Organizers: Giovanni Catino and Fabio Cipriani

**Francesco Esposito**, Universita' della Calabria,

*Some adaptations of the moving planes method to singular elliptic problems*, Wednesday, June 09, 2021, time 15:15 o'clock, https://polimi-it.zoom.us/j/89350397352?pwd=NGd4THV6UFR6VEU1VTVtaWxmOFZiQT09

**Abstract:****Abstract:**
In this talk we will consider positive singular solutions to some quasilinear elliptic problems under zero Dirichlet boundary conditions. We will study qualitative properties of the solutions via the moving plane procedure, that goes back to the celebrated papers of Alexandrov and Serrin. In particular, exploiting a fine adaptation of the moving plane procedure and a careful choice of the cutoff functions, we deduce symmetry and monotonicity properties of the solutions in bounded smooth domains.
**Marco Caroccia**, Dipartimento di Matematica, Politecnico di Milano,

*Contact surface of Cheeger sets*, Wednesday, May 26, 2021, time 15:15 o'clock, https://polimi-it.zoom.us/j/89745265407

**Abstract:****Abstract:**
Geometrical properties of Cheeger sets have been deeply studied by many authors since their introduction, as a way of bounding from below the first Dirichlet (p)-Laplacian eigenvualue. They represent, in some sense, the first eigenfunction of the Dirichlet (1)-Laplacian of a domain. In this talk we will introduce a recent property, studied in collaboration with Simone Ciani, concerning their contact surface with the ambient space. In particular we will show that the contact surface cannot be too small, with a lower bound on the (Hasudorff) dimension strictly related to the regularity of the ambient space. The talk will focus on the introduction of the problem and on the proof of the dimensional bounds. Functional to the whole argument is the notion of removable singularity, as a tool for extending solutions of pdes under some regularity constraint. Finally examples providing the sharpness of the bounds in the planar case are briefly treated.
**Giulia Meglioli**, Dipartimento di Matematica Politecnico di Milano,

*Smoothing effects and infinite time blowup for reaction-diffusion equations: an approach via Sobolev and PoincarĂ© inequalities*, Tuesday, May 18, 2021, time 11:15 o'clock, https://polimi-it.zoom.us/j/87010654658?pwd=dURNSzY4WE1VL0M5YWxwMzFLeTJSdz09

**Abstract:****Abstract:**
The talk is concerned with the reaction-diffusion equation $u_t=\Delta(u^m)+u^p$, on a complete noncompact Riemannian manifold $M$. We consider the particularly delicate case when $p$ is less than $m$; moreover, we assume that the PoincarĂ© and the Sobolev inequalities hold on $M$. We prove global existence in time of solutions for $L^m$ initial data. Furthermore, solutions are bounded for all positive times and their $L^\infty$ norm satisfy a certain quantitative bound. We also see that on a special class of Riemannian manifolds, solutions corresponding to sufficiently large $L^m$ data give rise to solutions that blow up in infinite time, a fact that cannot happen in the Euclidean setting.
The results have been recently obtained jointly with Gabriele Grillo and Fabio Punzo.
**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.