Comptes Rendus
no. 12
A criterion of the continuous spectrum for elasticity and other self-adjoint systems on sharp peak-shaped domains
[Un critère de spectre continu pour l'élasticité et d'autres systèmes auto-adjoints pour des domaines contenant des pointes]

Présenté par : Évariste Sanchez-Palencia

Sergey A. Nazarov1
1 Institute of Mechanical Engineering Problems, V.O., Bol'shoi pr. 61, 199178 St.-Petersburg, Russia
Comptes Rendus. Mécanique, Volume 335 (2007) no. 12, pp. 751-756.

Les spectres de l'élasticité et de systèmes piezo-electriques pour un solide avec une pointe sur la frontière, sans traction, ne sont pas discrets. Un critère algébrique de spectre continu non-vide est établi pour le problème de Neumann pour des systèmes elliptiques formellements auto-adjoints arbitraires d'équations differentielles du deuxième ordre dans un domaine de forme pointue.

The spectra of the elasticity and piezo-electricity systems for a solid with a sharp peak point on the boundary, which is free of traction, are not discrete. An algebraic criterion of non-empty continuous spectrum is found for the Neumann problem for rather arbitrary formally self-adjoint elliptic systems of second-order differential equations on a sharp peak-shaped domain.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crme.2007.10.019
Keywords: Computational solid mechanics, Elasticity system, Peak, Cusp, Essential, Continuous and discrete spectra
Mot clés : Mécanique des solides numérique, Système de l'elasticité, Pic, Pointe, Essentiel, Spectre continu et discret
Sergey A. Nazarov 1

1 Institute of Mechanical Engineering Problems, V.O., Bol'shoi pr. 61, 199178 St.-Petersburg, Russia 
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Sergey A. Nazarov. A criterion of the continuous spectrum for elasticity and other self-adjoint systems on sharp peak-shaped domains. Comptes Rendus. Mécanique, Volume 335 (2007) no. 12, pp. 751-756. doi : 10.1016/j.crme.2007.10.019. https://comptes-rendus.academie-sciences.fr/mecanique/articles/10.1016/j.crme.2007.10.019/

[1] S.A. Nazarov Asymptotic Theory of Thin Plates and Rods. Vol. 1. Dimension Reduction and Integral Estimates, Nauchnaya Kniga, Novosibirsk, 2001

[2] M.Sh. Birman; M.Z. Solomyak Spectral Theory of Selfadjoint Operators in Hilbert Space, D. Reidel Publ. Co., Dordrecht, 1987

[3] S.A. Nazarov Math. Notes, 62 (1997), pp. 629-641 (Erratum: Math. Notes, 63, 1998, pp. 565)

[4] E. Sanchez-Palencia C. R. Acad. Sci. Paris, Ser. 2, 311 (1990), pp. 909-916

[5] S.A. Nazarov Siberian Math. J., 41 (2000), pp. 744-759

[6] J. Nečas Les méthodes in théorie des équations elliptiques, Masson–Academia, Paris–Prague, 1967

[7] S.A. Nazarov J. Math. Sci., 92 (1998) no. 6, pp. 4338-4353

[8] V.A. Kozlov; V.G. Maz'ya; J. Rossmann Elliptic Boundary Value Problems in Domains with Point Singularities, Amer. Math. Soc., Providence, 1997

[9] S.A. Nazarov Russ. Math. Surveys, 54 (1999) no. 5, pp. 947-1014

[10] S.A. Nazarov J. Math. Sci., 114 (2003), pp. 1657-1725

Cité par Sources :

⁎ The author gratefully acknowledges the support by N.W.O., the Netherlands Organization for Scientific Research.

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