Organizers: Giovanni Catino and Fabio Cipriani

**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.
**William Borrelli**, Politecnico di Milano,

*Classification of ground states for critical Dirac equations*, Thursday, October 27, 2022, time 15:15 o'clock, Aula Seminari III Piano

**Abstract:****Abstract:**
In this talk I will present a classification result for critical Dirac equations on the Euclidean space, appearing naturally in conformal spin geometry and in variational problems related to Dirac equations on spin manifolds. Exploiting the conformal invariance, ground state solutions can be classified. This is the spinorial counterpart of the well-known result for the Yamabe equation.
Time permitting, I will also present similar results obtained for conformal Dirac-Einstein equations, consisting of a Dirac equation coupled with a Laplace- type equation.
Joint work with Ali Maalaoui, Andrea Malchiodi and Ruijun Wu.
**Fabio Cipriani**, Politecnico di Milano,

*Noncommutative Geometry by example*, Tuesday, June 07, 2022, time 14:15, Aula Seminari III Piano e online: https://polimi-it.zoom.us/j/95645334076?pwd=M1R1WmUvNno5bTduZkRZVk5RRSt0dz09

**Abstract:****Abstract:**
We will illustrate various examples of spaces which appear to be singular when analyzed by the classical tools of topology, measure theory, analysis and geometry, but which can be naturally approached by instruments of Noncommutative Geometry. Among them we will find instances of duals of non abelian groups, Clifford algebras of Riemannian manifolds, leafs space of foliations, orbit spaces of dynamical systems, orbifolds, spaces of observables and states in Quantum Mechanics, quasi-crystals, quantum information channels and quantum groups.
**Luigi De Rosa**, Università di Basilea,

*Hölder solutions, fractal singularities and turbulence for the incompressible Euler equations*, Wednesday, May 18, 2022, time 14:15, Aula Seminari III Piano

**Abstract:****Abstract:**
The motion of fluids in a turbulent regime turns out to be extremely chaotic. This makes their precise mathematical description very difficult and technical. Relying on the relatively new groundbreakings techniques introduced in this context by De Lellis and Székelyhidi I will explain how to date we can answer several mathematical questions that had a tremendous impact on both the mathematical and physical community. The talk will touch anomalous energy dissipation, the role of Hölder continuous weak solutions and fractal singularities in fully developed turbulence.