26 jun 2015
Aula 2001, Dip. Matematica, Università degli Studi di Roma “Tor Vergata”, Roma
To cherish Nirenberg’s 90th birthday and to celebrate his award of the Abel Prize 2015 the following mathematicians have accepted to deliver a talk:
Xavier Cabré (ICREA and UPC Barcelona, Spain)
Maria J. Esteban (Paris Dauphine, France)
Michael Struwe (ETH Zurich, Switzerland)
Claude Viterbo (ENS Paris, France)
Organizers: Gabriella Tarantello and Daniele Castorina
Program:
14:30 - Xavier Cabré, ICREA and UPC, Barcelona-Spain
Curves and surfaces with constant nonlocal mean curvature: meeting Alexandrov and Delaunay
Abstract: This is a joint work with Mouhamed M. Fall, Joan Sol-Morales and Tobias Weth. It concerns hypersurfaces of $\mathbb{R}^N$ with constant nonlocal (or fractional) mean curvature. This is the equation associated to critical points of the fractional perimeter under a volume constraint. Our results are twofold. First we prove the nonlocal analogue of the Alexandrov result characterizing spheres as the only closed embedded hypersurfaces in $\mathbb{R}^N$ with constant mean curvature. Here we use the moving planes method. Our second result establishes the existence of periodic bands or ``cylinders'' in $\mathbb{R}^2$ with constant nonlocal mean curvature and bifurcating from a straight band. These are Delaunay type bands in the nonlocal setting. Here we use a Lyapunov-Schmidt procedure for a quasilinear type fractional elliptic equation.
15:15 - Maria J. Esteban, Universite' de Paris Dauphine, Paris-France
Optimal symmetry results for the optimizers of Caffarelli-Kohn-Nirenberg inequalities
Abstract: In this talk I will present recent results, obtained in collaboration with J. Dolbeault and M. Loss, proving the radial symmetry of the optimizers of Caffarelli-Kohn-Nirenberg inequalities whenever they are local minima in the full functional space. These results are optimal and close a series of works proving partial results. The method used to obtain this result, which actually proves the uniqueness of positive solutions for the corresponding Euler-Lagrange equations, is based on a nonlinear fast diffusion flow which is applied around any positive solution to explore the energy landscape around it.
16:00 - Coffee break
16:30 - Michael Struwe, E.T.H., Zurich-Switzerland
The supercritical Lane-Emden equation and its gradient flow.
Abstract:
In joint work with Simon Blatt we study the Lane-Emden heat flow
$u_t-\Delta u = u
u
^{p-2}$ in the supercritical regime when
$p>\frac{2n}{n-2}$
and establish Morrey estimates and partial regularity results up to the
blow-up time.
17:15 - Claude Viterbo, ENS, Paris-France
Symplectic Homogenization.
Seminars:
26 Jun 2015
14:30 -
X. Cabré:
Curves and surfaces with constant nonlocal mean curvature: meeting Alexandrov and Delaunay
15:15 -
M. J. Esteban:
Optimal symmetry results for the optimizers of Caffarelli-Kohn-Nirenberg inequalities
16:30 -
M. Struwe:
The supercritical Lane-Emden equation and its gradient flow
17:15 -
C. Viterbo:
Symplectic Homogenization