Comptes Rendus
no. 4
Singular perturbations in shape optimization for the Dirichlet Laplacian
[Perturbations singulières en optimisation des formes pour le Laplacien avec conditions de Dirichlet]

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 mathematiques, université Henri Poincare Nancy I, BP 239, 54506 Vandoeuvre-les-Nancy cedex, France
Comptes Rendus. Mécanique, Volume 333 (2005) no. 4, pp. 305-310.

Un problème d'optimisation de forme est posé pour l'énergie du Laplacien avec conditions de Dirichlet. Des formes optimales obtenues par l'analyse asymptotique sont données par une perturbation singulière du domain initial régulier.

A shape optimization problem is considered for the Dirichlet Laplacian. Asymptotic analysis is used in order to characterise the optimal shapes which are finally given by a singular perturbation of the smooth initial domain.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crme.2005.02.006
Keywords: Computational solid mechanics, Dirichlet Laplacian, Shape optimization problem
Mot clés : Mécanique des solides numérique, Laplacien de Dirichlet, Problème d'optimisation de forme
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 mathematiques, université Henri Poincare Nancy I, BP 239, 54506 Vandoeuvre-les-Nancy cedex, France 
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Serguei A. Nazarov; Jan Sokolowski. Singular perturbations in shape optimization for the Dirichlet Laplacian. Comptes Rendus. Mécanique, Volume 333 (2005) no. 4, pp. 305-310. doi : 10.1016/j.crme.2005.02.006. https://comptes-rendus.academie-sciences.fr/mecanique/articles/10.1016/j.crme.2005.02.006/

[1] J. Sokołowski; J.-P. Zolesio Introduction to Shape Optimization. Shape Sensitivity Analysis, Springer-Verlag, 1992

[2] S.A. Nazarov; S.A. Nazarov Derivation of the variational inequality for small increase of mode-one crack, Mekh. Tverd. Tela, Volume 2 (1989), pp. 152-160 (English transl. Mech. Solids, 24, 1989, pp. 145-152)

[3] S.A. Nazarov; O.R. Polyakova On the equivalence of the fracture criteria for a mode-one crack in an elastic space, Mekh. Tverd. Tela, Volume 2 (1992), pp. 101-113 (in Russian)

[4] L.H. Kolton; S.A. Nazarov Quasistatic propagation of a mode-I crack in an elastic space, C. R. Acad. Sci. Paris. Sér. II, Volume 315 (1992), pp. 1453-1457

[5] M. Bach; S.A. Nazarov; W.L. Wendland Stable propagation of a mode-1 crack in an isotropic elastic space. Comparison of the Irwin and the Griffith approaches, Problemi attuali dell'analisi e della fisica matematica, Aracne, Roma, 2000, pp. 167-189

[6] V.I. Lebedev; V.I. Agoshkov Poincare–Steklov Operators and Their Applications in Analysis, Akad. Nauk SSSR, Vychisl. Tsentr, Moscow, 1983 18 pp. (in Russian)

[7] S.A. Nazarov; O.R. Polyakova Deformation and tear-off of a thin gasket from hardly compressible material, Mekh. Tverd. Tela, Volume 5 (1993), pp. 123-134 (in Russian)

[8] L.H. Kolton; S.A. Nazarov Variation of the shape of the front of plane mode-one crack which is not in equilibrium locally, Mekh. Tverd. Tela, Volume 3 (1997), pp. 125-133 (in Russian)

[9] M. Bach; S.A. Nazarov Smoothness properties of solutions to variational inequalities describing propagation of mode-1 cracks, Mathematical Aspects of Boundary Element Method (Palaiseau, 1998), CRC Res. Notes Math., vol. 414, Chapman & Hall, London, 2000, pp. 23-32

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