Organizers: Giovanni Catino and Fabio Cipriani
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.
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.