Vector and scalar potentials, Poincaré's theorem and Korn's inequality

Chérif Amrouche; Philippe G. Ciarlet; Patrick Ciarlet

doi:10.1016/j.crma.2007.10.020

Partial Differential Equations

Vector and scalar potentials, Poincaré's theorem and Korn's inequality

Presented by: Philippe G. Ciarlet

Chérif Amrouche ¹; Philippe G. Ciarlet ²; Patrick Ciarlet ³

¹ Laboratoire de mathématiques appliquées, CNRS UMR 5142, Université de Pau et des pays de l'Adour, IPRA, avenue de l'université, 64000 Pau, France
² Department of Mathematics, City University of Hong Kong, 83, Tat Chee Avenue, Kowloon, Hong Kong
³ Laboratoire POEMS, UMR 2706 CNRS/ENSTA/INRIA, École nationale supérieure de techniques avancées, 32, boulevard Victor, 75739 Paris cedex 15, France

Comptes Rendus. Mathématique, Volume 345 (2007) no. 11, pp. 603-608.

Abstract
Résumé

In this Note, we present several results concerning vector potentials and scalar potentials in a bounded, not necessarily simply-connected, three-dimensional domain. In particular, we consider singular potentials corresponding to data in negative order Sobolev spaces. We also give some applications to Poincaré's theorem and to Korn's inequality.

Dans cette Note, nous présentons plusieurs résultats concernant les potentiels vecteurs et les potentiels scalaires dans des domaines bornés tridimensionnels, éventuellement multiplement connexes. En particulier, on considère des potentiels singuliers correspondant à des données dans des espaces de Sobolev d'exposant négatif. On donne également des applications au théorème de Poincaré et à l'inégalité de Korn.

Accepted: 2007-10-06
Published online: 2007-11-26

DOI: 10.1016/j.crma.2007.10.020

Author's affiliations:

Chérif Amrouche ¹; Philippe G. Ciarlet ²; Patrick Ciarlet ³

¹ Laboratoire de mathématiques appliquées, CNRS UMR 5142, Université de Pau et des pays de l'Adour, IPRA, avenue de l'université, 64000 Pau, France
² Department of Mathematics, City University of Hong Kong, 83, Tat Chee Avenue, Kowloon, Hong Kong
³ Laboratoire POEMS, UMR 2706 CNRS/ENSTA/INRIA, École nationale supérieure de techniques avancées, 32, boulevard Victor, 75739 Paris cedex 15, France

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     author = {Ch\'erif Amrouche and Philippe G. Ciarlet and Patrick Ciarlet},
     title = {Vector and scalar potentials, {Poincar\'e's} theorem and {Korn's} inequality},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {603--608},
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     year = {2007},
     doi = {10.1016/j.crma.2007.10.020},
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Chérif Amrouche; Philippe G. Ciarlet; Patrick Ciarlet. Vector and scalar potentials, Poincaré's theorem and Korn's inequality. Comptes Rendus. Mathématique, Volume 345 (2007) no. 11, pp. 603-608. doi : 10.1016/j.crma.2007.10.020. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2007.10.020/

References
Cited by

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[2] C. Amrouche; P.G. Ciarlet; L. Gratie; S. Kesavan On the characterization of matrix fields as linearized strain tensor fields, J. Math. Pures Appl., Volume 86 (2006), pp. 116-132

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[4] C. Amrouche; V. Girault Decomposition of vector spaces and application to the Stokes problem in arbitrary dimension, Czech. Math. J., Volume 44 (1994), pp. 109-140

[5] C. Bernardi, V. Girault, Espaces duaux des domaines des opérateurs divergence et rotationnel avec trace nulle, Publications du Laboratoire Jacques-Louis Lions R 03017, 2003

[6] P.G. Ciarlet; P. Ciarlet Another approach to linearized elasticity and a new proof of Korn's inequality, Math. Models Methods Appl. Sci., Volume 15 (2005), pp. 259-271

[7] P. Fernandes; G. Gilardi Magnetostatic and electrostatic problems in inhomogeneous anisotropic media with irregular boundary and mixed boundary conditions, Math. Models Methods Appl. Sci., Volume 7 (1997), pp. 957-991

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