Comptes Rendus
Asymptotic behavior of the spectrum of an elliptic problem in a domain with aperiodically distributed concentrated masses
Comptes Rendus. Mécanique, Volume 345 (2017) no. 10, pp. 671-677.

In this paper, we consider a spectral problem with singular perturbation of density located near the boundary of the domain, depending on a small parameter. We prove the compactness theorem and study the behavior of eigenelements to the given problem, as the small parameter tends to zero.

Dans cet article, nous considérons un problème spectral avec une perturbation singulière de la densité située près de la limite du domaine, dépendant d'un petit paramètre. Nous prouvons le théorème de la compacité et étudions le comportement des éléments génériques du problème donné, lorsque le petit paramètre tend vers zéro.

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DOI: 10.1016/j.crme.2017.06.010
Keywords: Concentrated masses, Homogenization, Random functions
Mot clés : Masses concentrées, Homogénéisation, Fonctions aléatoires
Gregory A. Chechkin 1; Tatiana P. Chechkina 2

1 Department of Differential Equations, Faculty of Mechanics and Mathematics, M.V.Lomonosov Moscow State University, Moscow 119991, Russia
2 National Research Nuclear University MEPhI (Moscow Engineering Physics Institute), Moscow 115409, Russia
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Gregory A. Chechkin; Tatiana P. Chechkina. Asymptotic behavior of the spectrum of an elliptic problem in a domain with aperiodically distributed concentrated masses. Comptes Rendus. Mécanique, Volume 345 (2017) no. 10, pp. 671-677. doi : 10.1016/j.crme.2017.06.010. https://comptes-rendus.academie-sciences.fr/mecanique/articles/10.1016/j.crme.2017.06.010/

[1] A.N. Krylov Some differential equations of mathematical physics having applications in technical questions, Rep. Nikolaev Naval Acad., Volume 2 (1913), pp. 325-348 (in Russian)

[2] É. Sánchez-Palencia Perturbation of eigenvalues in thermoelasticity and vibration of systems with concentrated masses, Trends and Applications of Pure Mathematics to Mechanics, Lecture Notes in Phys., vol. 195, Springer-Verlag, Berlin, 1984, pp. 346-368

[3] Yu.D. Golovaty; S.A. Nazarov; O.A. Oleinik; T.S. Soboleva On eigenoscillations of a string with an attached mass, Sib. Math. J., Volume 29 (1989), pp. 744-760

[4] J. Sanchez-Hubert; É. Sánchez-Palencia Vibration and Coupling of Continuous Systems. Asymptotic Methods, Springer-Verlag, Berlin, 1989

[5] D. Gómez; M. Lobo; E. Pérez On the eigenfunctions associated with the high frequencies in systems with a concentrated mass, J. Math. Pures Appl., Volume 78 (1999) no. 9, pp. 841-865

[6] G.A. Chechkin; E. Pérez; E.I. Yablokova Non-periodic boundary homogenization and “light” concentrated masses, Indiana Univ. Math. J., Volume 54 (2005), pp. 321-348

[7] T.A. Mel'nyk; G.A. Chechkin Eigenvibrations of thick cascade junctions with “Super Heavy” concentrated masses, Izv. Math., Volume 79 (2015) no. 3, pp. 467-511

[8] B.A. Yu; C.G.A. Averaging Operators with boundary conditions of fine-scaled structure, Math. Notes, Volume 65 (1999) no. 4, pp. 418-429

[9] G.A. Chechkin; A.L. Piatnitski; S.A.S. Homogenization Methods and Applications, American Mathematical Society, Providence, RI, USA, 2007

[10] O.A. Oleinik; A.S. Shamaev; G.A. Yosifian Mathematical Problems in Elasticity and Homogenization, North-Holland, Amsterdam, 1992

[11] V.V. Jikov; S.M. Kozlov; O.A. Oleinik Homogenization of Differential Operators and Integral Functionals, Springer–Verlag, Berlin, 1994

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The paper was partially supported by RFBR grant 15-01-07920.

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