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.

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.