Organizers: Giovanni Catino and Fabio Cipriani

**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
**Franco Tomarelli**, Politecnico di Milano,

*Pure Traction Problems between Linear and Finite Elasticity*, Wednesday, June 19, 2019, time 15:15, Aula seminari 3° piano

**Abstract:****Abstract:**
A limit elastic energy for pure traction problem is derived from re-scaled nonlinear energies of a hyperelastic material body subject to an equilibrated force field.
We show that the strains of minimizing sequences associated to re-scaled nonlinear energies weakly converge, up to subsequences, to the strains of minimizers of a limit energy, provided an additional compatibility condition is fulfilled by the force field. The limit energy functional exhibits a gap that makes it different from the classical linear elasticity functional; nevertheless the compatibility condition entails the coincidence of related minima and minimizers. A strong violation of this condition provides a limit energy which is unbounded
from below, while a mild violation may produce unboundedness of strains and a limit energy which has infinitely many extra minimizers which are not minimizers of standard linear elastic energy. A consequence of this analysis is that a rigorous validation of linear elasticity fails for compressive force fields that infringe such compatibility condition.
**Francesca Colasuonno**, Università degli Studi di Torino,

*Symmetry preservation for fourth order eigenvalue optimization problems*, Wednesday, June 12, 2019, time 15:15, Aula seminari 3° piano

**Abstract:****Abstract:**
In this talk, we will discuss some recent results on two eigenvalue optimization problems governed by the biharmonic operator, under Dirichlet or Navier boundary conditions. From a physical point of view, in two dimensions our problem corresponds to building a plate, of prescribed shape and mass, out of materials with different densities --varying in a certain range of values-- in such a way to minimize the lowest frequency of the body. This problem is also referred to as composite plate problem. Both for the clamped plate (i.e., Dirichlet b.c.) and for the hinged plate (i.e., Navier b.c.), we will prove the existence of an optimal configuration and give an explicit representation of the minimizing densities in terms of sublevel sets of their corresponding first eigenfunctions. Finally, we will discuss symmetry preservation properties of the optimal configurations, in the presence of some symmetry and convexity of the domain. The tools used differ depending on the boundary conditions: while the hinged plate problem inherits the maximum principles for second order elliptic systems, allowing us to exploit the moving plane method to get symmetry preservation in more general domains, the situation is more complicated for the clamped plate problem, where we will use the polarization technique and the properties of the Green's function to deal with radial symmetry preservation in a ball.
This talk is based on two joint papers with Eugenio Vecchi (Trento).