Gradient stability for the Sobolev inequality: the case $p\geq 2$

Alessio Figalli; Robin Neumayer

doi:10.4171/jems/837

JournalsjemsVol. 21, No. 2pp. 319–354

Gradient stability for the Sobolev inequality: the case $p \geq 2$

Alessio Figalli
ETH Zürich, Switzerland
Robin Neumayer
Northwestern University, Evanston, USA

$Gradient stability for the Sobolev inequality: the case $p\geq 2$ cover$

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Abstract

We show a strong form of the quantitative Sobolev inequality in $R^{n}$ for $p \geq 2$ , where the deficit of a function $u \in \dot{W}^{1, p}$ controls $∥\nabla u - \nabla v ∥_{L^{p}}$ for an extremal function $v$ in the Sobolev inequality.

Cite this article

Alessio Figalli, Robin Neumayer, Gradient stability for the Sobolev inequality: the case $p \geq 2$ . J. Eur. Math. Soc. 21 (2019), no. 2, pp. 319–354

DOI 10.4171/JEMS/837