Comptes Rendus
no. 10
Self adjoint extensions of differential operators in application to shape optimization
[Extensions autoadjointes d'opérateurs différentiels et application à l'optimisation de forme]

Présenté par : Évariste Sanchez-Palencia

Serguei A. Nazarov1  ; Jan Sokolowski2
1 Institute of Mechanical Engineering Problems, Laboratory of Mathematical Methods, Russian Academy of Sciences, V.O. Bol'shoi 61, 199178 St. Petersburg, Russia
2 Institut Elie Cartan, laboratoire de mathématiques, Universite Henri Poincaré, Nancy I, BP 239, 54506 Vandoeuvre-les-Nancy cedex, France
Comptes Rendus. Mécanique, Volume 331 (2003) no. 10, pp. 667-672.

On propose deux approches permettant de modéliser des problèmes avec des singularités géométriques. La première approche repose sur des extensions autoadjointes d'opérateurs différentiels avec conditions asymptotiques. Dans la seconde approche, on introduit des espaces fonctionnels avec développements asymptotiques séparés puis on établit la formulation variationnelle. On obtient des estimations montrant que la même précision est atteinte pour ces deux approches. Enfin, on considère des problèmes spectraux et on donne des estimations pour les valeurs propres.

Two approaches are proposed for the modelling of problems with small geometrical defects. The first approach is based on the theory of self adjoint extensions of differential operators. In the second approach function spaces with separated asymptotics and point asymptotic conditions are introduced, and the variational formulation is established. For both approaches the accuracy estimates are derived. Finally, the spectral problems are considered and the error estimates for eigenvalues are given.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crme.2003.07.003
Keywords: Computational solid mechanics, Self adjoint extensions, Variational formulation, Error estimates
Mot clés : Mécanique des solides numérique, Extensions autoadjointes, Formulation variationnelle, Estimations d'erreur
Serguei A. Nazarov 1 ; Jan Sokolowski 2

1 Institute of Mechanical Engineering Problems, Laboratory of Mathematical Methods, Russian Academy of Sciences, V.O. Bol'shoi 61, 199178 St. Petersburg, Russia
2 Institut Elie Cartan, laboratoire de mathématiques, Universite Henri Poincaré, Nancy I, BP 239, 54506 Vandoeuvre-les-Nancy cedex, France 
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Serguei A. Nazarov; Jan Sokolowski. Self adjoint extensions of differential operators in application to shape optimization. Comptes Rendus. Mécanique, Volume 331 (2003) no. 10, pp. 667-672. doi : 10.1016/j.crme.2003.07.003. https://comptes-rendus.academie-sciences.fr/mecanique/articles/10.1016/j.crme.2003.07.003/

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