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

Abstract

We prove a sharp quantitative version of the p-Sobolev inequality for any 1

Degenerate stability of some Sobolev inequalities

“…In the last two decades there has been an abundance of stability results for various functional inequalities. Examples include, for instance, isoperimetric inequalities [31,25,17,21], L p -Sobolev inequalities [11,27,37,28], fractional Sobolev inequalities [13], Gagliardo-Nirenberg inequalities [6], Brunn-Minkowski, concentration and rearrangement inequalities [24,23,26,15,30], eigenvalue inequalities [36,10,7,33,1], solutions to elliptic equations with critical exponents [12,22,18], Young's inequality [16], Hausdorff-Young inequality [14], etc. Many of these works use strategies inspired by the paper of Bianchi-Egnell and in essentially all works (exceptions being [28,23] and one version of a refined Hölder inequality in [10]) the remainder term is quadratic in the distance to the optimizers.…”

The sharp Gagliardo–Nirenberg–Sobolev inequality in quantitative form

“…For example, Bianchi and Egnell [4] established a stability version for the L 2 −Sobolev inequality in whole space R n which answers affirmatively a question of Brezis and Lieb in [7]. The quantitative form of the L p −Sobolev inequality with p = 2 was proved by Cianchi, Fusco, Maggi and Pratelli [16] and recently by Figalli and Neumayer [32]. The stability version of the Sobolev inequality on functions of bounded variation were studied by Cianchi [14], Fusco, Maggi and Pratelli [34], and by Figalli, Maggi and Pratelli [29].…”

A note on strong-form stability for the Sobolev inequality

“…The argument combines symmetrization arguments in the spirit of [FMP08] with a mass transportation argument in one dimension. More recently, in [FN19], Figalli and the author strengthened this result in the case p ≥ 2 by showing that the deficit of a function controls a power of A(u). The main idea there was to view W 1,p (R n ) as a weighted Hilbert space and to establish a spectral gap for the linearized operator in the second variation as in [BE91].…”