2013/09/13 by Cyril Tintarev, Tintarev, Cyril
Mathematics · #Advanced Banach Space Theory #Advanced Operator Algebra Research #Advanced Topics in Algebra #Analysis of PDEs (math.AP) #FOS: Mathematics #Functional Analysis (math.FA) #math.AP #math.FA
paper · pdf · doi:10.48550/arxiv.1309.3431
arxiv created 2013/09/13 · openalex publication_date 2013/09/13 · arxiv updated 2013/09/16 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 gkw, gk∈ D, w∈ 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.