Symmetry breaking of extremals for the Caffarelli-Kohn-Nirenberg inequalities in a non-Hilbertian setting

TL;DR

This paper establishes a necessary condition indicating when extremals for the best constant in the Caffarelli-Kohn-Nirenberg inequalities are not radially symmetric, advancing understanding of symmetry properties in these inequalities.

Contribution

It introduces an explicit necessary condition for symmetry breaking of extremals in the Caffarelli-Kohn-Nirenberg inequalities in a non-Hilbertian setting.

Findings

Identifies conditions under which extremals lose radial symmetry

Provides explicit criteria for symmetry breaking

Enhances understanding of extremal behavior in weighted inequalities

Abstract

We provide an explicit necessary condition to have that no extremal for the best constant in the Caffarelli-Kohn-Nirenberg inequality is radially symmetric.