Optimal Hardy-type inequalities for elliptic operators

Baptiste Devyver; Martin Fraas; Yehuda Pinchover

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

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

Baptiste Devyver ¹ ; Martin Fraas ² ; Yehuda Pinchover 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

@article{CRMATH_2012__350_9-10_475_0,
     author = {Baptiste Devyver and Martin Fraas and Yehuda Pinchover},
     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},
}

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Baptiste Devyver; Martin Fraas; Yehuda Pinchover. Optimal Hardy-type inequalities for elliptic operators. Comptes Rendus. Mathématique, Volume 350 (2012) no. 9-10, pp. 475-479. doi : 10.1016/j.crma.2012.04.020. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2012.04.020/

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