Convegno
“ASK CONFERENCE 2024: INTERACTION BETWEEN DIFFERENT AREAS OF MATHEMATICAL ANALYSIS AND PROBABILITY”

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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