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Stability estimate for the Lane-Emden inequality

Eric Carlen, Mathieu Lewin, Elliott H. Lieb, Robert Seiringer

Abstract

The Lane-Emden inequality controls $\iint_{\mathbb{R}^{2d}}ρ(x)ρ(y)|x-y|^{-λ}\,dx\,dy$ in terms of the $L^1$ and $L^p$ norms of $ρ$. We provide a remainder estimate for this inequality in terms of a suitable distance of $ρ$ to the manifold of optimizers.

Stability estimate for the Lane-Emden inequality

Abstract

The Lane-Emden inequality controls in terms of the and norms of . We provide a remainder estimate for this inequality in terms of a suitable distance of to the manifold of optimizers.

Paper Structure

This paper contains 10 sections, 6 theorems, 119 equations.

Table of Contents

  1. Introduction and main result
  2. Lane-Emden inequality
  3. Application of the stability bound
  4. Proof of the main result
  5. Euler-Lagrange equation
  6. Computation of the Hessian
  7. Stability
  8. Proof of Theorem \ref{['thm:scattering']}
  9. Local case $s=1$
  10. Non-local case $0<s<1$

Key Result

Theorem 1

Assume that and let $a=a(\lambda,p,d)$ denote the optimal Lane-Emden constant in LEp. Then there exists a constant $c = c(\lambda,p,d) > 0$ such that for all non-negative $\rho \in L^p(\mathbb{R}^d)$ with $\int_{\mathbb{R}^d}\rho= 1$, where the infimum is over all optimizers of the Lane-Emden inequality eq:Lane-Emden with mass $\int_{\mathbb{R}^d}\ell= 1$.

Theorems & Definitions (13)

  • Theorem 1: Stability for the Lane-Emden inequality
  • Theorem 2: No radial decaying solution if there is a scattering solution
  • Remark 3: Another proof of uniqueness
  • Corollary 4
  • proof
  • Proposition 5: Non-degeneracy
  • proof
  • Remark 6
  • Remark 7
  • Lemma 8
  • ...and 3 more