Organizers: Giovanni Catino and Fabio Cipriani

**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.
**Flavia Smarrazzo**, Università Campus Bio-medico di Roma,

*Radon measure-valued solutions to a class of noncoercive diffusion equations with singular initial data*, Tuesday, May 03, 2022, time 14:15, https://polimi-it.zoom.us/j/96038004554?pwd=aitXT1BCTjBjRHh0L0dYUm1TcnN5QT09

**Abstract:****Abstract:**
Initial-boundary value problems for nonlinear parabolic equations $u_t = \Delta \phi(u)$ with a Radon measure as initial condition have been widely investigated, in general looking for solutions which for positive times take values in some function space. On the other hand, if the diffusivity degenerates too fast at infinity, it is well known that function-valued solutions may not exist, and in these singular cases it looks very natural to consider Radon measure-valued solutions.
The aim of this talk is to address existence and regularity results in the above framework, depending on whether or not the initial data charge sets of suitable capacity (determined by the growth order of $\phi$), and on suitable compatibility conditions, describing the behaviour of the singular part of solutions. The diffusion function $\phi$ is only assumed to be continuous, nondecreasing and at most powerlike: no assumptions about existence or estimates from below of the diffusivity $\phi'$ are made (except for some regularization results). The proof of existence is constructive and, in particuar, relies on a suitable approximation of the initial measure. Finally, the possible occurrence or lack of instantaneous $M-L^1$ regularizing effects for the constructed solutions, as well as partial uniqueness results, will also be discussed.
**Filippo Giuliani**, Politecnico di Milano (nuovo rtd-b del DMAT),

*Some results and open problems in Hamiltonian PDEs*, Thursday, March 17, 2022, time 12:15 o'clock, !! CAMBIO D'ORARIO !! Aula Seminari MOX VI Piano, 12:15-13:15

**Abstract:****Abstract:**
We present some recent results concerning the dynamics of Hamiltonian partial differential equations on compact manifolds, especially fluid dynamics models. We talk about both stability and instability issues, like the existence of quasi-periodic in time solutions and transfers of energy between Fourier modes of solutions of nonlinear PDEs.
We also discuss some open problems and possible new research directions.