Organizers: Giovanni Catino and Fabio Cipriani

**Laura Abatangelo**, Dipartimento di Matematica, Politecnico di Milano,

*Perturbation theory for Dirichlet eigenvalues in perforated domains*, Wednesday, May 12, 2021, time 15:15 o'clock, https://polimi-it.zoom.us/j/89595030984?pwd=UG5wWmd1MjJkU1c1UnFRMmNaZUFMQT09

**Abstract:****Abstract:**
In this talk I will present some recent results on asymptotics of eigenvalues of the Dirichlet Laplacian when a small compact set is removed from the initial domain. If the small set is concentrating at a point in some sense, the eigenvalue variation is proved to be strictly related to the vanishing order of one of the relative eigenfunctions at that point. A good understanding of this asymptotics leads to new issues, for instance optimal location or optimal shape of the hole (open problem) as well as possible ramification of multiple eigenvalues.
**Clara Patriarca**, Dipartimento di Matematica, Politecnico Milano,

*Existence and uniqueness result for a fluid-structure-interaction evolution problem in an unbounded 2D channel*, Tuesday, April 27, 2021, time 11:15 o'clock, https://polimi-it.zoom.us/j/89665138535?pwd=MnJnRWJScU5oeStUUEJkV0MxYndvdz09

**Abstract:****Abstract:**
In an unbounded 2D channel, we consider the vertical displacement of a rectangular obstacle in a regime of small flux for the incoming flow field, modelling the interaction between the cross-section of the deck of a suspension bridge and the wind. We prove an existence and uniqueness result for a fluid-structure-interaction evolution problem set in this channel, where at infinity the velocity field of the fluid has a Poiseuille flow profile. We introduce a suitable definition of weak solutions and we make use of a penalty method. In order to prevent collisions of the obstacle with the boundary of the channel, we introduce a strong force in the differential equation governing the motion of the rigid body and we find a unique global-in-time solution.
**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).