Stability estimate for the Lane-Emden inequality

TL;DR

This paper establishes a stability estimate for the Lane-Emden inequality, quantifying how close a density function is to the optimal form based on the inequality's residual.

Contribution

It provides the first quantitative stability estimate for the Lane-Emden inequality involving a remainder term and a distance to the optimizer manifold.

Findings

Quantitative stability estimate derived for the Lane-Emden inequality

Remainder estimate expressed in terms of a distance to the optimizer

Enhanced understanding of the inequality's equality cases

Abstract

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