Organizers: Giovanni Catino and Fabio Cipriani

**Tobias Weth**, Goethe-Universitat Frankfurt,

*Serrin's overdetermined problem on the sphere*, Monday, April 23, 2018, time 16:00, Aula 6° piano

**Abstract:****Abstract:**
In this talk, I will discuss Serrin's overdetermined boundary value problem
\begin{equation*}
-\Delta\, u=1 \quad \text{ in $\Omega$},\qquad u=0, \; \partial_\eta u=\textrm{const} \quad \text{on $\partial \Omega$}
\end{equation*}
in subdomains $\Omega$ of the round unit sphere $S^N \subset {\mathbb R}^{N+1}$, where $\Delta$ denotes the Laplace-Beltrami operator on $S^N$. We call a subdomain $\Omega$ of $S^N$ a Serrin
domain if it admits a solution of this overdetermined problem. In our main result, we construct Serrin domains in $S^N$, $N \ge 2$ which bifurcate from symmetric straight tubular neighborhoods of the
equator. By this we complement recent rigidity results for Serrin domains on the sphere. This is joint work with M.M.Fall and I.A.Minlend (AIMS Senegal).
**Lorenzo Mazzieri**, Università degli Studi di Trento,

*Monotonicity formulas in potential theory with applications*, Wednesday, April 18, 2018, time 15:15, Aula seminari 3° piano

**Abstract:****Abstract:**
We present an overview of some new monotonicity formulas, holding in the realm of linear and nonlinear potential theory, together with their main applications to several domains of investigation. These are ranging from the theory of overdetermined boundary value problems in the classical Euclidean setting, to the classification of static black holes in general relativity and to the geometry of manifolds with nonnegative Ricci curvature. (Joint works with V. Agostiniani, M. Fogagnolo and A. Pinamonti).
**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.
**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 $Q(x)$.