Calculus of Variations and Geometric Measure Theory

A. Figalli - Y. R. Y. Zhang

Sharp gradient stability for the Sobolev inequality

created by figalli on 05 Aug 2021
modified on 10 Feb 2023


Published Paper

Inserted: 5 aug 2021
Last Updated: 10 feb 2023

Journal: Duke Mathematical Journal
Year: 2022


We prove a sharp quantitative version of the $p$-Sobolev inequality for any $1<p<n$, with a control on the strongest possible distance from the class of optimal functions. Surprisingly, the sharp exponent is constant for $p \leq 2$, while it depends on $p$ for $p>2$.