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Comptes Rendus Mathématique
Volume 350, n° 9-10
pages 475-479 (mai 2012)
Doi : 10.1016/j.crma.2012.04.020
Received : 20 April 2012 ;  accepted : 27 April 2012
Optimal Hardy-type inequalities for elliptic operators
Sur des inégalités de Hardy optimales
 

Baptiste Devyver a , Martin Fraas b , Yehuda Pinchover a
a Department of Mathematics, Technion - Israel Institute of Technology, Haifa, 32000, Israel 
b Theoretische Physik ETH Zürich, Wolfgang-Pauli-Str. 27, 8093 Zürich, Switzerland 

Abstract

For a general second order elliptic operator P in a domain Ω , we construct a Hardy weight W in the punctured domain   such that   is subcritical in   for  , null-critical in   for  , and supercritical near infinity and near 0 for  . 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.

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Résumé

Soit P un opérateur elliptique du second ordre sur un domaine Ω . On construit un poids W , tel que si   est un domaine épointé, alors   est sous-critique sur   pour  , nul-critique dans   pour  , et supercritique à lʼinfini et en 0 pour  . 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.

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© 2012  Published by Elsevier Masson SAS de la part de Académie des sciences.
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