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

TL;DR

This paper studies the lack of compactness in the critical Sobolev embedding on the Heisenberg group, showing energy concentration phenomena and extending Struwe's global compactness result using new methods.

Contribution

It introduces a novel approach to extend Struwe's global compactness theorem to the Heisenberg group and analyzes energy concentration in Sobolev embeddings.

Findings

Energy concentrates at one point in subcritical approximations.

Extended global compactness result to the Heisenberg group.

Applied De Giorgi's $ ext{Gamma}$-convergence techniques.

Abstract

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 [37].