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Concentration Analysis and Cocompactness

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Concentration Analysis and Applications to PDE

Part of the book series: Trends in Mathematics ((TM))

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.

Mathematics Subject Classification (2010). Primary 35J20, 35J60, 35B44, 35H20; Secondary 46B50, 58J05, 58J70, 74H35.

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Authors and Affiliations

  1. Department of Mathematics, Uppsala University, 480, S-751 06, Uppsala, Sweden

    Cyril Tintarev

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  1. Cyril Tintarev
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Correspondence to Cyril Tintarev .

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Editors and Affiliations

  1. Centre for Applicable Mathematics Post Bag No. 6503, GKVK Post Office, Tata Institute of Fundamental Research, Bangalore, Karnataka, India

    Adimurthi

  2. Centre for Applicable Mathematics Post Bag No. 6503, GKVK Post Office, Tata Institute of Fundamental Research, Bangalore, Karnataka, India

    K. Sandeep

  3. CeReMath, Université Toulouse I, Toulouse, France

    Ian Schindler

  4. Department of Mathematics, Uppsala University, Uppsala, Sweden

    Cyril Tintarev

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Tintarev, C. (2013). Concentration Analysis and Cocompactness. In: Adimurthi, ., Sandeep, K., Schindler, I., Tintarev, C. (eds) Concentration Analysis and Applications to PDE. Trends in Mathematics. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-0373-1_7

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