Archivio 2024 381 seminari

I will address some regularity properties of almost-minimizers of a perimeter functional with a weight that degenerates at the boundary of a domain. These objects arise from the heavy surfaces problem (surfaces that minimise the gravitational potential energy) and have connections with free boundary problems and classical minimal surfaces with rotational symmetries. This talk is based on joint work with Carlo Gasparetto and Bozhidar Velichkov.

The barycentric isoperimetric inequality has been proved by B. Fuglede in the 90’s for convex sets and more recently by C. Bianchini, G. Croce and A. Henrot in the planar case for connected sets and by myself and A. Pratelli for bounded sets in any dimension. We will discuss a conjecture by C. Bianchini, G. Croce and A. Henrot on the set that optimizes the constant in the planar case among connected sets, providing an equivalent formulation.

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Cross-diffusion systems are non-linear parabolic systems describing the evolution of densities or concentrations of multicomponent populations in interaction. Namely in population dynamics, cross-diffusion systems play a key role in modelling the spatial segregation of competing species. In this talk we analyse the role of cross-diffusion terms in pattern formation and in the derivation of the cross-diffusion model, obtained as the singular limit of a parabolic system with linear diffusion and fast reaction. [1] Brocchieri, E., Corrias, L., Dietert, H. and Kim, Y-J. Evolution of dietary diversity and a starvation driven cross-diffusion system as its singular limit, J. Math. Biol., 83 (2021). https://doi.org/10.1007/s00285-021-01679-y [2] N. Shigesada, K. Kawasaki, E. Teramoto. Spatial segregation of interacting species, J. Theor. Biol. 79.1 (1979): 83-99. https://www.sciencedirect.com/science/article/abs/pii/0022519379902583

By means of the Malliavin-Stein method, we will discuss central and non-central limit theorems for p-domain functionals of stationary Gaussian random fields. The talk is based on a joint work with N. Leonenko, L. Maini and I. Nourdin.

In this talk we present a new result about the compactness with respect to the H-convergence for a class of non-symmetric and nonlocal linear operators in fractional divergence form, where the oscillations of the matrices are prescribed outside the reference domain. The compactness argument presented today bypasses the failure of the classical localisation techniques, that mismatch with the nonlocal nature of the operators. In the second part of the presentation, we assume symmetry and show an equivalence between the H-convergence of the nonlocal operators and the Γ-convergence of the corresponding energies. At the end of the talk a list of some open problems and new research directions drawn from this work is presented. This research is carried out in collaboration with Maicol Caponi (University of L'Aquila) and Alessandro Carbotti (University of Salento).

We study spectral properties of generalized MIT bag models. These are Dirac operators Hτ (τ ∈R) acting on domains of R3 with confining boundary conditions. Their lowest positive eigenvalue is of special interest, and it is conjectured to be minimal for a ball among all domains with fixed volume. Studying the resolvent convergence of Hτ in the limits τ →±∞, some spectral properties of the limiting operators H±∞ are inherited throughout the parameterization.

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Geophysical fluid dynamics refers to the fluid dynamics of naturally occurring flows, such as oceans and planetary atmospheres on Earth and other planets. These flows are primarily characterized by two elements: stratification and rotation. In this article we investigate the effects of rotation on the dynamics, by neglecting stratification, in a 2D model. We consider the well-known 2D β-plane Navier-Stokes equations in the statistically forced case. Our problem addresses energy-related phenomena associated with the solution of the equations. To maintain the fluid in a turbulent state, we introduce energy into the system through a stochastic force. In the 2D case, a scaling analysis argument indicates a direct cascade of enstrophy and an inverse energy cascade. Following the evolution of the so-called third-order structure function, we compare the behavior of the direct/inverse cascade with the 2D model lacking the Coriolis force, observing that at small scales, the enstrophy flux from larger to smaller scales remains unaffected by the planetary rotation, in contrast to the large scales where the energy flux from smaller to larger scales is dominated by the Coriolis parameter, confirming experimental and numerical observations. In fact, to the best of our knowledge this is the first mathematically rigorous study of the above equations. This is a joint work with Amirali Hannani and Gigliola Staffilani.

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In this talk I consider a surface embedded in a 3D contact sub-Riemannian manifold. Such a surface inherits a field of direction (with norm) from the ambient space. This field of directions is singular at characteristic points (i.e., where the surface is tangent to the set of admissible directions). In this talk we will study when the normed field of directions permits to give to the surface the structure of metric space (of ``SNCF'' type). I will also study how to define the heat and the Schroedinger equation on such a structure and if the singular points are ``accessible'' or not. When the singular points are accessible we will study self-adjoint extensions with Kirchhoff like boundary conditions.

The talk presents a collection of results with Pascal Heid, Giacomo Sodini, and Giuseppe Savaré. We start the talk by presenting general results of strong density of sub-algebras of bounded Lipschitz functions in metric Sobolev spaces. We apply such results to show the density of smooth cylinder functions in Sobolev spaces of functions on the Wasserstein space $\mathcal P_2$ endowed with a finite positive Borel measure. As a byproduct, we obtain the infinitesimal Hilbertianity of Wassertein Sobolev spaces. By taking advantage of these results, we further address the challenging problem of the numerical approximation of Wassertein Sobolev functions defined on probability spaces. Our particular focus centers on the Wasserstein distance function, which serves as a relevant example. In contrast to the existing body of literature focused on approximating efficiently pointwise evaluations, we chart a new course to define functional approximants by adopting three machine learning-based approaches: 1. Solving a finite number of optimal transport problems and computing the corresponding Wasserstein potentials. 2. Employing empirical risk minimization with Tikhonov regularization in Wasserstein Sobolev spaces. 3. Addressing the problem through the saddle point formulation that characterizes the weak form of the Tikhonov functional's Euler-Lagrange equation. As a theoretical contribution, we furnish explicit and quantitative bounds on generalization errors for each of these solutions. In the proofs, we leverage the theory of metric Sobolev spaces introduced above and we combine it with techniques of optimal transport, variational calculus, and large deviation bounds. In our numerical implementation, we harness appropriately designed neural networks to serve as basis functions. Consequently, our constructive solutions significantly enhance at equal accuracy the evaluation speed, surpassing that of state-of-the-art methods by several orders of magnitude.

