$ {L}^{p}$-theory for vector potentials and Sobolevʼs inequalities for vector fields

Chérif Amrouche; Nour El Houda Seloula

doi:10.1016/j.crma.2011.04.008

Partial Differential Equations

L^{p}

-theory for vector potentials and Sobolevʼs inequalities for vector fields
[Théorie

L^{p}

pour les potentiels vecteurs et inégalités de Sobolev pour des champs de vecteurs]

Présenté par : Philippe G. Ciarlet

Chérif Amrouche ¹ ; Nour El Houda Seloula ^1,²

¹ Laboratoire de mathématiques appliquées, CNRS UMR 5142, université de Pau et des Pays de lʼAdour, IPRA, avenue de lʼuniversité, 64000 Pau, France
² EPI Concha, LMA UMR CNRS 5142, INRIA Bordeaux-Sud-Ouest, 64000 Pau, France

Comptes Rendus. Mathématique, Volume 349 (2011) no. 9-10, pp. 529-534.

Résumé
Abstract

Dans un ouvert borné tridimensionnel, éventuellement multiplement connexe, nous prouvons lʼexistence et lʼunicité des potentiels vecteurs en théorie $L^{p}$ , associés à des champs de vecteurs à divergence nulle et vérifiant plusieurs conditions aux limites. On présente également des résultats concernant les potentiels scalaires et les potentiels vecteurs faibles. De plus, plusieurs inégalités de Sobolev sont données.

In a three-dimensional bounded possibly multiply-connected domain, we prove the existence and uniqueness of vector potentials in $L^{p}$ -theory, associated with a divergence-free function and satisfying some boundary conditions. We also present some results concerning scalar potentials and weak vector potentials. Furthermore, various Sobolev-type inequalities are given.

Reçu le : 2010-07-07
Accepté le : 2011-03-31
Publié le : 2011-05-01

DOI : 10.1016/j.crma.2011.04.008

Affiliations des auteurs :

Chérif Amrouche ¹ ; Nour El Houda Seloula ^{1, 2}

¹ Laboratoire de mathématiques appliquées, CNRS UMR 5142, université de Pau et des Pays de lʼAdour, IPRA, avenue de lʼuniversité, 64000 Pau, France
² EPI Concha, LMA UMR CNRS 5142, INRIA Bordeaux-Sud-Ouest, 64000 Pau, France

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     title = {$ {L}^{p}$-theory for vector potentials and {Sobolev's} inequalities for vector fields},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {529--534},
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     volume = {349},
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     year = {2011},
     doi = {10.1016/j.crma.2011.04.008},
     language = {en},
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Chérif Amrouche; Nour El Houda Seloula. $ {L}^{p}$-theory for vector potentials and Sobolevʼs inequalities for vector fields. Comptes Rendus. Mathématique, Volume 349 (2011) no. 9-10, pp. 529-534. doi : 10.1016/j.crma.2011.04.008. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2011.04.008/

Bibliographie
Cité par

[1] C. Amrouche; C. Bernardi; M. Dauge; V. Girault Vector potentials in three-dimensional nonsmooth domains, Math. Methods Appl. Sci., Volume 21 (1998), pp. 823-864

[2] C. Amrouche; V. Girault Decomposition of vector spaces and application to the Stokes problem in arbitrary dimension, Czechoslovak Math. J., Volume 119 (1994) no. 44, pp. 109-140

[3] C. Amrouche, N. Seloula, $L^{p}$ -theory for vector potentials and Sobolevʼs inequalities for vector fields. Application to the Stokes problemʼs with pressure boundary conditions, submitted for publication.

[4] M. Costabel A remark on the regularity of solutions of Maxwellʼs equations on Lipschitz domains, Math. Methods Appl. Sci. Theory, Volume 12 (1990), pp. 365-368

[5] V. Girault; P.A. Raviart Finite Element Methods for the Navier–Stokes Equations, Theory and Algorithms, Springer, Berlin, 1986

[6] W. von Wahl Estimating ∇u by div u, $curl u$ , Math. Methods Appl. Sci., Volume 15 (1992), pp. 123-143

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