Organizers: Stefano Biagi, Filippo Dell’Oro, Filippo Giuliani.
Matteo Fogato, Dipartimento di Matematica, Politecnico di Milano,
Modal analysis of some nonlinear beam equations, Wednesday, April 14, 2021, time 15:15 o'clock, https://polimi-it.zoom.us/j/82401456185?pwd=MS90c0hET3dXV0xxcmN5RTlqUGhQQT09
Abstract:Abstract:
We consider the equation $u_{tt}+\delta u_t +\|A^{\theta/2}u\|^2 A^\theta u=g$ where $A^2$ is a diagonal, self-adjoint and positive-definite operator, $\theta\in [0,1]$ and we study some finite-dimensional approximations of the problem. First, we analyze the dynamics in the case when the forcing term $g$ is a combination of a finite number of modes. Next, we estimate the error we commit by neglecting the modes larger than a given $N$. We then prove, for a particular class of forcing terms, a theoretical result allowing to study the distribution of the energy among the modes and, with this background, we refine the results.
Some generalizations and applications to the study of the stability of suspension bridges are given.
Alessio Falocchi, Dipartimento di Scienze Matematiche, Politecnico di Torino,
Optimization of eigenvalues of partially hinged rectangular composite plates, Tuesday, March 30, 2021, time 11:15 o'clock, https://polimi-it.zoom.us/j/85332796977?pwd=QlhtRDZSWXkxaDhUTjFQQlEyL0NOUT09
Abstract:Abstract:
We study the spectrum of non-homogeneous partially hinged rectangular plates having structural engineering applications. A possible way to prevent instability phenomena is to optimize the frequencies of certain oscillating modes with respect to the density function of the plate. In striking contrast to what happens under Dirichlet boundary conditions, we prove a result of positivity preserving property for the biharmonic operator of the related problem through fine estimates of the Fourier expansion of the corresponding Green function. This is useful in order to get qualitative properties, e.g. symmetry and monotonicity, of the eigenfunction corresponding to the density minimizing the first eigenvalue.
This is a joint work with Elvise Berchio (Politecnico di Torino).
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
