Organizers: Giovanni Catino and Fabio Cipriani

**Giuseppe Maria Coclite**, Politecnico di Bari,

*Nonlinear Peridynamic Models*, Wednesday, January 22, 2020, time 15:15, Sala Consiglio 7° piano

**Abstract:****Abstract:**
Some materials may naturally form discontinuities such as cracks as a result of scale effects and long range interactions. Peridynamic models such behavior introducing a new nonlocal framework for the basic equations of continuum mechanics. In this lecture we consider a nonlinear peridynamic model and discuss its well-posedness in suitable fractional Sobolev spaces.
Those results were obtained in collaboration with S. Dipierro (Perth), F. Maddalena (Bari) and E. Valdinoci (Perth).
**Matteo Cozzi**, University of Bath,

*Long-time asymptotics for evolutionary crystal dislocations models*, Tuesday, December 17, 2019, time 15:30, Aula seminari 3° piano

**Abstract:****Abstract:**
In this talk, I will discuss a recent result concerning the long-time behavior of solutions to evolutionary Peierls-Nabarro type equations, related to crystal dislocations.
I will present the construction of solutions that, at large times, behave like a superposi- tion of an arbitrary finite number of fundamental dislocations, equally oriented and centered near points that evolve according to a repulsive dynamical system.
This result has been obtained in collaboration with J. D ?avila and M. del Pino (University of Bath).
**Matteo Caggio**, Università degli Studi dell'Aquila,

*On the highly compressible limit for the Navier-Stokes-Korteweg model with density dependent viscosity*, Tuesday, November 12, 2019, time 14:30, Aula seminari 3° piano

**Abstract:****Abstract:**
We investigate the regime of high Mach number flows for compressible barotropic fluids with density dependent viscosity. The Korteweg model as an isothermal model of capillary and quantum compressible fluids is considered. A weak-strong uniqueness analysis is also discussed.
**Hugo Tavares**, Universidade de Lisboa,

*Sharp concentration estimates near criticality for sign-changing solutions of Dirichlet and Neumann problems*, Tuesday, November 12, 2019, time 15:30, Aula seminari 3° piano

**Abstract:****Abstract:**
Consider the slightly subcritical problem $-\Delta u_\varepsilon = |u_\varepsilon|^{\frac{4}{n-2}-\varepsilon}u_\varepsilon$ either on $\mathbb{R}^n$ ($n\geq 3$) or in a ball $B$ satisfying Dirichlet or Neumann boundary conditions. For radial solutions, we provide sharp rates and constants describing the asymptotic behavior (as $\varepsilon\to 0$) of all local minima and maxima of $u_\varepsilon$ as well as its derivative at roots. As corollaries, we complement a known asymptotic approximation of the Dirichlet nodal solution in terms of a tower of bubbles and present a similar formula for the Neumann problem.
Moreover, we analyse the nonradial case with Neumann boundary conditions, namely the existence of least energy solutions and their dependence on the exponent $p$ up to the Sobolev critical exponent.
These are joint works with Alberto Saldaña and Massimo Grossi.
**Giulio Ciraolo**, Università degli Studi di Milano,

*Symmetry results for critical $p$-Laplace equations*, Wednesday, October 23, 2019, time 15:15, Aula seminari 3° piano

**Abstract:****Abstract:**
It is known that positive solutions to $\Delta_p u + u^{p^*-1}=0$ in $\mathbb{R}^n$, with $n \geq 3$ and $1

**Antonio Marigonda**, University of Verona,

*Control problems in Wasserstein space*, Monday, October 14, 2019, time 15:15, Aula seminari 6° piano

**Abstract:****Abstract:**
In this talk we present recent results about the existence and uniqueness of the viscosity solution for a certain classes on Hamilton-Jacobi Equations in the Wasserstein space of probability measure, arising in problem of mean field control of multi-agent systems. We consider a multi-agent system subject to a centralized controller
aiming to minimize a cost function. The microscopic dynamics of each agent is given by a differential inclusion. We model the distribution of agents by a probability measure, and formulate the minimization problem
as a Mayer problem for a dynamics in the Wasserstein space represented by a controlled continuity equation describing the macroscopical evolution of the system. We prove that the value function V of the
problem solves a Hamilton-Jacobi equation in the Wasserstein space in a suitable viscosity sense, and prove a comparison principle for such an equation, thus characterizing V as the unique viscosity solution of the
Hamilton-Jacobi equation associated to the problem.