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St. Petersburg Mathematical Journal
St. Petersburg Mathematical Journal
ISSN 1547-7371(online) ISSN 1061-0022(print)

 

On the existence of extremal functions in Sobolev embedding theorems with critical exponents


Authors: A. V. Demyanov and A. I. Nazarov
Translated by: A. I. Nazarov
Original publication: Algebra i Analiz, tom 17 (2005), nomer 5.
Journal: St. Petersburg Math. J. 17 (2006), 773-796
MSC (2000): Primary 49J10, 35J20, 35J60
Published electronically: July 20, 2006
MathSciNet review: 2241425
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Abstract | References | Similar Articles | Additional Information

Abstract: Sufficient conditions for the existence of extremal functions in Sobolev-type inequalities on manifolds with or without boundary are established. Some of these conditions are shown to be sharp. Similar results for embeddings in some weighted $ L_q$-spaces are obtained.


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Additional Information

A. V. Demyanov
Affiliation: St. Petersburg State University, Russia
Email: alex@ad9503.spb.edu

A. I. Nazarov
Affiliation: St. Petersburg State University, Russia
Email: an@AN4751.spb.edu

DOI: http://dx.doi.org/10.1090/S1061-0022-06-00929-0
PII: S 1061-0022(06)00929-0
Keywords: Minimizers, critical exponent, Sobolev inequality, Sobolev--Poincar\'e inequality, Hardy--Sobolev inequality, $p$-Laplacian
Received by editor(s): November 30, 2004
Published electronically: July 20, 2006
Additional Notes: Partially supported by the RF Ministry of Education (project no. 4733), and by RFBR (grant no. 05–01–01063).
Dedicated: In memory of Ol$′$ga Aleksandrovna Ladyzhenskaya
Article copyright: © Copyright 2006 American Mathematical Society