Stability with explicit constants of the critical points of the fractional Sobolev inequality and applications to fast diffusion

TL;DR

This paper establishes quantitative stability bounds for critical points of the fractional Sobolev inequality with explicit constants and applies these results to determine the extinction rate in fractional fast diffusion equations.

Contribution

It provides explicit stability bounds for fractional Sobolev critical points and determines the sharp asymptotic constants, with applications to fractional fast diffusion.

Findings

Bounded the deviation of functions from fractional Talenti bubbles using energy norms.

Derived explicit polynomial extinction rates for solutions to fractional fast diffusion equations.

Established stability with explicit constants for fractional Sobolev inequality critical points.

Abstract

We study the quantitative stability of critical points of the fractional Sobolev inequality. We show that, for a non-negative function $u \in \dot H^s(\mathbb R^N)$ whose energy satisfies $$\tfrac{1}{2} S^\frac{N}{2s}_{N,s} \le \|u\|_{\dot H^s(\mathbb R^N)} \le \tfrac{3}{2}S_{N,s}^\frac{N}{2s},$$ where $S_{N,s}$ is the optimal Sobolev constant, the bound $$ \|u -U[z,\lambda]\|_{\dot{H}^s(\mathbb R^N)} \lesssim \|(-\Delta)^s u - u^{2^*_s-1}\|_{\dot{H}^{-s}(\mathbb R^N)}, $$ holds for a suitable fractional Talenti bubble $U[z,\lambda]$. {For functions $u$ which are close to Talenti bubbles, we give the sharp asymptotic value of the implied constant in this inequality.} As an application {of this}, we derive an explicit polynomial extinction rate for positive solutions to a fractional fast diffusion equation.