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

Hernán Castro; Juan Dávila; Hui Wang

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

Hernán Castro ¹ ; Juan Dávila ² ; Hui Wang ^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, Volume 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 :

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

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     author = {Hern\'an Castro and Juan D\'avila and Hui Wang},
     title = {A {Hardy} type inequality for $ {W}_{0}^{2,1}(\Omega )$ functions},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {765--767},
     publisher = {Elsevier},
     volume = {349},
     number = {13-14},
     year = {2011},
     doi = {10.1016/j.crma.2011.06.026},
     language = {en},
}

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Hernán Castro; Juan Dávila; Hui Wang. A Hardy type inequality for $ {W}_{0}^{2,1}(\Omega )$ functions. Comptes Rendus. Mathématique, Volume 349 (2011) no. 13-14, pp. 765-767. doi : 10.1016/j.crma.2011.06.026. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2011.06.026/

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Cité par

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[2] Haim Brezis Functional Analysis, Sobolev Spaces and Partial Differential Equations, Universitext, Springer, New York, 2011 (MR 2759829)

[3] Haim Brezis; Moshe Marcus Hardyʼs inequalities revisited, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), Volume 25 (1997) no. 1–2, pp. 217-237 (1998), Dedicated to Ennio De Giorgi. MR 1655516 (99m:46075)

[4] Hernán Castro; Hui Wang A Hardy type inequality for $W^{m, 1} (0, 1)$ functions, Calc. Var. Partial Differential Equations, Volume 39 (2010) no. 3–4, pp. 525-531 (MR 2729310)

[5] Hernán Castro, Juan Dávila, Hui Wang, A Hardy type inequality for $W_{0}^{m, 1} (Ω)$ functions, in press.

[6] David Gilbarg; Neil S. Trudinger Elliptic Partial Differential Equations of Second Order, Classics in Mathematics, Springer-Verlag, Berlin, 2001 Reprint of the 1998 edition. MR MR1814364 (2001k:35004)

[7] Moshe Marcus; Laurent Veron Removable singularities and boundary traces, J. Math. Pures Appl. (9), Volume 80 (2001) no. 9, pp. 879-900 MR 1865379 (2002j:35124)

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