Optimal Hardy-type inequalities for elliptic operators

Devyver, Baptiste; Fraas, Martin; Pinchover, Yehuda

doi:10.1016/j.crma.2012.04.020

Partial Differential Equations

Optimal Hardy-type inequalities for elliptic operators
[Sur des inégalités de Hardy optimales]

Présenté par : Haïm Brézis

Devyver, Baptiste ¹ ; Fraas, Martin ² ; Pinchover, Yehuda ¹

¹ Department of Mathematics, Technion - Israel Institute of Technology, Haifa, 32000, Israel
² Theoretische Physik ETH Zürich, Wolfgang-Pauli-Str. 27, 8093 Zürich, Switzerland

Comptes Rendus. Mathématique, Tome 350 (2012) no. 9-10, pp. 475-479.

Résumé
Abstract

Soit P un opérateur elliptique du second ordre sur un domaine Ω. On construit un poids W, tel que si $Ω^{⋆} : = Ω ∖ {0}$ est un domaine épointé, alors $P - λ W$ est sous-critique sur $Ω^{⋆}$ pour $λ < 1$ , nul-critique dans $Ω^{⋆}$ pour $λ = 1$ , et supercritique à lʼinfini et en 0 pour $λ > 1$ . Notre approche repose sur la théorie des solutions positives dʼun opérateur elliptique du second ordre, et sʼapplique à la fois au cas symétrique et non symétrique. Le poids est de plus donné par une formule explicite faisant intervenir la fonction de Green de P et son gradient.

For a general second order elliptic operator P in a domain Ω, we construct a Hardy weight W in the punctured domain $Ω^{⋆} : = Ω ∖ {0}$ such that $P - λ W$ is subcritical in $Ω^{⋆}$ for $λ < 1$ , null-critical in $Ω^{⋆}$ for $λ = 1$ , and supercritical near infinity and near 0 for $λ > 1$ . Our method is based on the theory of positive solutions and applies to both symmetric and nonsymmetric operators. The constructed Hardy weight is given by an explicit formula involving the Green function of P and its gradient.

Reçu le : 2012-04-20
Accepté le : 2012-04-27
Publié le : 2012-05-01

DOI : 10.1016/j.crma.2012.04.020

Affiliations des auteurs :

Devyver, Baptiste ¹ ; Fraas, Martin ² ; Pinchover, Yehuda ¹

¹ Department of Mathematics, Technion - Israel Institute of Technology, Haifa, 32000, Israel
² Theoretische Physik ETH Zürich, Wolfgang-Pauli-Str. 27, 8093 Zürich, Switzerland

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     author = {Devyver, Baptiste and Fraas, Martin and Pinchover, Yehuda},
     title = {Optimal {Hardy-type} inequalities for elliptic operators},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {475--479},
     publisher = {Elsevier},
     volume = {350},
     number = {9-10},
     year = {2012},
     doi = {10.1016/j.crma.2012.04.020},
     language = {en},
     url = {http://www.numdam.org/articles/10.1016/j.crma.2012.04.020/}
}

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Devyver, Baptiste; Fraas, Martin; Pinchover, Yehuda. Optimal Hardy-type inequalities for elliptic operators. Comptes Rendus. Mathématique, Tome 350 (2012) no. 9-10, pp. 475-479. doi : 10.1016/j.crma.2012.04.020. http://www.numdam.org/articles/10.1016/j.crma.2012.04.020/

Bibliographie
Cité par

[1] Adimurthi; Sekar, A. Role of the fundamental solution in Hardy–Sobolev-type inequalities, Proc. Roy. Soc. Edinburgh Sect. A, Volume 136 (2006), pp. 1111-1130

[2] Ancona, A. Some results and examples about the behavior of harmonic functions and Greenʼs functions with respect to second order elliptic operators, Nagoya Math. J., Volume 165 (2002), pp. 123-158

[3] Barbatis, G.; Filippas, S.; Tertikas, A. A unified approach to improved $L^{p}$ Hardy inequalities with best constants, Trans. Amer. Math. Soc., Volume 356 (2004), pp. 2169-2196

[4] Carron, G. Inégalités de Hardy sur les variétés riemaniennes non-compactes, J. Math. Pures Appl., Volume 76 (1997), pp. 883-891

[5] Cowan, C. Optimal Hardy inequalities for general elliptic operators with improvements, Commun. Pure Appl. Anal., Volume 9 (2010), pp. 109-140

[6] Filippas, S.; Tertikas, A.; Tidblom, J. On the structure of Hardy–Sobolev–Mazʼya inequalities, J. Eur. Math. Soc. (JEMS), Volume 11 (2009), pp. 1165-1185

[7] Li, P.; Wang, J. Weighted Poincaré inequality and rigidity of complete manifolds, Ann. Sci. École Norm. Sup. (4), Volume 39 (2006), pp. 921-982

[8] Pinchover, Y. Topics in the theory of positive solutions of second-order elliptic and parabolic partial differential equations (Gesztesy, F. et al., eds.), Spectral Theory and Mathematical Physics: A Festschrift in Honor of Barry Simonʼs 60th Birthday, Proc. of Sympos. in Pure Math., vol. 76, Amer. Math. Society, Providence, RI, 2007, pp. 329-356

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