On the defect of compactness in Sobolev embeddings on Riemannian manifolds

Tintarev, C.

doi:10.1090/spmj/1606

On the defect of compactness in Sobolev embeddings on Riemannian manifolds
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by C. Tintarev

St. Petersburg Math. J. 31 (2020), 421-434

DOI: https://doi.org/10.1090/spmj/1606

Published electronically: April 30, 2020

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Abstract:

The defect of compactness for an embedding $E\hookrightarrow F$ of two Banach spaces is the difference between a weakly convergent sequence in $E$ and its weak limit, taken modulo terms vanishing in $F$. We discuss the structure of the defect of compactness for (noncompact) Sobolev embeddings on manifolds, giving a brief outline of the theory based on isometry groups, followed by a summary of recent studies of the structure of bounded sequences without invariance assumptions.

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