Comptes Rendus
On the vibration of a partially fastened membrane with many ‘light’ concentrated masses on the boundary
Comptes Rendus. Mécanique, Volume 332 (2004) no. 12, pp. 949-954.

We consider a partially fastened membrane with many concentrated masses near the boundary. Masses have the diameter (aɛ); the density is O(1) outside the masses and O((aɛ)m), 0<m<2, in the masses. We assume that the distance between masses is O(ɛ) and a is fixed. We obtain the leading terms of the asymptotic expansion of eigenvalues and eigenfunctions of the respective spectral problems for the Laplacian in such a domain.

Nous considérons une membrane partiellement attachée avec plusieurs masses concentrées prés de la frontière. Le diamètre des masses est ègal à (aɛ) ; la densité est d'ordre O(1) en dehors des masses et la densité des masses d'ordre O((aɛ)m), 0<m<2. Nous supposons que la distance entre les masses est d'ordre O(ɛ) et que a est fixé. Nous obtenons les termes principaux du développement asymptotique des valeurs propres et des fonctions propres du Laplacian dans un domaine de ce type.

Published online:
DOI: 10.1016/j.crme.2004.08.002
Keywords: Vibrations, Homogenization, Small parameter, Eigenvalues, Asymptotics
Mot clés : Vibrations, Homogénéisation, Petit paramètre, Valeur propre, Asymptotique
Gregory A. Chechkin 1

1 Department of Differential Equations, Faculty of Mechanics and Mathematics, Moscow State University, Moscow 119992, Russia
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Gregory A. Chechkin. On the vibration of a partially fastened membrane with many ‘light’ concentrated masses on the boundary. Comptes Rendus. Mécanique, Volume 332 (2004) no. 12, pp. 949-954. doi : 10.1016/j.crme.2004.08.002.

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