Rigidity results with applications to best constants and symmetry of Caffarelli-Kohn-Nirenberg and logarithmic Hardy inequalities

TL;DR

This paper leverages a rigidity result to establish symmetry properties of extremal functions in Caffarelli-Kohn-Nirenberg and logarithmic Hardy inequalities, extending known symmetry ranges.

Contribution

It introduces a novel approach using reparametrization and estimates based on rigidity to prove symmetry for a broader set of inequalities.

Findings

Symmetry results for a wider parameter range than previous methods

Extension of symmetry results beyond symmetrization and comparison techniques

Application to best constants in inequalities

Abstract

We take advantage of a rigidity result for the equation satisfied by an extremal function associated with a special case of the Caffarelli-Kohn-Nirenberg inequalities to get a symmetry result for a larger set of inequali-ties. The main ingredient is a reparametrization of the solutions to the Euler-Lagrange equations and estimates based on the rigidity result. The symmetry results cover a range of parameters which go well beyond the one that can be achieved by symmetrization methods or comparison techniques so far.