Data: 08/04/2019

Ora: 14:30:00

Descrizione:

I will report on some recent results - obtained in joint work with Huyuan Chen - on Dirichlet problems for the Logarithmic Laplacian Operator, which arises as formal derivative of fractional Laplacians at order s= 0. I will discuss the functional analytic framework for these problems and show how it allows to characterize the asymptotics of principal Dirichlet eigenvalues and eigenfunctions of fractional Laplacians as the order tends to zero. Furthermore, I will discuss necessary and sufficient conditions on domains giving rise to weak and strong maximum principles for the logarithmic Laplacian. If time permits, I will also discuss regularity estimates for solutions to corresponding Poisson problems.

]]>Data: 15/03/2019

Ora: 14:30:00

Descrizione:

We study a geometric flow driven by the fractional mean curvature. The notion of fractional mean curvature arises naturally when performing the first variation of the fractional perimeter functional. More precisely, we show the existence of surfaces which develope neckpinch singularities in any dimension n ≥ 2. Interestingly, in dimension n = 2 our result gives a counterexample to Greyson Theorem for the classical mean curvature flow. We also present a very recent result, in the volume preserving case, establishing convergence to a sphere. The results have been obtained in collaboration with C. Sinestrari and E. Valdinoci.

Data: 22/02/2019

Ora: 11:30:00

Descrizione:

The physical life of a comet near the Sun is relatively short. Therefore, in order to provide the actual flow of comets, there should be cometary reservoirs in the solar system where comets do not feel the heat of the Sun. For long period comets, i.e. comets with an orbital period greater than 200 years, this reservoir is believed to be at more than 10,000 times the Earth-Sun distance, known as the Oort Cloud. However, the shape, density and formation of this Oort cloud remains an open question.

The problem is that cometary dynamics is highly chaotic, so it is useless to investigate the origin of comets by backward propagation of their movement. We are therefore reduced to massive simulations where huge samples of synthetic comets are propagated forward over a long time span. The Oort cloud informations are then obtained by comparing the final results with the observations.

I will first introduce how we have built our model for the long-term propagation of long-lived comets. This model takes into account the gravitational attraction of the whole galaxy on the Sun and the comets, which induces an quasi-integrable dynamic, the passage of the stars close to the Sun and the planetary scattering by the four giant planets of our solar system. These last two effects are stochastic.

Then, I will present how this model has allowed us to investigate the memory of the Oort cloud and should give us crucial informations about its formation.

Data: 18/12/2018

Ora: 15:00:00

Descrizione: ]]>

Data: 30/10/2018

Ora: 09:00:00

Descrizione:

After giving some basic examples of Lie groupoids, we will explain the rule of convolution of distributions on a Lie groupoid G and its relationship with the symplectic groupoid structure on T^*G found by Coste-Dazord-Weinstein. Secondly, we will explain how to develop a calculus for Fourier Integral Operators on a groupoid G. Finally, we will show that the one parameter group e^{itP}, t ∈ **R**, where P is a first order elliptic positive G-pseudodifferential operator, consists of G-FIOs in a weaker sense.