A criterion of the continuous spectrum for elasticity and other self-adjoint systems on sharp peak-shaped domains

Sergey A. Nazarov

doi:10.1016/j.crme.2007.10.019

A criterion of the continuous spectrum for elasticity and other self-adjoint systems on sharp peak-shaped domains

Presented by: Évariste Sanchez-Palencia

Sergey A. Nazarov ¹

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.

Abstract
Résumé

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.

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.

Received: 2007-06-04
Accepted: 2007-10-30
Published online: 2007-11-26

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

Author's affiliations:

Sergey A. Nazarov ¹

¹ Institute of Mechanical Engineering Problems, V.O., Bol'shoi pr. 61, 199178 St.-Petersburg, Russia

@article{CRMECA_2007__335_12_751_0,
     author = {Sergey A. Nazarov},
     title = {A criterion of the continuous spectrum for elasticity and other self-adjoint systems on sharp peak-shaped domains},
     journal = {Comptes Rendus. M\'ecanique},
     pages = {751--756},
     publisher = {Elsevier},
     volume = {335},
     number = {12},
     year = {2007},
     doi = {10.1016/j.crme.2007.10.019},
     language = {en},
}

TY  - JOUR
AU  - Sergey A. Nazarov
TI  - A criterion of the continuous spectrum for elasticity and other self-adjoint systems on sharp peak-shaped domains
JO  - Comptes Rendus. Mécanique
PY  - 2007
SP  - 751
EP  - 756
VL  - 335
IS  - 12
PB  - Elsevier
DO  - 10.1016/j.crme.2007.10.019
LA  - en
ID  - CRMECA_2007__335_12_751_0
ER  -

%0 Journal Article
%A Sergey A. Nazarov
%T A criterion of the continuous spectrum for elasticity and other self-adjoint systems on sharp peak-shaped domains
%J Comptes Rendus. Mécanique
%D 2007
%P 751-756
%V 335
%N 12
%I Elsevier
%R 10.1016/j.crme.2007.10.019
%G en
%F CRMECA_2007__335_12_751_0

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/

References
Cited by

[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

Cited by Sources:

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

Comments - Politique