Preprint

Inserted: 30 jul 2023
Last Updated: 5 apr 2024

Year: 2023

Abstract:

We investigate some effects of the lack of compactness in the critical Sobolev embedding by proving that a famous conjecture of Brezis & Peletier (Essays in honor of Ennio De Giorgi – Progr. Differ. Equ. Appl. 1989) does still hold in the Heisenberg framework: optimal functions for a natural subcritical approximations of the Sobolev quotient concentrate energy at one point which can be localized via the Green function associated to the involved domain and in clear accordance with the underlying sub-Riemannian geometry – and consequently a new suitable definition of domains geometrical regular near their characteristic set is given. In order to achieve the aforementioned result, we need to combine proper estimates and tools to attack the related CR Yamabe equation (Jerison & Lee, J. Diff. Geom. 1987) with novel feasible ingredients in PDEs and Calculus of Variations which also aim to constitute general independent results in the Heisenberg framework, as e. g. a fine asymptotic control of the optimal functions via the Jerison & Lee extremals realizing the equality in the critical Sobolev inequality (J. Amer. Math. Soc. 1988).

Keywords: Heisenberg group, Sobolev embeddings, CR Yamabe

Download:

• Palatucci-Piccinini_ArXiv_2023.pdf