Nous développons un cadre général pour l’utilisation d’une dualité permettant de “transférer” des résultats de stabilité pour une inégalité fonctionnelle à son inégalité duale. Comme application, nous donnons un résultat de stabilité pour l’inégalité de Hardy–Littlewood–Sobolev qui est lié, par la dualité et les résultats prouvés ici, à une inégalité de stabilité pour l’inégalité de Sobolev prouvée par Bianchi et Egnell, et prolongée par Chen, Frank et Weth. Nous discutons également comment les résultats donnés ici peuvent étre combinés à la preuve d’inégalités fonctionnelles utilisant des flots, afin de démontrer les limites de stabilité avec des constantes calculables.
We develop a general framework for using duality to “transfer” stability results for a functional inequality to its dual inequality. As an application, we prove a stability bound for the Hardy–Littlewood–Sobolev inequality, which is related by duality, and the results proved here, to a stability inequality for the Sobolev inequality proved by Bianchi and Egnell, and extended by Chen, Frank and Weth. We also discuss how the results proved here can be combined with the proof of functional inequalities by means of flows to prove stability bounds with computable constants.
Mots clés : uniform convexity, entropy
@article{AFST_2017_6_26_2_319_0,
author = {Carlen, Eric A.},
title = {Duality and {Stability} for {Functional} {Inequalities}},
journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques},
pages = {319--350},
publisher = {Universit\'e Paul Sabatier, Toulouse},
volume = {Ser. 6, 26},
number = {2},
year = {2017},
doi = {10.5802/afst.1535},
language = {en},
url = {http://www.numdam.org/articles/10.5802/afst.1535/}
}
TY - JOUR AU - Carlen, Eric A. TI - Duality and Stability for Functional Inequalities JO - Annales de la Faculté des sciences de Toulouse : Mathématiques PY - 2017 SP - 319 EP - 350 VL - 26 IS - 2 PB - Université Paul Sabatier, Toulouse UR - http://www.numdam.org/articles/10.5802/afst.1535/ DO - 10.5802/afst.1535 LA - en ID - AFST_2017_6_26_2_319_0 ER -
%0 Journal Article %A Carlen, Eric A. %T Duality and Stability for Functional Inequalities %J Annales de la Faculté des sciences de Toulouse : Mathématiques %D 2017 %P 319-350 %V 26 %N 2 %I Université Paul Sabatier, Toulouse %U http://www.numdam.org/articles/10.5802/afst.1535/ %R 10.5802/afst.1535 %G en %F AFST_2017_6_26_2_319_0
Carlen, Eric A. Duality and Stability for Functional Inequalities. Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 26 (2017) no. 2, pp. 319-350. doi : 10.5802/afst.1535. http://www.numdam.org/articles/10.5802/afst.1535/
[1] Problèmes isoperimétriques et espaces de Sobolev, J. Differ. Geom., Volume 11 (1976), pp. 573-598 | DOI
[2] Sharp uniform convexity and smoothness inequalities for trace norms, Invent. Math., Volume 115 (1997) no. 3, pp. 463-482
[3] A note on the Sobolev inequality, J. Funct. Anal., Volume 100 (1991) no. 1, pp. 18-24 | DOI
[4] Asymptotics of the fast diffusion equation via entropy estimates, Arch. Ration. Mech. Anal., Volume 191 (2009) no. 3, pp. 347-385 | DOI
[5] Hardy–Littlewood–Sobolev inequalities via fast diffusion flows, Proc. Natl. Acad. Sci. USA, Volume 107 (2010) no. 46, pp. 19696-19701 | DOI
[6] Stability for a GNS inequality and the log-HLS inequality, with application to the critical mass Keller–Segel equation, Duke Math. J., Volume 162 (2013) no. 3, pp. 579-625 | DOI
[7] Stability estimates for the lowest eigenvalue of a Schrödinger operator, Geom. Funct. Anal., Volume 24 (2014) no. 1, pp. 63-84 | DOI
[8] Competing symmetries, the logarithmic Hardy–Littlewood–Sobolev inequality and Onofri’s inequality on , Geom. Funct. Anal., Volume 2 (1992) no. 1, pp. 90-104 | DOI
[9] Remainder terms in the fractional Sobolev inequality, Indiana Univ. Math. J., Volume 62 (2013) no. 4, pp. 1381-1397 | DOI
[10] Porous media equation and Sobolev inequalities under negative curvature, Bull. Sci. Math., Volume 129 (2005) no. 10, pp. 804-830 | DOI
[11] The Cauchy problem for when , Trans. Am. Math. Soc., Volume 291 (1985), pp. 145-158
[12] Sharp constants in the Hardy–Littlewood–Sobolev and related inequalities, Ann. Math., Volume 118 (1983), pp. 349-374 | DOI
[13] Analysis, Graduate Studies in Mathematics, 14, American Mathematical Society, 1996, 278 pages
[14] Convex Analysis, Princeton University Press, 1970, xviii+451 pages
[15] An Extension of the Bianchi–Egnell Stability Estimate to Bakry, Gentil, and Ledoux’s Generalization of the Sobolev Inequality to Continuous Dimensions (2016) (https://arxiv.org/abs/1512.06121)
[16] Best constant in Sobolev inequality, Ann. Mat. Pura Appl., Volume 110 (1976), pp. 353-372 | DOI
[17] Asymptotic behaviour for the porous medium equation posed in the whole space, J. Evol. Equ., Volume 3 (2003) no. 1, pp. 67-118 | DOI
[18] The Porous Medium Equation. Mathematical theory, Oxford Mathematical Monographs, Oxford University Press, 2007, xxii+624 pages
Cité par Sources :
