Symmetry for extremal functions in subcritical Caffarelli-Kohn-Nirenberg inequalities

TL;DR

This paper establishes the range of parameters for which extremal functions in subcritical Caffarelli-Kohn-Nirenberg inequalities are symmetric, using R{\'e}nyi entropies and a variational approach to prove a rigidity theorem.

Contribution

It extends symmetry results from critical to subcritical inequalities by developing new tools involving R{\'e}nyi entropies and a modified variational method.

Findings

Symmetry holds in a specific parameter range for extremal functions.

The method confirms radial symmetry is unique up to scalings and multiplications.

The results are sharp and complement known symmetry-breaking conditions.

Abstract

We use the formalism of the R{\'e}nyi entropies to establish the symmetry range of extremal functions in a family of subcriti-cal Caffarelli-Kohn-Nirenberg inequalities. By extremal functions we mean functions which realize the equality case in the inequalities, written with optimal constants. The method extends recent results on critical Caffarelli-Kohn-Nirenberg inequalities. Using heuristics given by a nonlinear diffusion equation, we give a variational proof of a symmetry result, by establishing a rigidity theorem: in the symmetry region, all positive critical points have radial symmetry and are therefore equal to the unique positive, radial critical point, up to scalings and multiplications. This result is sharp. The condition on the parameters is indeed complementary of the condition which determines the region in which symmetry breaking holds as a consequence of the linear instability of radial optimal functions. Compared to the critical case, the subcritical range requires new tools. The Fisher information has to be replaced by R{\'e}nyi entropy powers, and since some invariances are lost, the estimates based on the Emden-Fowler transformation have to be modified.