Hynek Kovarik, Università degli Studi di Brescia, Resolvent expansion and time decay of the wave functions of magnetic Hamiltonians in dimension two, Thursday, May 25, 2023, time 14:30 o'clock, Aula Seminari III piano
Abstract:Abstract:
I this talk I will present some results on the resolvent expansions of magnetic Hamiltonians at the threshold of the essential spectrum. I will show, in particular, that the nature of the expansion of a two-dimensional magnetic Hamiltonian is completely determined by the flux of the associated magnetic field. Applications to time decay of the wave functions will be discussed as well.

Edoardo Bocchi, Politecnico di Milano, Asymmetric equilibrium configurations of a body immersed in a 2D laminar flow, Thursday, May 18, 2023, time 15:15, Aula Seminari III piano
Abstract:Abstract:
We will study the equilibrium configurations of a fluid-structure interaction problem where a body is immersed in a fluid confined in a bounded planar channel and governed by the stationary Navier-Stokes equations with laminar inflow and outflow. The body is subject to both the lift force from the fluid and to some external elastic force. Motivated by an application to suspension bridges, asymmetry is taken into account. This requires the introduction of suitable assumptions to prevent collisions of the body with the boundary. We will present an existence and uniqueness result for sufficiently small inflow/outflow. This talk is based on a recent joint work with F. Gazzola.

Paolo Luzzini, Università degli Studi di Padova, Shape sensitivity and optimization of Grushin eigenvalues, Thursday, May 11, 2023, time 15:15, Aula Seminari III piano
Abstract:Abstract:
It is well known that among all domains of a fixed volume, the ball minimizes the first eigenvalue of the Dirichlet Laplacian. The counterpart of this result for the degenerate operator known as the Grushin Laplacian is instead an open problem. In this talk I will present some results in the direction of understanding such a problem. I will first consider the shape sensitivity of Grushin eigenvalues on general domains with the aim of characterizing critical domains under isovolumetric perturbations. Next I will pass to the simplified case of cartesian product domains, showing that in this class the first eigenvalue admits a unique minimizer and providing some estimates on the minimum. Finally I will discuss some numerical experiments and some open problems. The talk is based on joint works with Pier Domenico Lamberti (Università degli Studi di Padova), Paolo Musolino (Università Ca' Foscari Venezia) Luigi Provenzano (Sapienza Università di Roma), and Joachim Stubbe (EPFL).

Fabio Cavalletti, SISSA, Optimal transport between algebraic hypersurfaces, Wednesday, May 03, 2023, time 15:00, Aula Seminari III piano
Abstract:Abstract:
What is the optimal way to deform a projective hypersurface into another one? We will answer this question adopting the point of view of measure theory, introducing the optimal transport problem between complex algebraic projective hypersurfaces. First, a natural topological embedding of the space of hypersurfaces of a given degree into the space of measures on the projective space is constructed. Then, the optimal transport problem between hypersurfaces is defined through a constrained dynamical formulation, minimizing the energy of absolutely continuous curves which lie on the image of this embedding. In this way an inner Wasserstein distance on the projective space of homogeneous polynomials is introduced. We will show the main properties of this distance and discuss applications on the regularity of the zeroes of a family of multivariate polynomials and on the condition number of polynomial systems solving.

Giovanni Siclari, Università degli Studi di Milano-Bicocca, Unique continuation for the fractional heat operator, Thursday, April 13, 2023, time 15:15, Aula Seminari III piano
Abstract:Abstract:
We study unique continuation properties and the asymptotic behaviour for a class of equations involving the fractional heat operator with an Hardy-type potential. Our methods are based on a Almgren-Poon monotonicity formula combined with a blow-up argument. Since the operator has a global nature we will also need suitable extension results in the spirit of Caffarelli-Silvestre extension. Key words: Parabolic partial differential equations, unique continuation, blow-up, asymptotics, monotonicity formula, Hardy potential.

Francesco Esposito, Università della Calabria, A classification result for a Gross-Pitaevskii type system, Thursday, April 13, 2023, time 16:15, Aula seminari III piano
Abstract:Abstract:
This talk will be focused on the study of a family of semilinear elliptic systems defined in $ R^n $, which is doubly critical since it involves Sobolev critical exponents and Hardy-type potentials. We aim to provide qualitative properties of positive solutions for these Gross-Pitaevskii type systems. In particular, we shall deduce that solutions are symmetric about the origin. In order to do it, we apply a suitable version of the moving planes technique for cooperative singular systems. Finally, we are able to provide a classification result for these kind of problems. This is based on a joint work with Rafael López-Soriano (University of Granada, Spain) and Berardino Sciunzi (University of Calabria, Italy).