A Hardy type inequality for $ {W}_{0}^{2,1}(\Omega )$ functions

Castro, Hernán; Dávila, Juan; Wang, Hui

doi:10.1016/j.crma.2011.06.026

Partial Differential Equations/Functional Analysis

A Hardy type inequality for

W_{0}^{2, 1} (Ω)

functions
[Une inégalité de type Hardy pour les fonctions de

W_{0}^{2, 1} (Ω)

]

Présenté par : Haïm Brezis

Castro, Hernán ¹ ; Dávila, Juan ² ; Wang, Hui ^1,³

¹ Department of Mathematics, Rutgers University, 110 Frelinghuysen Road, Piscataway, NJ 08854, USA
² Departamento de Ingeniería Matemática and Centro de Modelamiento Matemático (UMI 2807 CNRS), Universidad de Chile, Casilla 170 Correo 3, Santiago, Chile
³ Department of Mathematics, Technion, Israel Institute of Technology, 32000 Haifa, Israel

Comptes Rendus. Mathématique, Tome 349 (2011) no. 13-14, pp. 765-767.

Résumé
Abstract

Nous considérons des fonctions $u \in W_{0}^{2, 1} (Ω)$ , où $Ω \subset R^{N}$ est un domaine régulier borné. Nous prouvons que $\frac{u (x)}{d (x)} \in W_{0}^{1, 1} (Ω)$ avec

‖ \nabla (\frac{u (x)}{d (x)}) ‖_{L^{1} (Ω)} ⩽ C ‖ u ‖_{W^{2, 1} (Ω)},

où d est une fonction régulière positive qui coïncide avec

dist (x, \partial Ω)

près de ∂Ω et C est une constante ne dépendant que de d et Ω.

We consider functions $u \in W_{0}^{2, 1} (Ω)$ , where $Ω \subset R^{N}$ is a smooth bounded domain. We prove that $\frac{u (x)}{d (x)} \in W_{0}^{1, 1} (Ω)$ with

‖ \nabla (\frac{u (x)}{d (x)}) ‖_{L^{1} (Ω)} ⩽ C ‖ u ‖_{W^{2, 1} (Ω)},

where d is a smooth positive function which coincides with

dist (x, \partial Ω)

near ∂Ω and C is a constant depending only on d and Ω.

Reçu le : 2011-06-06
Accepté le : 2011-06-30
Publié le : 2011-07-01

DOI : 10.1016/j.crma.2011.06.026

Affiliations des auteurs :

Castro, Hernán ¹ ; Dávila, Juan ² ; Wang, Hui ^{1, 3}

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     title = {A {Hardy} type inequality for $ {W}_{0}^{2,1}(\Omega )$ functions},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {765--767},
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Castro, Hernán; Dávila, Juan; Wang, Hui. A Hardy type inequality for $ {W}_{0}^{2,1}(\Omega )$ functions. Comptes Rendus. Mathématique, Tome 349 (2011) no. 13-14, pp. 765-767. doi : 10.1016/j.crma.2011.06.026. http://www.numdam.org/articles/10.1016/j.crma.2011.06.026/

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