Jean Dolbeault, Ceremade, Entropy methods for parabolic and elliptic equations, Tuesday, Febraury 06, 2018, time 15:15, Aula seminari 3° piano
Abstract:Abstract:
Entropy methods associated with linear and nonlinear parabolic equations are powerful tools for establishing results of symmetry, unique- ness and optimality in functional inequalities. Optimality cases can be identified by considering asymptotic regimes and appropriate lineariza- tions. Entropy methods raise various issues of regularity and rely on integrations by parts which are not straightforward to justify. On the other hand, these methods give optimal criteria in some fundamental examples. This lecture will be devoted to an overview of the state of the art and list some open questions.

Alessio Falocchi, Politecnico di Milano, Torsional instability in two nonlinear isolated models for suspension bridges, Tuesday, January 16, 2018, time 15:15, Aula seminari 6° piano
Abstract:Abstract:
Many observations throughout history have recorded that torsional instability afflicts suspension bridges. In the scientific community there is not an unanimously accepted explanation about the origin of instability. We present two new dynamical isolated models derived from variational principles and we study two nonlinear hyperbolic partial differential equations, involving the vertical displacement and the torsional rotation of the deck. In the first model we consider three different nonlinearities to model the hangers slackening mechanisms; in the second one we deal with the nonlinearities due to the geometric configuration of the main cables, supposed deformable, and we prove the existence and uniqueness of a weak solution. We show some numerical experiments, highlighting the torsional instability phenomena with respect to the longitudinal modes excited. Our results suggest that the origin of the instability is hidden in the nonlinear behavior of these structures.

Matteo Muratori, Politecnico di Milano, Sobolev-type inequalities on Cartan-Hadamard manifolds and applications to some nonlinear diffusions, Wednesday, January 10, 2018, time 15:15, Sala del Consiglio, 7° piano
Abstract:Abstract:
It is well known that the classical Sobolev inequality not only holds on Euclidean space, but also on any Cartan-Hadamard manifold, namely a complete and simply connected Riemannian manifold with everywhere nonpositive sectional curvatures. On the other hand, the Poincaré (or spectral gap) inequality fails on Euclidean space but holds on hyperbolic space or more in general on any Cartan-Hadamard manifold with sectional curvatures bounded from above by a negative constant. However, almost nothing seems to be known in between, that is when curvatures are negative but allowed to vanish at infinity. Here we show some partial results in this direction, mainly restricted to radial functions, and discuss related consequences concerning smoothing effects for certain nonlinear diffusions of porous medium type. This is a joint work with A. Roncoroni.

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.