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.

Kenneth DeMason, The University of Texas at Austin, A Strong Form of the Quantitative Wulff Inequality for Crystalline Norms, Tuesday, March 19, 2024, time 15:00, Aula seminari - III piano
Abstract:Abstract:
The anisotropic perimeter is a natural functional appearing in the mathematical framework for determining equilibrium states of crystals in media. As with the usual isotropic perimeter there is an analogous anisotropic isoperimetric inequality, known as the Wulff inequality, where minimizers of the volume constrained anisotropic perimeter problem, known as Wulff shapes, are characterized. In view of statistical mechanics, almost-minimizers are the most likely observable states; as such their identification is just as important as the absolute minimizers. In this talk we will explore a recent result by the speaker which proves quantitative control on almost-minimizers in an H^1 sense when the Wulff shape is a polytope, an upgrade from the previous L^2 control via the so-called Fraenkel asymmetry.

Ermanno Lanconelli, Università di Bologna, Sulla caratterizzazione armonica delle sfere: una disuguaglianza di stabilita' per domini C^1-pericentrici, Thursday, March 14, 2024, time 15:00, Aula seminari MOX - VI piano
Abstract:Abstract:
Nel 2002 Lewis e Vogel dimostrarono che le pseudo sfere armoniche, cioè le frontiere dei domini limitati sui quali vale, per le funzioni armoniche, la formula di media di superficie rispetto ad un loro punto interno, sono sfere euclidee se i domini sono Dirichlet-regolari e in più il loro bordo ha misura (n- 1)-dimensionale con crescita al più euclidea. Il risultato di Lewis e Vogel, nelle stesse ipotesi, può essere riformulato nel modo seguente: se il nucleo di Poisson di un dominio, con polo in un suo punto interno x_0, è costante sul bordo, allora il dominio è una sfera euclidea di centro x_0. Nel 2007 Preiss e Toro, assumendo le stesse ipotesi, dimostrarono che il risultato di Lewis e Vogel è stabile, nel senso seguente: se il nucleo di Poisson di un dominio, con polo in un suo punto interno, è quasi costante sul bordo, allora il bordo del dominio è geometricamente vicino ad una sfera centrata in quel punto. Con Giovanni Cupini abbiamo dimostrato che il risultato di rigidità di Lewis e Vogel, e una disuguaglianza di stabilità alla Preiss e Toro, valgono assumendo ''soltanto'' una regolarità C^1 del bordo vicino ad almeno un suo punto pericentrale, cioè un punto del bordo avente distanza minima dal fissato centro della pseudosfera, senza nulla richiedere sul resto della frontiera, neppure l'esistenza del nucleo di Poisson. Le nostre tecniche sono dirette, e non usano gli elevati metodi di Analisi armonica e di frontiera libera utilizzate da Lewis e Vogel e da Preiss e Toro.

Cristopher Hermosilla, Universidad Técnica Federico Santa María, Valparaíso - CHILE, A Minimality Property of the Value Function in Optimal Control over the Wasserstein Space, Tuesday, March 05, 2024, time 15:00, Aula seminari - III piano
Abstract:Abstract:
In this talk we study an optimal control problem with (possibly) unbounded terminal cost in the space of Borel probability measures with finite second moment. We consider the metric geometry associated with the Wasserstein distance, and a suitable weak topology rendering this space locally compact. In this setting, we show that the value function of a control problem is the minimal viscosity supersolution of an appropriate Hamilton-Jacobi-Bellman equation. Additionally, if the terminal cost is bounded and continuous, we show that the value function is the unique viscosity solution of the HJB equation.

Filippo Giuliani, Politecnico di Milano, Arbitrarily large growth of Sobolev norms for a quantum Euler system, Thursday, Febraury 15, 2024, time 15:00, Aula seminari MOX VI piano
Abstract:Abstract:
In this talk we present a result of existence of solutions to the quantum hydrodynamic (QHD) system, under periodic boundary conditions, which undergo an arbitrarily large growth of higher order Sobolev norms in polynomial times. The proof is based on the connection between the QHD system and the cubic NLS equation, provided by the Madelung transform. We show that the cubic NLS equation on the two dimensional torus possesses solutions which starts close to plane waves and undergo an arbitrarily large growth of higher order Sobolev norms in polynomial times. This is an improvement of the result by Guardia-Hani-Haus-Maserp-Procesi (JEMS 2023) and it is achieved by a refined normal form approach. Then we show that the existence of such solutions to NLS implies the existence of solutions to the QHD system exhibiting a large growth in Sobolev norms.

Eduard Feireisl, Institute of Mathematics of the Czech Academy of Sciences, Praha, Statistical stationary solutions to the compressible Rayleigh-Benard convection problem, Monday, Febraury 05, 2024, time 11:30, Aula seminari MOX - VI piano
Abstract:Abstract:
Stationary statistical solutions (invariant measures) represent a standard tool for describing turbulent phenomena in fluid mechanics. We consider the complete Navier-Stokes-Fourier system describing the motion of a compressible, viscous and heat conducting fluid driven by boundary temperature fluctuations (Rayleigh-Benard problem). We show that any omega-limit set associated to a global in time (weak) solution supports an invariant measure - stationary statistical solution. The proof is based on careful analysis of propagation of density oscillations (joint work with A.Swierczewska-Gwiazda (Warsaw)).