Organizers: Stefano Biagi, Filippo Dell’Oro, Filippo Giuliani.

**Eduardo Munoz Hernandez**, Universidad Complutense de Madrid,

*Existence and multiplicity of coexistence states in a heterogeneous predator-prey model with saturation*, Tuesday, May 14, 2024, time 11:30, Aula Seminari - III Piano

**Abstract:****Abstract:**
In this talk we analyze a predator-prey model coming from the heterogeneous counterparts of the classical Lotka-Volterra
and Holling-Tanner models by considering a non-negative saturation term that can vanish inside the domain. By the use of the maximum principle and bifurcation theory the regions of coexistence can be exactly determined. Furthermore, by varying the amplitude of the saturation different multiplicity results will be obtained. This is a joint work with Julián López-Gómez.
**Alessio Falocchi**, Politecnico di Milano,

*On the long-time behaviour of solutions to unforced evolution Navier-Stokes equations under Navier boundary conditions*, Thursday, May 09, 2024, time 16:30, Aula Seminari - III Piano

**Abstract:****Abstract:**
We study the asymptotic behaviour of the solutions to Navier-Stokes unforced equations under Navier boundary conditions in a wide class of merely Lipschitz domains of physical interest that we call sectors. The main motivations come from the celebrated results by Foias-Saut related to the long time behaviour of the solutions to Navier-Stokes equations under Dirichlet conditions.
Here the choice of the boundary conditions requires carefully considering the geometry of the domain, due to the possible lack of the Poincaré inequality in presence of axial symmetries. In non-axially symmetric domains we show the validity of the Foias-Saut result about the limit at infinity of the Dirichlet quotient, in axially symmetric domains we provide two invariants of the flow which completely characterize the motion and we prove that the Foias-Saut result holds for initial data belonging to one of the invariants.
This is a joint work with Prof. Elvise Berchio (Politecnico di Torino, Italy) and Clara Patriarca (Université libre de Bruxelles, Belgium).
**Marco Caroccia**, Politecnico di Milano,

*On the singular planar plateau problem*, Tuesday, May 07, 2024, time 15:00, Aula Seminari - III Piano

**Abstract:****Abstract:**
The classical Plateau problem asks which surface in three-dimensional space spans the least area among all the surfaces with boundary given by an assigned curve S. This problem has many variants and generalizations, along with (partial) answers, and has inspired numerous new ideas and techniques. In this talk, we will briefly introduce the problem in both its classical and modern contexts, and then we will focus on a specific vectorial (planar) type of the Plateau problem. Given a curve S in the plane, we can ask which diffeomorphism T of the disk D maps the boundary of D to S and spans the least area, computed as the integral of the Jacobian of T, among competitors with the same boundary condition. For simply connected curves, the answer is provided by the Riemann map, and the minimal area achieved is the Lebesgue measure of the region enclosed by S. For more complex curves, possibly self-intersecting, new analysis is required. I will present a recent result in this sense, obtained in collaboration with Prof. Riccardo Scala from the University of Siena, where the value of the minimum area is computed with an explicit formula that depends on the topology of S.
**Elena Danesi**, Università di Padova,

*Strichartz estimates for the Dirac equation on compact manifolds without boundary*, Thursday, April 18, 2024, time 15:00, Aula seminari - III piano

**Abstract:****Abstract:**
The Dirac equation on Rn can be listed within the class of dispersive equations, together with, e.g., the wave and Klein-Gordon equations. In the years a lot of tools have been developed in order to quantify the dispersion of a system. Among these one finds the Strichartz estimates, that are a priori estimates of the solutions in mixed Lebesgue spaces. For the flat case Rn they are known, as they are derived from the ones that hold for the wave and Klein-Gordon equations. However, when passing to a curved spacetime domain, very few results are present in the literature. In this talk I will firstly introduce the Dirac equation on curved domains. Then, I will discuss the validity of this kind of estimates in the case of Dirac equations on compact Riemannian manifolds without boundary. This is based on a joint work with Federico Cacciafesta (Università di Padova) and Long Meng (CERMICS-École des ponts ParisTech).
**Giovanni Cupini**, Università di Bologna,

*The Leray-Lions existence theorem under (p,q)-growth conditions*, Thursday, April 11, 2024, time 15:00, Aula Seminari - III Piano

**Abstract:****Abstract:**
In this talk I will describe recent results obtained in collaboration with P. Marcellini and E. Mascolo. In particular, we proved an existence result of weak solutions to a Dirichlet problem associated to second order elliptic equations in divergence form satisfying (p,q)-growth conditions. This is a first attempt to extend to (p,q)-growth the well known Leray-Lions existence theorem, which holds under the so-called natural growth conditions.
Our existence result is obtained "via regularity", i.e., by using new local regularity results (boundedness, Lipschitz continuity and higher differentiability) for the weak solutions of the associated equation.
**Elodie Pozzi**, St. Louis University, USA,

*Some Examples of Persistence of Superoscillations*, Wednesday, April 10, 2024, time 15:00, Aula Seminari - III Piano

**Abstract:****Abstract:**
When combining N bandlimited functions with low frequencies and allowing N to approach infinity, one would typically anticipate obtaining a function with low frequency. However, for certain bandlimited functions with low frequency, the resulting function exhibits high frequencies. Such functions are termed superoscillating functions, and the phenomenon described is known as super oscillation. In this presentation, we will delve into the behavior of superoscillating functions under the time-dependent Schrödinger equation and its variations. Specifically, we will explore existing methods for demonstrating the persistence of superoscillations, applying them to illustrate the persistence of superoscillations under certain partial differential equations. Our focus will be on the Schrödinger equation governing spin particles and the Schrödinger equation pertaining to the Aharonov-Bohm effect. It is based on joint works with Fabrizio Colombo, Irene Sabadini and Brett D. Wick.
This initiative is part of the "PhD Lectures" activity of the project "Departments of Excellence 2023-2027" of the Department of Mathematics of Politecnico di Milano. This activity consists of seminars open to PhD students, followed by meetings with the speaker to discuss and go into detail on the topics presented at the talk.