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

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 j (cid:13) j 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. 1. We show that symmetry breaking occurs provided that a parameter j (cid:13) j is large enough. In the argument we effectively employ equivalence between the CKN-type inequalities with p = 1 and isoperimetric inequalities with weights.