Organizers: Giovanni Catino and Fabio Cipriani

**Riccardo Molle**, Università di Roma "Tor Vergata",

*Nonlinear scalar field equations with competing potentials*, Tuesday, April 17, 2018, time 15:15, Aula 6° piano

**Abstract:****Abstract:**
In this talk a class of nonlinear elliptic equations on R^N is presented. These equations come from physical problems, where a potential interacts with the bumps in the solutions. We first discuss the interactions and present some old and new results; then we consider competing potentials, showing in particular a theorem on the existence of infinitely many positive bound state solutions. Finally, we present some examples on existence/non existence of a ground state solution, when the theorem applies.
**Enrico Serra**, Politecnico di Torino,

*Nonlinear Schrödinger equations on branched structures: the role of topology in the existence of ground states*, Wednesday, April 11, 2018, time 11:15, Aula seminari 3° piano

**Abstract:****Abstract:**
We describe some results on the existence or nonexistence of ground states of prescribed mass for the nonlinear Schrödinger equation on noncompact metric graphs. We will highlight the role played by the topology of the graph in all the existence results, in the cases of L^2 subcritical and critical power nonlinearity. In particular, in the critical case, a key role is played by a thorough analysis of the Gagliardo-Nirenberg inequalities on metric graphs and by estimates of their best constants. Most of the techniques are new and suited to the investigation of variational problems on metric graphs.
**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.