Organizers: Giovanni Catino and Fabio Cipriani

**Gianmarco Sperone**, Dept. of Mathematical Analysis, Faculty of Mathematics and Physics, Charles University in Prague,

*Explicit bounds for the generation of a lift force exerted by steady-state Navier-Stokes flows over a fixed obstacle*, Wednesday, March 24, 2021, time 15:15 o'clock, Url: https://polimi-it.zoom.us/j/87365124649?pwd=SjZOcDgrQU9qUGZCM3FZRmxCTUhlUT09

**Abstract:****Abstract:**
We analyze the steady motion of a viscous incompressible fluid in a two- and three-dimensional channel containing an obstacle through the Navier-Stokes equations under different types of boundary conditions. In the 2D case we take constant non-homogeneous Dirichlet boundary data in a (virtual) square containing the obstacle, and emphasize the connection between the appearance of lift and the unique solvability of Navier-Stokes equations. In the 3D case we consider mixed boundary conditions: the inflow is given by a fairly general datum and the flow is assumed to satisfy a constant traction boundary condition on the outlet. In the absence of external forcing, explicit bounds on the inflow velocity guaranteeing existence and uniqueness of such steady motion are provided after estimating some Sobolev embedding constants and constructing a suitable solenoidal extension of the inlet velocity. In the 3D case, this solenoidal extension is built through the Bogovskii operator and explicit bounds on its Dirichlet norm (in terms of the geometric parameters of the obstacle) are found by solving a variational problem involving the infinity-Laplacian. The talk accounts for results obtained in collaboration with Filippo Gazzola and Ilaria Fragalà (both at Politecnico di Milano)
**Eugenio Vecchi**, Politecnico di Milano,

*Symmetry and rigidity for composite membranes and plates*, Tuesday, Febraury 16, 2021, time 15:15, Link: https://polimi-it.zoom.us/j/86140582546?pwd=U0hoTEhhalo1Q2NLeFAvTWd2TTdUUT09

**Abstract:****Abstract:**
The composite membrane problem is an eigenvalue optimization problem that can be formulated as follows:
Build a body of prescribed shape out of given materials (of varying densities) in such a way that the body has a prescribed mass and so that the basic frequency of the resulting membrane (with fixed boundary) is as small as possible.
In the first part of the talk we will review the known results and present a Faber-Krahn-type result obtained in collaboration with G. Cupini (Università di Bologna).
A natural extension of the above problem to the case of plates is the composite plate problem, which is an eigenvalue optimization problem involving the bilaplacian operator. The Euler-Lagrange equation associated to it is a fourth-order PDE that is coupled with Navier boundary conditions (for the hinged plate). In the second part of the talk we will focus on symmetry properties of optimal pairs. These results have been obtained in collaboration with F. Colasuonno (Università di Bologna).
**Stefano Biagi**, Politecnico di Milano,

*Some global results for homogeneous Hormander sums of squares*, Wednesday, Febraury 10, 2021, time 11:15, Link: https://polimi-it.zoom.us/j/81317327136?pwd=S1VuZ1NQVmxmckZuUWVzbytlZVQrUT09

**Abstract:****Abstract:**
In this talk we present several global results concerning the class of the homogeneous Hörmander sums of squares. As the name suggests, the operators falling in this class are sums of squares of smooth vector fields which are homogeneous of degree 1 with respect to a family of non-isotropic diagonal maps (usually called dilations); moreover, these operators intervene in several contexts of interest (Lie group Theory, sub-Riemannian manifolds, Mathematical Finance, etc.).
After a brief introduction on general sub-elliptic operators (of which any homogeneous sum of squares is a particular case), we properly introduce the class of the homogeneous Hörmander sums of squares and we discuss some global qualitative aspects regarding these operators: global lifting on Carnot groups; existence/global estimates for the associated fundamental solution and heat kernel; maximum principles on unbounded domains.
The results presented in this talk are contained in several papers in collaboration with A. Bonfiglioli, M. Bramanti and E. Lanconelli.
**Elsa Marchini**, Politecnico di Milano,

*Minimal time optimal control for the moon lander problem*, Tuesday, Febraury 02, 2021, time 11:15, https://polimi-it.zoom.us/j/83891252320?pwd=aS9Oa3V5aktwWGExa3lwZnFBYm9kUT09

**Abstract:****Abstract:**
We study a variant of the classical safe landing optimal control problem in aerospace, introduced by Miele in the Sixties, where the target was to land a spacecraft on the moon by minimizing the consumption of fuel. Assuming that the spacecraft has a failure and that the thrust (representing the control) can act in both vertical directions, the new target becomes to land safely by minimizing time, no matter of what the consumption is. In dependence of the initial data (height, velocity, and fuel), we prove that the optimal control can be of four different kinds, all being piecewise constant. Our analysis covers all possible situations, including the nonexistence of a safe landing strategy due to the lack of fuel or for heights/velocities for which also a total braking is insufficient to stop the spacecraft.
This talk is based on a joint work with Filippo Gazzola
**Élise Delphine Le Mélédo**, University of Zurich,

*A conservative correction procedure for Level Set methods in evolving interfaces*, Friday, January 29, 2021, time 17:30, online

**Abstract:****Abstract:**
Evolving interfaces is a key feature of many practical applications as e.g. pattern dynamics, multi-phase flows, material design, and their approximation raise tremendous interesting numerical issues.
In particular, incompressible two-phase flows require an accurate description of the interface in order to select the proper phase properties and dynamic to consider on each subdomain. Indeed, while the volume of each phase remains constant in time, the interface may develop heavy distortion and change topology across time. There, a small error in the interface location may have dreadful repercussions on the state dynamic (e.g. when vortices are encountered).
Therefore, the designed numerical scheme has to pay a particular attention to the representation of the interface(s) and the description of its motion.
Using an implicit representation of the interface, the Level Set method is a popular technique that allows a complex interface shape and is suited to topology changes in time. However, it is not natively conservative, that is the quantity of each fluid is not preserved as the interface is evolved.
In this talk, we propose a correction technique to the Level Set method that enforces mass conservation while preserving the continuity of each connected component of the interface. Flexible, it can be used in combination with any meshed scheme updating the non-corrected Level Set field.
**Davide Macera**, Università degli Studi Roma Tre,

*Full spectrum Anderson localization for a general model of a disordered quantum wire*, Friday, January 29, 2021, time 18:15, online

**Abstract:****Abstract:**
Mathematicians have long been interted in rigorously understanding the conductivity properties of disordered materials at the quantum level, in particular after the work of the Nobel Prize winning American physicist Philip W. Anderson (1923-2020)
In 1990, Klein, Lacroix and Speis analyzed a well studied random operator model for an electron moving on a portion of lattice of the form Z × [0, W], W ? N and subject to a random potential, called Anderson model on the strip. They showed, in particular, that such a model boasts spectral localization on all of its energy spectrum, a well defined mathematical property that is a very powerful signature of the electron getting trapped in a region by the potential.
In thie present work, we focus on a more general model of a quantum particle with internal degrees of freedom moving in a quasi 1D random medium (disordered quantum wire), that we call "generalized Wegner Orbital Model".
In particular, we prove spectral and dynamical localization at all energies for such a model suggesting that the disordered materials belonging to the wide class described by this model are all perfect insulators.
In this talk, I will start by introducing basic concepts related to Anderson Localization in general, then move to the specific model considered in this work, and outline our proof of its spectral localization. The proof combines techniques from probability theory, spectral theory of selfadjoint operator and ergodic theory, with an unexpected algebraic twist...