Organizers: Giovanni Catino and Fabio Cipriani
Matteo Bonforte, Universidad Autonoma de Madrid,
Nonlinear and Nonlocal Degenerate Diffusions on Bounded Domains, Thursday, March 16, 2017, time 11:00 o'clock, Aula seminari 3° piano
Abstract:Abstract:
We investigate quantitative properties of nonnegative solutions to a nonlinear fractional diffusion equation, of degenerate type, posed in a bounded domains, with appropriate homogeneous Dirichlet boundary conditions. The diffusion is driven by a quite general class of linear operators that includes the three most common versions of the fractional Laplacian in a bounded domain with zero Dirichlet boundary conditions; many other examples are included. The nonlinearity is assumed to be increasing and is allowed to be degenerate, the prototype being convex powers. We will present some recent results about existence, uniqueness and a priori estimates for a quite large class of very weak solutions, that we call weak dual solutions.
We will devote special attention to the regularity theory: decay and positivity, boundary behavior, Harnack inequalities, interior and boundary regularity, and asymptotic behavior. All this is done in a quantitative way, based on sharp a priori estimates. Although our focus is on the fractional models, our techniques cover also the local case, and provide new results even in this setting.
A surprising instance of this problem is the possible presence of nonmatching powers for the sharp upper and lower boundary behavior. This unexpected phenomenon is a completely new feature of the nonlocal nonlinear structure of this model, and it is not present in the elliptic case.
The above results are contained on a series of recent papers in collaboration with A. Figalli, Y. Sire, X. Ros-Oton and J. L. Vazquez.
Andrzej Zuk , C.N.R.S. France et Université Paris 7,
Spectra, automata and discrete analogues of the KdV equation, Monday, March 13, 2017, time 11:30 o'clock, Aula seminari 6° piano
Abstract:Abstract:
Box-Ball systems are discrete analogues of the KdV equation. We prove that their evolution can be described by automata. With these automata we associate self-adjoint operators. We relate spectral properties of these operators with L^2 Betti numbers of closed manifolds.
Genni Fragnelli, Università degli Studi di Bari,
Null controllability in degenerate and singular parabolic problems, Wednesday, March 08, 2017, time 15:00 o'clock, Aula seminari 6° piano
Abstract:Abstract:
In this talk we will present the problem of null controllability for parabolic systems and we will focus on some recent results for degenerate and singular problems coming from real-world models, such as Biology, Climatology, Medicine.
Dimitri Mugnai, Università di Perugia,
Fractional Hartree equations, Wednesday, March 08, 2017, time 16:15 o'clock, Aula seminari 6° piano
Abstract:Abstract:
We will report on some recent results on fractional Hartree equations having various forms of nonlinearities.
Elisabetta Chiodaroli, EPFL, Lausanne,
Surprising solutions to the Euler system of isentropic gas dynamics, Wednesday, Febraury 15, 2017, time 17:00 o'clock, Aula seminari 6° piano
Abstract:Abstract:
In this talk we discuss some applications of the method of convex integration to the compressible Euler system of gas dynamics in several space dimensions. This leads to the construction of
non-standard solutions which disprove the efficiency of classical admissibility criteria proposed in the literature to select unique solutions. We also show some recent studies meant at visualising numerically such non-standard solutions.
Fabio Cavalletti, SISSA - Scuola Internazionale Superiore di Studi Avanzati,
Isoperimetric inequality for metric measure spaces, Wednesday, Febraury 01, 2017, time 13:15, Aula seminari 3° piano
Abstract:Abstract:
By using an L^1 localization argument, we prove that in metric measure spaces satisfying lower Ricci curvature bounds (more precisely RCD*(K,N) or more generally essentially non branching CD*(K,N) the classical Lévy-Gromov isoperimetric inequality holds with the associated rigidity and almost rigidity statements.