On the stability constant of Caffarelli-Kohn-Nirenberg inequality

TL;DR

This paper investigates the stability constants of the Caffarelli-Kohn-Nirenberg inequality, showing non-validity of certain inequalities on specific curves and establishing the existence of minimizers near a new curve, extending prior results.

Contribution

It introduces a spectral analysis approach to identify where the inequality fails and proves the existence of minimizers near a newly identified curve, expanding previous work.

Findings

Inequality does not hold on the Felli-Schneider curve for orders less than 4.

Existence of minimizers of the sharp stability constant near a new curve.

Extension of previous results to a larger parameter region.

Abstract

By using a spectral analysis, we first show that the Caffarelli--Kohn--Nirenberg inequality with gradient remainder term of any order less than $4$ does not hold on the {\em Felli-Schneider} curve $b_{\mathrm{FS}}(a)$. Furthermore, we prove the existence of minimizers of sharp stability constant of Caffarelli--Kohn--Nirenberg inequality near the new curve $b^*_{\mathrm{FS}}(a)(>b_{\mathrm{FS}}(a))$, which extends the work of Wei and Wu [Math. Z., 2024] to a sightly larger region.