Organizers: Giovanni Catino and Fabio Cipriani

**Simone Dovetta**, Politecnico di Torino,

*Action versus energy ground states in nonlinear Schrödinger equations*, Thursday, January 26, 2023, time 15:15, Aula Seminari III Piano

**Abstract:****Abstract:**
The talk investigates the relation between normalized critical points of the nonlinear Schrödinger energy functional and critical points of the corresponding action functional on the associated Nehari manifold. First, we show that the ground state levels are strongly related by the following duality result: the (negative) energy ground state level is the Legendre–Fenchel transform of the action ground state level. Furthermore, whenever an energy ground state exists at a certain frequency, then all action ground states with that frequency have the same mass and are energy ground states too. We see that the converse is in general false and that the action ground state level may fail to be convex. Next we analyze the differentiability of the ground state action level and we provide an explicit expression involving the mass of action ground states. Finally we show that similar results hold also for local minimizers, and we exhibit examples of domains where our results apply.
This is a joint work with Enrico Serra and Paolo Tilli.
**Andrea Giorgini**, Politecnico di Milano,

*Recent results for the Navier-Stokes-Cahn-Hilliard model with unmatched densities*, Thursday, January 19, 2023, time 15:15, Aula Seminari III Piano

**Abstract:****Abstract:**
We consider the initial-boundary value problem for the incompressible Navier-Stokes-Cahn-Hilliard system with non-constant density proposed by Abels, Garcke and Grün in 2012. This model arises in the diffuse interface theory for binary mixtures of viscous incompressible fluids. In particular, this system is a generalization of the well-known Model H in the case of fluids with unmatched densities. In this talk, I will present some recent results concerning the propagation of regularity of global weak solutions (for which uniqueness is not known) and their longtime convergence towards an equilibrium state in three dimensional bounded domains.
**Andrea Signori**, Politecnico di Milano,

*Chemotaxis model for tumour growth*, Thursday, December 15, 2022, time 15:15, Aula Seminari III Piano

**Abstract:****Abstract:**
We discuss analytic results for a new diffuse interface model describing the evolution of a tumour mass under the effects of a chemical substance (e.g., a nutrient). The process is described by utilising an order parameter representing the local proportion of tumour cells, and a variable describing the concentration of the chemical. The order parameter is assumed to satisfy a suitable form of the Cahn–Hilliard equation with mass source and logarithmic potential of Flory–Huggins type, whereas the chemical concentration satisfies a reaction-diffusion equation where the cross-diffusion term has the same expression as in the celebrated
Keller–Segel model. Weak well-posedness, regularity, and continuous dependence results are presented.
This is a joint work with E. Rocca (University of Pavia) and G. Schimperna (University of Pavia).
**Andrea Giorgini**, Politecnico di Milano,

*Recent results for the Navier-Stokes-Cahn-Hilliard model with unmatched densities*, Thursday, December 01, 2022, time 15:15, Aula Seminari III Piano

**Abstract:****Abstract:**
SI AVVISA CHE IL SEMINARIO IN OGGETTO È ANNULLATO PER INDISPONIBILITÀ DELLO SPEAKER
**Alessandro Giacomini**, Università di Brescia,

*A free discontinuity approach to optimal profiles in Stokes flows*, Tuesday, November 22, 2022, time 15:15 o'clock, Aula Seminari III Piano

**Abstract:****Abstract:**
In the talk I will consider the problem of finding the optimal shape of an obstacle which minimizes the drag force in an incompressible Stokes flow under Navier conditions at the boundary. I will propose a relaxation of the problem within the framework of free discontinuity problems, modeling the obstacle as a set of finite perimeter and the velocity field as a special function of bounded deformation (SBD): within this approach, the optimal obstacle may develop naturally geometric features of co-dimension 1.
**Alberto Roncoroni**, Politecnico di Milano,

*On the critical $p-$Laplace equation*, Thursday, November 10, 2022, time 15:15, Aula Seminari III Piano

**Abstract:****Abstract:**
The starting point of the seminar is the well-known generalized Lane-Emden equation
\begin{equation}
\Delta_p u + \vert u \vert^{q-1} u =0 \quad \text{ in $\mathbb{R}^n$ } \, , \quad \quad \text{(LE)}
\end{equation}
where $\Delta_p$ is the usual $p-$Laplace operator with $ 1 < p < n $ and $ q > 1 $. I will discuss several non-existence and classification results for positive solutions of (LE) in the subcritical ($q < p^\ast - 1 $) and in the critical case ($q = p^\ast - 1 $). In the critical case, it has been recently shown, exploiting the moving planes method, that positive solutions to the critical $ p-$ Laplace equation (i.e. (LE) with $ q = p^\ast -1 $) and with finite energy, i.e. such that $ u \in L^{p^\ast}(\mathbb{R}^n) $ and $ \nabla u \in L^p(\mathbb{R}^n) $, can be completely classified. In this talk, I will present some recent classification results for positive solutions to the critical $p-$Laplace equation with (possibly) infinite energy satisfying suitable conditions at infinity. Moreover, if time permits I will discuss analogous results in the anisotropic, conical and Riemannian settings.
This is based on a recent joint work with G. Catino and D. Monticelli.