Convergence of functional sequences, in many cases, cannot be directly obtained from compactness properties, but requires a structural analysis of the defect of compactness. While concentration compactness techniques have been widely adopted since the 1980’s, more powerful methods have been developed in the last 15 years and applied by mathematicians working in different disciplines of analysis. The concentration argument plays a central technical role in elliptic PDE theory, geometric analysis, as well as in the analysis of NLS, wave, and Navier-Stokes equations. The original description of concentration in terms of singular weak limits for sequences of measures has been supplemented by more detailed profile decompositions. Formalization of the latter on the functional analytic level in terms of wavelet bases and cocompact imbeddings relative to a given group has been used to investigate increasingly diverse non-compact invariances involved in the loss and recovery of compactness. Con...
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