Characterization of the kernel of the operator CURL CURL

Philippe G. Ciarlet; Patrick Ciarlet; Giuseppe Geymonat; Françoise Krasucki

doi:10.1016/j.crma.2007.01.001

Functional Analysis/Mathematical Problems in Mechanics

Characterization of the kernel of the operator CURL CURL
[Caractérisation du noyau de l'opérateur CURL CURL]

Présenté par : Philippe G. Ciarlet

Philippe G. Ciarlet ¹ ; Patrick Ciarlet ² ; Giuseppe Geymonat ³ ; Françoise Krasucki ³

¹ Department of Mathematics, City University of Hong Kong, 83 Tat Chee Avenue, Kowloon, Hong Kong
² Laboratoire POEMS, UMR 2706 CNRS/ENSTA/INRIA, École nationale supérieure de techniques avancées, 32, boulevard Victor, 75739 Paris cedex 15, France
³ Laboratoire de mécanique et de génie civil, UMR 5508, Université Montpellier II, place Eugène-Bataillon, 34695 Montpellier cedex 5, France

Comptes Rendus. Mathématique, Volume 344 (2007) no. 5, pp. 305-308.

Résumé
Abstract

Dans un domaine simplement connexe Ω de $R^{3}$ , le noyau de l'opérateur $CURL CURL$ agissant sur des champs de matrices symétriques de $L_{s}^{2} (Ω)$ dans $H_{s}^{−2} (Ω)$ , coïncide avec l'espace des champs de tenseurs de déformation linéarisés. Dans le cas de domaines non simplement connexes, Volterra a caractérisé ce noyau pour des champs réguliers. Dans cette Note, nous étendons ce résultat pour un domaine à frontière lipschitzienne et pour des champs dans $L_{s}^{2} (Ω)$ .

In a simply-connected domain Ω in $R^{3}$ , the kernel of the operator $CURL CURL$ acting on symmetric matrix fields from $L_{s}^{2} (Ω)$ to $H_{s}^{−2} (Ω)$ coincides with the space of linearized strain tensor fields. For not simply-connected domains, Volterra has characterized this kernel for smooth fields. Here we extend this result for domains with a Lipschitz-continuous boundary for fields in $L_{s}^{2} (Ω)$ .

Accepté le : 2006-12-22
Publié le : 2007-02-09

DOI : 10.1016/j.crma.2007.01.001

Affiliations des auteurs :

Philippe G. Ciarlet ¹ ; Patrick Ciarlet ² ; Giuseppe Geymonat ³ ; Françoise Krasucki ³

¹ Department of Mathematics, City University of Hong Kong, 83 Tat Chee Avenue, Kowloon, Hong Kong
² Laboratoire POEMS, UMR 2706 CNRS/ENSTA/INRIA, École nationale supérieure de techniques avancées, 32, boulevard Victor, 75739 Paris cedex 15, France
³ Laboratoire de mécanique et de génie civil, UMR 5508, Université Montpellier II, place Eugène-Bataillon, 34695 Montpellier cedex 5, France

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     author = {Philippe G. Ciarlet and Patrick Ciarlet and Giuseppe Geymonat and Fran\c{c}oise Krasucki},
     title = {Characterization of the kernel of the operator {CURL\,CURL}},
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     pages = {305--308},
     publisher = {Elsevier},
     volume = {344},
     number = {5},
     year = {2007},
     doi = {10.1016/j.crma.2007.01.001},
     language = {en},
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Philippe G. Ciarlet; Patrick Ciarlet; Giuseppe Geymonat; Françoise Krasucki. Characterization of the kernel of the operator CURL CURL. Comptes Rendus. Mathématique, Volume 344 (2007) no. 5, pp. 305-308. doi : 10.1016/j.crma.2007.01.001. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2007.01.001/

Bibliographie
Cité par

[1] R.A. Adams; J.J.F. Fournier Sobolev Spaces, Academic Press, 2003

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

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

[4] D.N. Arnold; R.S. Falk; R. Winther Finite element exterior calculus, homological techniques, and applications, Acta Numer., Volume 15 (2006), pp. 1-155

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

[6] M. Dauge Elliptic Boundary Value Problems on Corner Domains, Lecture Notes in Mathematics, vol. 1341, Springer-Verlag, 1988

[7] G. Geymonat; F. Krasucki Some remarks on the compatibility conditions in elasticity, Rend. Accad. Naz. Sci. XL, Volume 123 (2005), pp. 175-182

[8] G. Geymonat; F. Krasucki Beltrami's solutions of general equilibrium equations in continuum mechanics, C. R. Acad. Sci. Paris, Ser. I, Volume 342 (2006), pp. 359-363

[9] P.W. Gross; P.R. Kotiuga Electromagnetic Theory and Computation: A Topological Approach, MSRI Publications Series, Cambridge University Press, Cambridge, 2004

[10] V. Volterra Sur l'équilibre des corps élastiques multiplement connexes, Annales Scientifiques de l'Ecole Normale Supérieure, 3ème Série, Volume 24 (1907), pp. 401-517

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