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Winter 2012 On the Caffarelli–Kohn–Nirenberg-type inequalities involving critical and supercritical weights
Toshio Horiuchi, Peter Kumlin
Kyoto J. Math. 52(4): 661-742 (Winter 2012). DOI: 10.1215/21562261-1728839

Abstract

The main purpose of this article is to establish the Caffarelli–Kohn– Nirenberg-type (CKN-type) inequalities for all αR and to study the related matters systematically. Roughly speaking, we discuss the characterizations of the CKN-type inequalities for all αR as the variational problems, the existence and nonexistence of the extremal solutions to these variational problems in proper spaces, and the exact values and the asymptotic behaviors of the best constants in both the noncritical case and the critical case.

In the study of the CKN-type inequalities, the presence of weight functions on both sides prevents us from employing effectively the so-called spherically symmetric rearrangement. Further the invariance of Rn by the group of dilatations creates some possible loss of compactness. As a result we see that the existence of extremals, the values of best constants, and their asymptotic behaviors essentially depend upon the relations among parameters in the inequality.

Citation

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Toshio Horiuchi. Peter Kumlin. "On the Caffarelli–Kohn–Nirenberg-type inequalities involving critical and supercritical weights." Kyoto J. Math. 52 (4) 661 - 742, Winter 2012. https://doi.org/10.1215/21562261-1728839

Information

Published: Winter 2012
First available in Project Euclid: 15 November 2012

zbMATH: 1267.46051
MathSciNet: MR2872207
Digital Object Identifier: 10.1215/21562261-1728839

Subjects:
Primary: 35J70
Secondary: 35J60

Rights: Copyright © 2012 Kyoto University

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Vol.52 • No. 4 • Winter 2012
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