LOCAL AND NONLOCAL OPERATORS

AND SUB-RIEMANNIAN GEOMETRIES

Modena, July 13th 2023

Program

Aula M1.7, Edificio Matematica
chair Sergio Polidoro

15:00
Francesca Anceschi (Università Politecnica delle Marche)

Boundedness estimates for a class of nonlocal kinetic Kolmogorov-Fokker-Planck equations

Abstract – I will present boundedness estimates for weak solutions to a wide class of kinetic Kolmogorov-Fokker-Planck equations with fractional diffusion term, from a joint work with M. Piccinini (Univ. Parma).

15:40
Carlo Mercuri (Università degli Studi di Modena e Reggio Emilia)

On some $p$-Laplacian problems involving critical nonlinearities

Abstract – I will discuss a class of quasilinear elliptic equations involving the $p$-Laplace operator and nonlinearities of Sobolev-critical growth, focusing on existence, non-existence, and compactness issues related to their variational formulation.

16:30
Mirco Piccinini (Università degli Studi di Parma)

Effects of the lack of compactness in the critical Sobolev embedding in the Heisenberg group

Abstract – In the sub-Riemannian setting of the Heisenberg group we present some results on the effects of the lack of compactness in the critical Sobolev embedding. In particular, via variational techniques, we show an energy subcritical approximation in bounded set (without any further regularity assumptions) and that the enegry of the related extremal functions do concentrate at one point. Under natural regularity and geometrical hypothesis on the domain, in clear accordance with the underlying geometrical framework, we show that the concentration point can be characterized via the Green function, proving that a famous conjecture of Brezis & Peletier (Essays in honor of Ennio De Giorgi 1989) still holds in the Heisenberg group. The proof of such localization result relies on several new and independent results, such as an asymptotic control of the subcritical extremal functions via the Jerison & Lee optimal function; Caccioppoli and Schauder estimates for CR Yamabe-type equations; the “Global Compactness” in the Heisenberg group established via a completely new proof than the original one by Struwe.