On radial symmetry and its breaking in the Caffarelli-Kohn-Nirenberg type inequalities for p = 1

Naoki Chiba; Toshio Horiuchi

doi:10.5036/mjiu.47.49

Mathematical Journal of Ibaraki University

Online ISSN : 1883-4353
Print ISSN : 1343-3636
ISSN-L : 1343-3636

On radial symmetry and its breaking in the Caffarelli-Kohn-Nirenberg type inequalities for p = 1

Naoki Chiba, Toshio Horiuchi

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Keywords: CKN-type inequality, Hardy-Sobolev inequality, Weighted Hardy inequality, Degenerate elliptic equation, Best constant

JOURNAL FREE ACCESS

2015 Volume 47 Pages 49-63

DOI https://doi.org/10.5036/mjiu.47.49

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Abstract

The main purpose of this article is to study the Caffarelli-Kohn-Nirenberg type inequalities (1.2) with p = 1. We show that symmetry breaking of the best constants occurs provided that a parameter |γ| is large enough. In the argument we effectively employ equivalence between the Caffarelli-Kohn-Nirenberg type inequalities with p = 1 and isoperimetric inequalities with weights.

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