Organizers: Giovanni Catino and Fabio Cipriani

**Alessandro Savo**, Università La Sapienza Roma,

*Heat content asymptotics of bounded domains*, Tuesday, May 08, 2018, time 15:15, Aula Seminari 3° piano

**Abstract:****Abstract:**
For a bounded domain in a Riemannian manifold, we consider the solution of the heat equation with unit initial data and Dirichlet boundary conditions. Integrating the solution with respect to the space variable one obtains the function of time known in the literature as the "heat content" of the given domain. In this talk we show how the geometry of the boundary affects heat diffusion by examining the small time behavior of the heat content. In particular, we study a three term asymptotic expansion for polyhedral Euclidean domains, and give a recursive algorithm for the calculation of the entire asymptotic series when the boundary is smooth.
**Adriano Pisante**, Università degli Studi di ROMA "La Sapienza" ,

*Large deviations for the stochastic Allen-Cahn approximation of the mean curvature flow*, Thursday, May 03, 2018, time 15:15, Aula seminari 3° piano

**Abstract:****Abstract:**
We consider the sharp interface limit for the Allen-Cahn equation on the three dimensional torus with deterministic initial condition and deterministic or stochastic forcing terms. In the deterministic case, we discuss the convergence of solutions to the mean curvature flow, possibly with a forcing term, in the spirit of the pioneering work of Tom Ilmanen (JDG '93). In addition we analyze the convergence of the corresponding action functionals to a limiting functional described in terms of varifolds. In the second part I will comment on related results for the stochastic case, describing how this limiting functional enters in the large deviation asymptotics for the laws of the corresponding processes in the joint sharp interface and small noise limit.
**Zindine Djadli**, Université Grenoble Alpes,

*A review on some fourth order problems on manifolds*, Tuesday, April 24, 2018, time 15:15, Sala del Consiglio 7° piano

**Abstract:****Abstract:**
I will review some recent works on some non linear problem on manifolds, mostly in conformal geometry.
**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.