Quantitative stability for minimizing Yamabe metrics

We show that the Caffarelli–Kohn–Nirenberg (CKN) inequality holds with a remainder term that is quartic in the distance to the set of optimizers for the full parameter range of the Felli–Schneider (FS) curve. The fourth power is best possible. This is due to the presence of non-trivial zero modes of the Hessian of the deficit functional along the FS-curve. Following an iterated Bianchi–Egnell strategy, the heart of our proof is verifying a ‘secondary non-degeneracy condition’. Our result completes the stability analysis for the CKN-inequality to leading order started by Wei and Wu. Moreover, it is the first instance of degenerate stability for non-constant optimizers and for a non-compact domain.

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Sharp quantitative stability of Poincaré-Sobolev inequality in the hyperbolic space and applications to fast diffusion flows

Bhakta,

Ganguly,

Karmakar

et al. 2024

Calc. Var.

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Consider the Poincaré-Sobolev inequality on the hyperbolic space: for every $$n \ge 3$$ n ≥ 3 and $$1 < p \le \frac{n+2}{n-2},$$ 1 < p ≤ n + 2 n - 2 , there exists a best constant $$S_{n,p, \lambda }({\mathbb {B}}^{n})>0$$ S n , p , λ ( B n ) > 0 such that $$\begin{aligned} S_{n, p, \lambda }({\mathbb {B}}^{n})\left( ~\int \limits _{{\mathbb {B}}^{n}}|u|^{{p+1}} \, \textrm{d}v_{{\mathbb {B}}^n} \right) ^{\frac{2}{p+1}} \le \int \limits _{{\mathbb {B}}^{n}}\left( |\nabla _{{\mathbb {B}}^{n}}u|^{2}-\lambda u^{2}\right) \, \textrm{d}v_{{\mathbb {B}}^n}, \end{aligned}$$ S n , p , λ ( B n ) ∫ B n | u | p + 1 d v B n 2 p + 1 ≤ ∫ B n | ∇ B n u | 2 - λ u 2 d v B n , holds for all $$u\in C_c^{\infty }({\mathbb {B}}^n),$$ u ∈ C c ∞ ( B n ) , and $$\lambda \le \frac{(n-1)^2}{4},$$ λ ≤ ( n - 1 ) 2 4 , where $$\frac{(n-1)^2}{4}$$ ( n - 1 ) 2 4 is the bottom of the $$L^2$$ L 2 -spectrum of $$-\Delta _{{\mathbb {B}}^n}.$$ - Δ B n . It is known from the results of Mancini and Sandeep (Ann. Sc. Norm. Super. Pisa Cl. Sci. 7 (4): 635–671, 2008) that under appropriate assumptions on n, p and $$\lambda $$ λ there exists an optimizer, unique up to the hyperbolic isometries, attaining the best constant $$S_{n,p,\lambda }({\mathbb {B}}^n).$$ S n , p , λ ( B n ) . In this article, we investigate the quantitative gradient stability of the above inequality and the corresponding Euler-Lagrange equation locally around a bubble. Our result generalizes the sharp quantitative stability of Sobolev inequality in $${\mathbb {R}}^{n}$$ R n by Bianchi and Egnell (J. Funct. Anal. 100 (1): 18–24. 1991) and Ciraolo, Figalli and Maggi (Int. Math. Res. Not. IMRN (21): 6780–6797, 2018) to the Poincaré-Sobolev inequality on the hyperbolic space. Furthermore, combining our stability results and implementing a novel and refined smoothing estimates in spirit of Bonforte and Figalli (Comm. Pure Appl. Math. 74 (4): 744–789, 2021), we prove a quantitative extinction rate towards its basin of attraction of the solutions of the sub-critical fast diffusion flow for radial initial data. In another application, we derive sharp quantitative stability of the Hardy-Sobolev-Maz’ya inequalities for the class of functions which are symmetric in the component of singularity.

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Quantitative stability for minimizing Yamabe metrics

Cited by 7 publications

References 49 publications

The Sharp Sobolev Inequality and Its Stability: An Introduction

The Sharp Sobolev Inequality and Its Stability: An Introduction

Degenerate stability of the Caffarelli–Kohn–Nirenberg inequality along the Felli–Schneider curve

Sharp quantitative stability of Poincaré-Sobolev inequality in the hyperbolic space and applications to fast diffusion flows

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