Serena Dipierro, Università degli Studi di Milano, Long-range phase transitions and minimal surfaces, Tuesday, December 19, 2017, time 15:15, Aula seminari 3° piano
Abstract:Abstract:
We discuss some recent results on nonlocal minimal surfaces and discuss their connections with nonlocal phase transitions. In particular, we will consider the "genuinely nonlocal regime" in which the diffusion operator is of order less than 1 and present some rigidity and symmetry results.

Alberto Farina, Université de Picardie, A sharp Bernstein-type theorem for entire minimal graphs, Tuesday, December 05, 2017, time 15:15, Sala Consiglio 7° piano
Abstract:Abstract:
We consider entire solutions $u$ to the minimal surface equation in $R^N$, with $N\ge8$, and we prove the following sharp result : if $N-7$ partial derivatives $ \frac{\partial u }{\partial {x_j}}$ are bounded on one side (not necessarily the same), then $u$ is necessarily an affine function.

Carlo Orrieri, Università di Roma La Sapienza, A variational approach to the Mean Field Planning Problem, Tuesday, November 21, 2017, time 15:15, Aula seminari 3° piano
Abstract:Abstract:
In the talk we deal with the so-called mean field planning problem: a coupled system of two PDEs, a forward continuity equation and a backward Hamilton-Jacobi equation. This problem has been introduced by P-L. Lions in a series of lectures held at Collège de France and can be viewed as a modification of the mean field games system as well as a generalization of the optimal transportation problem in its dynamic formulation à la Benamou-Brenier. We concentrate on the variational structure of the problem, from which a notion of "weak variational" solution can be given. In particular, we provide a well-posedness result for the system on the whole space in a $L^p$ framework under general assumptions on the coupling term. The talk is based on a joint work with G. Savaré and A. Porretta.

Matteo Santacesaria, Politecnico di Milano, Inverse problems for PDE via infinite dimensional compressed sensing, Tuesday, November 14, 2017, time 15:15, Aula seminari 3° piano
Abstract:Abstract:
Compressed sensing stands for a series of techniques whose aim is to recover a sparse signal from a small number of measurements. Since the 2006 seminal papers of Candes-Romberg-Tao and Donoho, which concerned the recovery of a sparse vector from few discrete Fourier coefficients, the subject has been extensively studied and generalized. In this talk we will present new results concerning generalization of compressed sensing in the framework of Hilbert spaces: in particular, the measurement operator does not need to be a orthonormal transformation and the unknown is assumed to be sparse in a frame. Applications to inverse problems for PDE, such as electrical impedance tomography, will be discussed as well. This is a joint work with Giovanni S. Alberti.

Fabio Punzo, Politecnico di Milano, The Porous Medium Equation with Large Initial Data on Negatively Curved Riemannian Manifolds , Wednesday, October 18, 2017, time 15:15, Aula seminari 3° piano
Abstract:Abstract:
We discuss existence and uniqueness of very weak solutions of the Cauchy problem for the porous medium equation on Cartan–Hadamard manifolds satisfying suitable lower bounds on Ricci curvature, with initial data that can grow at in?nity at a prescribed rate, that depends crucially on the curvature bounds. Furthermore, we give a precise estimate for the maximal existence time, and we show that in general solutions do not exist if the initial data grow at in?nity too fast. Such results have been recently obtained jointly with G. Grillo and M. Muratori.

Ciprian G. Gal, Florida International University, Miami, USA, Doubly nonlocal Cahn-Hilliard equations, Friday, July 14, 2017, time 11:00, Aula seminari 3° piano
Abstract:Abstract:
We consider a doubly nonlocal nonlinear parabolic equation which describes phase-segregation of a two-component material in a bounded domain. This model is a more general version than the recent nonlocal Cahn–Hilliard equation proposed by Giacomin and Lebowitz, such that it reduces to the latter under certain conditions. It turns out that there are four possible cases of double interaction in which the nonlocality is reflected, we shall discuss some of them.