Andrea Mondino, University of Warwick, A representation formula for the Laplacian of the distance function, Wednesday, March 21, 2018, time 15:15, Sala del Consiglio 7° piano
Abstract:Abstract:
In the seminar I will present a recent work in collaboration with Fabio Cavalletti (SISSA) where, using techniques from optimal transportation, we prove a rather explicit representation formula for the Laplacian of the distance function in spaces with Ricci curvature bounded below. Even if the paper deals with rather general non-smooth spaces, since some results seem new even for smooth manifolds, the seminar will be mostly focused on the smooth framework. This seminar is organized within the PRIN 2015 Research project «Variational methods, with applications to problems in mathematical physics and geometry» Grant Registration 2015KB9WPT_010, funded by MIUR – Project coordinator Prof. Gianmaria Verzini

Scott Rodney, Cape Breton University, Poincaré-Sobolev Inequalities and the p-Laplacian, Wednesday, Febraury 21, 2018, time 15:15, Aula seminari 3° piano
Abstract:Abstract:
It is well known that Poincar\'e-Sobolev inequalities play an important role in applications and in regularity theory for weak solutions of PDEs. In this talk I will discuss two new results connecting matrix weighted Poincar\'e-Sobolev estimates to the existence of regular weak solutions of Dirichlet and Neumann problems for a degenerate $p$-Laplacian: \begin{eqnarray} \Delta_{Q,p} \varphi(x) = \textrm{Div}\left(\big|Q(x)~\nabla \varphi(x)\big|^{p-2}~Q(x)~\nabla\varphi(x)\right).\nonumber \end{eqnarray} Degeneracy of $\Delta_{Q,p}$ is given by a measurable non-negative definite matrix-valued function $Q(x)$.

Daniele Cassani, Università degli Studi dell'Insubria, Critical aspects of Choquard type equations, Wednesday, Febraury 07, 2018, time 16:30, Aula seminari 3° piano
Abstract:Abstract:
We are concerned with a class of nonlocal Schroedinger equations which show up in many different applied contexts. In particular we consider the case in which the nonlinear interaction has critical features. In the classical Sobolev sense, when we are in presence of the Hardy-Littlewood-Sobolev upper-critical exponent, as well as new critical phenomena occur, the so-called 'bubbling at infinity', when the nonlinearity exhibits lower-critical growth. Assuming mild conditions on the nonlinearity and by using variational methods, we establish existence, non-existence and qualitative properties of finite energy solutions. Some partial results in the limiting case of dimension two will be also presented. (Joint works in collaboration with: J. Zhang; J. Van Schaftingen and J. Zhang; C. Tarsi and M. Yang).

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.