will give a review of classical and recent results on the ergodic theory of one-dimensional maps, with particular focus on the problem of the existence and properties of invariant measures which are absolutely continuous with respect to Lebesgue.

Andrea Marson (Università di Padova)**The converge rate of the Glimm scheme**

Abstract: In a celebrated paper appeared in the journal Communications on Pure and Applied Mathematics in 1965, J. Glimm proved the first general existence theorem for the Cauchy problem for systems of conservation laws in one space dimensions. Since then, several papers were published concerning the so-called Glimm scheme, both investigating its extensions, and the convergence rate. In this talk, after a review of the scheme, we will focus on this second topic, dealing with some recent results obtained in collaboration with F. Ancona.

Marcello Ponsiglione (Università di Roma "La Sapienza")**Variational equivalence between Ginzburg Landau energies and screw dislocations**

Vortices in superconductivity as well as screw dislocations in crystals can be represented, in a two dimensional setting, as point singularities. At higher resolution, such singularities have indeed a characteristic length scale $\epsilon$. In the Guinzburg Landau Theory, the scale $\epsilon$ is incorporated in a phenomenological free energy, expressed in terms of a complex order parameter $\psi$, which describes how deep into the superconducting phase the system is. Since the paper by Bethuel, Brezis and H\'elein, a lot of analysis has been done to study the asymptotic of the Ginzburg-Landau energy as the scale $\epsilon$ tends to zero. In this talk, we show that most of this analysis can be reinterpreted in the framework of screw dislocations, providing new interesting result in this context. The subject is a work in progress with R. Alicandro and M. Cicalese.

Franco Rampazzo (Università di Padova)**Curvature terms in the use of coordinates as controls in Classical Mechanics**

Enrico Serra (Politecnico di Torino)**Symmetry questions in Neumann problems with weight**

We discuss some recent results concerning symmetry properties of ground states of certain semilinear elliptic problems with Hénon-type weight and Neumann boundary conditions. The case of Dirichlet conditions has been the object of intense research in the last few years and the symmetry / symmetry breaking picture is fairly well understood. The same questions for Neumann problems are instead still partially open, and the structure of the problem appears to be quite richer, with links towards classical items such as trace inequalities. We will treat the case of power nonlinearities in dimension greater or equal to three and of Trudinger-Moser type nonlinearities in dimension two.

Mario Sigalotti (Institut Élie Cartan Nancy)**Generic controllability of the bilinear Schrödinger equation**

Susanna Terracini (Università di Milano Bicocca)**Uniform holder bounds and regularity properties of the limiting profile for highly competing nonlinear systems of Schroedinger equations**

We consider systems of k Gross-Pitaevskii equations, in the case of large interspecies competitive interactions, both in the focusing and defocusing cases. We prove a priori bounds in the space of Holder continuous functions and, as a consequence, convergence to a limiting space, which can be proved to be Lipschitz continuous. This problem arises in the study of k-mixtures of Bose-Einstein condensates. The technique is based upon perturbed monotinicity formulas. As a by product we can prove convergence of both the ground and excited states.

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