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Cocompact Imbeddings and Structure of Weakly Convergent Sequences

  • Kiril Tintarev
Part of the International Mathematical Series book series (IMAT, volume 8)

Abstract

The concentration compactness method is a powerful technique for establishing the existence of minimizers for inequalities and critical points of functionals. We give a functional-analytic formulation for the method in a Banach space. The key object is a dislocation space, i.e., a triple (X, F, D), where F is a convex functional defining a norm on a Banach space X and D is a group of isometries on X. Bounded sequences in dislocation spaces admit a decomposition into an asymptotic sum of “profiles” w(n)X dislocated by the actions of D. This decomposition allows to extend the weak convergence argument from variational problems with compactness to problems, where X is cocompactly (relatively to D) imbedded into a Banach space Y . We prove a general statement on the existence of minimizers in cocompact imbeddings that applies, in particular, to Sobolev imbeddings which lack compactness (an unbounded domain, a critical exponent) including the subelliptic Sobolev spaces and spaces over Riemannian manifolds.

Keywords

Banach Space Riemannian Manifold Unbounded Domain Convergent Sequence Carnot Group 
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Copyright information

© Springer Science+Business Media, LLC 2009

Authors and Affiliations

  • Kiril Tintarev
    • 1
  1. 1.Uppsala UniversityUppsalaSweden

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