A.A. 2023/24

Abstract

After the breakthrough paper by Caffarelli and Silvestre in 2007, the study of fractional Laplacian and more general nonlocal operators has gained increasing popularity, from both an analytical and a probabilistic point of view. The purpose of the seminar is to present such a class of operators, starting from basic notions and then focusing on the PDEs’ theory developing from them. First, the formula for the square root of the Laplacian (-∆)^½ is provided, together with a few immediate remarks and the motivation behind its name. Then, after introducing the definition of fractional Laplacian (-∆)^s, we give an overview of its main properties, highlighting some similarities and differences with the classical Laplacian (-∆). The second part of the talk is entirely devoted to nonlocal PDEs, reserving particular attention to some regularity issues. We begin by dealing with the possible notions of solutions and next various results available in the literature are outlined. Finally, after introducing the class of Reifenberg flat sets, we present the problem of boundary Hölder regularity of some nonlocal PDEs on this kind of sets, sketching some possible solution strategies if time permits.

Abstract

We introduce the Calogero-Moser derivative nonlinear Schrödinger equation. We study its semiclassical limit by establishing the existence of a weak limit solution when we neglect the dispersion part of the equation. This weak limit solution will be identified explicitly. In a subsequent step, we explore the relationship between this weak limit and the branches of the multivalued solution of the Burgers equation. Finally, we derive a maximal principle that governs this weak limit solution

Abstract

The Dirac vacuum is a non-linear polarisable medium rather than an empty space. This non-linear behaviour starts to be significant for extremely large electromagnetic fields such as the magnetic field on the surface of certain neutron stars. Even though the null temperature case was deeply studied in the past decades, the problem at non-zero temperature needs to be better understood.

In this talk, we will present the first rigorous derivation of the one-loop effective magnetic Lagrangian at positive temperature, a non-linear functional describing the free energy of quantum vacuum in a classical magnetic field. After introducing our model, we will properly define the free energy functional using the Pauli-Villars regularisation technique in order to remove the worst ultraviolet divergences, which represent a well known issue of the theory. The study of the properties of this functional will be addressed before focusing on the limit of slowly varying classical magnetic fields. In this regime, one can prove the convergence of this functional to the Euler-Heisenberg formula with thermal corrections, recovering the effective Lagrangian first derived by Dittrich in 1979. The talk is based on the work available at arXiv:2404.12733