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St. Petersburg Mathematical Journal

ISSN 1547-7371(online) ISSN 1061-0022(print)



On the defect of compactness in Sobolev embeddings on Riemannian manifolds

Author: C. Tintarev
Original publication: Algebra i Analiz, tom 31 (2019), nomer 3.
Journal: St. Petersburg Math. J. 31 (2020), 421-434
MSC (2010): Primary 46E35, 46B50, 46N20, 54D30, 43A99, 58E99
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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Additional Information

C. Tintarev
Affiliation: Sankt Olofsgatan 66B, 75330 Uppsala, Sweden

Keywords: Concentration compactness, profile decomposition, weak convergence, Sobolev spaces on manifolds
Received by editor(s): August 30, 2018
Published electronically: April 30, 2020
Dedicated: Dedicated to the memory of S. G. Mikhlin
Article copyright: © Copyright 2020 American Mathematical Society