Concentration analysis and cocompactness

TL;DR

This paper discusses the concept of cocompact embeddings in Banach spaces, their role in profile decompositions, and how they help analyze loss of compactness in PDE-related sequences.

Contribution

It surveys known cocompact embeddings and elucidates their significance in the framework of profile decompositions for PDE analysis.

Findings

Cocompact embeddings generalize compactness in Banach spaces.

Profile decompositions represent sequences as sums of elementary concentrations.

Cocompactness aids in understanding loss of compactness in PDE sequences.

Abstract

Loss of compactness that occurs in may significant PDE settings can be expressed in a well-structured form of profile decomposition for sequences. Profile decompositions are formulated in relation to a triplet $(X,Y,D)$, where $X$ and $Y$ are Banach spaces, $X\hookrightarrow Y$, and $D$ is, typically, a set of surjective isometries on both $X$ and $Y$. A profile decomposition is a representation of a bounded sequence in $X$ as a sum of elementary concentrations of the form $g_kw$, $g_k\in D$, $w\in X$, and a remainder that vanishes in $Y$. A necessary requirement for $Y$ is, therefore, that any sequence in $X$ that develops no $D$-concentrations has a subsequence convergent in the norm of $Y$. An imbedding $X\hookrightarrow Y$ with this property is called $D$-cocompact, a property weaker than, but related to, compactness. We survey known cocompact imbeddings and their role in profile decompositions.