Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Degenerate Stability of the Caffarelli-Kohn-Nirenberg Inequality along the Felli-Schneider Curve

Rupert L. Frank, Jonas W. Peteranderl

Abstract

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.

Degenerate Stability of the Caffarelli-Kohn-Nirenberg Inequality along the Felli-Schneider Curve

Abstract

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.
Paper Structure (17 sections, 12 theorems, 154 equations)

This paper contains 17 sections, 12 theorems, 154 equations.

Table of Contents

  1. Introduction and main result
  2. The CKN-inequality
  3. Stability for the CKN-inequality -- our main result
  4. A reformulation
  5. Strategy of the proof
  6. Projection on the trivial zero modes of the Hessian
  7. Proof of Proposition \ref{['prop1']}
  8. Projection on the non-trivial zero modes of the Hessian
  9. Degeneracy along the Felli--Schneider curve
  10. Proof of Proposition \ref{['prop2']}
  11. Non-vanishing of the quartic order
  12. Quartic expansion of the deficit functional
  13. Bounding $(A) + (B)$ from below
  14. Proof of Proposition \ref{['prop3']}
  15. Solving a certain inhomogeneous second order equation
  16. ...and 2 more sections

Key Result

Theorem 1

Let $(a,b)\in\mathbb{R}^2$ satisfy $a<0$ and $\Lambda=\Lambda_{FS}$ with $d\geq 2$ and $q$ given by q. Then there is a constant $c(q,d)>0$ such that for all $v\in \mathcal{D}^{1}_a (\mathbb{R}^d)$, Moreover, the inequality is best possible with respect to the quartic vanishing of the distance to $\mathcal{Z}$, that is, there is a sequence $(v_n)_n\subset\mathcal{D}^1_a(\mathbb{R}^d)\setminus\{0\}

Theorems & Definitions (22)

  • Theorem 1: Degenerate stability of the CKN-inequality along the FS-curve
  • Proposition 2: Projection on the trivial zero modes of the Hessian
  • Proposition 3: Projection on the non-trivial zero modes of the Hessian
  • Proposition 4: Non-vanishing of the quartic order
  • Definition 5
  • Corollary 6: Degenerate stability of a Sobolev inequality for a cylinder along the FS-curve
  • proof
  • Lemma 7: Spectral analysis of lower eigenvalues
  • proof
  • Lemma 8: Quartic order expansion of $\mathcal{F}$
  • ...and 12 more