Defect of compactness in spaces of bounded variation

TL;DR

This paper extends the profile decomposition for Sobolev spaces to the non-reflexive space of functions of bounded variation, providing new insights into the defect of compactness in these spaces and their applications.

Contribution

It generalizes the profile decomposition to BV spaces, including non-reflexive cases and Lie groups, and establishes existence of minimizers for related inequalities.

Findings

Extended profile decomposition to BV spaces.

Proved existence of minimizers for inequalities in BV.

Generalized results to BV spaces on Lie groups.

Abstract

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.