  Defect of compactness for non-compact imbeddings of Banach spaces can be
expressed in the form of a profile decomposition. This paper extends the
profile decomposition for Sobolev spaces proved by Solimini (AIHP 1995) to the
non-reflexive case p=1. Since existence of concentration profiles relies on
weak-star compactness, the corresponding result is set in a larger, conjugate,
space of functions of bounded variation. We prove existence of minimizers for
related inequalities and generalizations for to spaces of bounded variation on
Lie groups.