I will present a relative version of Gromov's Vanishing Theorem about amenable open covers with small multiplicity

By the Nielsen-Thurston classification theorem, there are three types of elements in the mapping class group of a surface: periodic, reducible, and pseudo-Anosov. We describe an efficient (i.e., polynomial-time) algorithm to distinguish pseudo-Anosov mapping classes

We will discuss a correlation theorem for pairs of locally Hölder continuous potentials with strong entropy gaps at infinity on a topologically mixing countable Markov shift with the BIP property. This extends a result of Lalley on shifts of finite type, and we will explain its application to the dynamics of (pairs of) Hitchin representations of a punctured surface. This talk is based on joint work in progress with Lien-Yung Nyima Kao.

We provide two examples of convex cocompact Kleinian groups whose limit set is a Pontryagin sphere. The examples are in dimension 4 and 6 respectively, and are obtained from reflection groups.

A discrete and faithful representation of a surface group in PSL(2,C) is said to be quasi-Fuchsian when it preserves a Jordan curve on the Riemann sphere. Classically these objects lie at the intersection of several areas of mathematics and have been studied (for example) using complex dynamics, Teichmüller theory, and 3-dimensional hyperbolic geometry. From a dynamical perspective, an important invariant of such representations is the Hausdorff dimension of the invariant Jordan curves (typically a very fractal object). It is elementary to see that this number is always at least 1. A celebrated result of Bowen establishes it is equal to 1 if and only if the quasi-Fuchsian representation is Fuchsian, that is, it is conjugate in PSL(2,R). I will first describe this classical picture and then report on recent joint work with James Farre and Beatrice Pozzetti where we prove a generalization of Bowen's result for the much larger class of hyperconvex representations of surface groups in PSL(d,C) (where d is arbitrary).

In this talk we will describe a generalisation of the Gordon-Luecke Theorem that says that knots in the 3-sphere are determined by their complements. We will show that the same holds when the ambient manifold is a circle bundle.

In this talk, we investigate qualitative properties of singular solutions to some elliptic problems. In the first part, we will focus our attention on semilinear and quasilinear elliptic problems under zero Dirichlet boundary conditions. In the second part, thanks to the previous analysis, we obtain some rigidity results for overdetermined boundary value problems for singular solutions in bounded domains.

In this talk, we consider n-dimensional Grushin spaces, where a Riemannian metric degenerates along a line in the space, resulting in a sub-Riemannian structure. We discuss the validity of (sub-)mean value property for (sub-)harmonic functions on hypersurfaces within Grushin spaces of dimension n>2. Our interest is driven by the classical counterpart: mean value formulas for harmonic functions on surfaces in the Euclidean setting are crucial for establishing the Bombieri-De Giorgi-Miranda gradient bound, which, in turn, plays a central role in the classical regularity theory. We conclude by presenting remarks and open questions about the regularity theory of minimal surfaces within this sub-Riemannian framework, which is yet to be established.

One of the oldest and most studied problems in the spectral theory of differential operators is the eigenvalue problem for the Dirichlet Laplacian. Classical questions about Laplacian eigenvalues concern their sensitivity analysis, optimization, asymptotic expansions, and many other more properties. On the other hand, similar questions remain open for an important class of degenerate operators, that is the Grushin Laplacians. In this talk I will present some recent results regarding the spectral theory of the Grushin Laplacian and in particular its shape sensitivity analysis, the optimization of the first eigenvalue, and a blow-up analysis.

We look at spreading phenomena under the action of singular potentials modeling repulsion between the liquidgas interface and the substrate. We mainly discuss the static case: depending on the form of the potential, the macroscopic profile of equilibrium configurations can be either droplet-like or pancake-like, with a transition profile between the two at zero spreading coefficient. These results generalize, complete, and give mathematical rigor to de Gennes’ formal discussion of spreading equilibria. Uniqueness and non-uniqueness phenomena are also discussed. Then we will briefly focus on the dynamics, assuming zero slippage at the contact line. Based on formal analysis arguments, we report that generic travelling-wave solutions exist and have finite rate of dissipation, indicating that singular potentials stand as an alternative solution to the contact-line paradox. In agreement with equilibrium configurations, travelling-wave solutions have microscopic contact angle equal to π/2 and, for mild singularities, finite energy. This is a joint work with Lorenzo Giacomelli.

The well-known isocapacitary inequality states that balls minimize the capacity among all sets of the same given volume. In the talk, we prove a sharp quantitative form of this classical result. Namely, we show that the difference between the capacity of a set and that of a ball with the same volume bounds the square of the Fraenkel asymmetry of the set. We then discuss some possible extensions.

The regularity of minimizers of the classical one-phase Bernoulli functional was deeply studied after the pioneering work of Alt and Caffarelli. More recently, the regularity of almost minimizers was investigated as well. We present a regularity result for almost minimizers for a one-phase Bernoulli-type functional in Carnot Groups of step two. Our approach is inspired by the methods introduced by De Silva and Savin in the Euclidean setting. Moreover, some recent intrinsic gradient estimates have been employed. Generalizations to the nonlinear framework will be discussed. Some of the results presented are obtained in collaboration with F. Ferrari (University of Bologna) and N. Forcillo (Michigan State University).