Organizers: Giovanni Catino and Fabio Cipriani

**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.
**Stefano Pigola**, Università dell’Insubria,

*Decay and Sobolev regularity properties for solutions at infinity of (nonlinear) PDEs*, Friday, September 20, 2019, time 11:15, Aula seminari 3° piano

**Abstract:****Abstract:**
I will present some recent results on the global behaviour of nonnegative and bounded subsolutions of $\Delta_p u = f(u)$ over an exterior domain of a complete Riemannian manifold. I shall discuss geometric conditions under which such a subsolution decays to zero at infinity. The main tools are represented by (a nonlinear version of) the Feller property and some global comparison results. These, in turn, are related to a new characterization of the ($p$-)stochastic completeness of the manifold in terms of the Sobolev space $W^{1,p}$.
**Abdelaziz Soufyane**, University of Sharjah,

*Stability of some coupled partial differential equations in both bounded and unbounded domains*, Thursday, September 12, 2019, time 15:15, Aula seminari 3° piano

**Abstract:****Abstract:**
This talk deals with some recent results on the stability of a coupled partial differential equations. We will present the energy decay rates for many systems (arising in many applications) in the bounded domain, different approaches will be used to establish the energy decay. Also, we will discuss the rate decay for some models in the unbounded domain using the Fourier transformation, the multipliers techniques in Fourier image. We conclude our talk by giving some remarks and open problems.
This seminar is organized within the PRIN 2017 Research project «Direct and inverse problems for partial differential equations: theoretical aspects and applications» Grant Registration number 201758MTR2, funded by MIUR - Project coordinator Prof. Filippo Gazzola
**Colette De Coster**, Univ. Valenciennes,

*A prescribed anisotropic mean curvature equation modeling the corneal shape: A paradigm of nonlinear analysis*, Tuesday, July 02, 2019, time 15:15, Aula seminari 3° piano

**Abstract:****Abstract:**
In this talk, we survey some recent results concerning the Dirichlet problem for the prescribed anisotropic mean curvature equation
\begin{equation*}
{\rm -div}\left({\nabla u}/{\sqrt{1 + |\nabla u|^2}}\right) = -au + {b}/{\sqrt{1 + |\nabla u|^2}},
\end{equation*}
in a bounded Lipschitz domain $\Omega \subset \mathbb{R}^n$, with $a,b>0$ parameters.
This equation appears in the description of the geometry of the human cornea, as well as in the modeling theory of capillarity phenomena for compressible fluids.
In this talk, we show how various techniques of nonlinear functional analysis can successfully be applied to derive a complete picture of the solvability patterns of the problem.
This seminar is organized within the PRIN 2015 Research project «Partial Differential Equations and related Analytic-Geometric Inequalities» Grant Registration number 2015HY8JCC _003, funded by MIUR – Project coordinator Prof. Filippo Gazzola