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 ↪ 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 gk w , gk ∈ D, w ∈ X, and a remainder that vanishes in Y . A necessary requirement for X is, therefore, that any sequence in X that develops no D-concentrations has a subsequence convergent in the norm of Y . An imbedding X ↪ 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.
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