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Description of the lack of compactness for the Sobolev imbedding

P. Gérard — 2010

ESAIM: Control, Optimisation and Calculus of Variations

We prove that any bounded sequence in a Hilbert homogeneous Sobolev space has a subsequence which can be decomposed as an almost-orthogonal sum of a sequence going strongly to zero in the corresponding Lebesgue space, and of a superposition of terms obtained from fixed profiles by applying sequences of translations and dilations. This decomposition contains in particular the various versions of the concentration-compactness principle.

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