Calculus of Variations and Geometric Measure Theory

G. Palatucci - M. Piccinini - L. Temperini

Struwe's Global Compactness and energy approximation of the critical Sobolev embedding in the Heisenberg group

created by palatucci on 03 Aug 2023
modified by piccinini on 24 Apr 2024


Submitted Paper

Inserted: 3 aug 2023
Last Updated: 24 apr 2024

Year: 2023

ArXiv: 2308.01153 PDF


We investigate some of the effects of the lack of compactness in the critical Folland-Stein-Sobolev embedding in very general (possible non-smooth) domains, by proving via De Giorgi's $\Gamma$-convergence techniques that optimal functions for a natural subcritical approximations of the Sobolev quotient concentrate energy at one point. In the second part of the paper, we try to restore the compactness by extending the celebrated Global Compactness result to the Heisenberg group via a completely different approach with respect to the original one by Struwe \cite{Str84}.

Keywords: Heisenberg group, profile decomposition, global compactness, Sobolev embeddings, CR Yamabe