Perturbation of the eigenvalues of a membrane with a concentrated mass
Authors:
C. Leal and J. Sanchez-Hubert
Journal:
Quart. Appl. Math. 47 (1989), 93-103
MSC:
Primary 73K12; Secondary 35B25
DOI:
https://doi.org/10.1090/qam/987898
MathSciNet review:
987898
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Abstract: We study a vibrating membrane with a distribution of density depending on $\varepsilon$, which converges, as $\varepsilon \searrow 0$, to a uniform density, plus a point mass at the origin. We establish local vibrations at the vicinity of the origin and global vibrations of the membrane. The asymptotic study for $\varepsilon \searrow 0$ is performed using the method of matched asymptotic expansions.
- E. Sánchez-Palencia, Perturbation of eigenvalues in thermoelasticity and vibration of systems with concentrated masses, Trends and applications of pure mathematics to mechanics (Palaiseau, 1983) Lecture Notes in Phys., vol. 195, Springer, Berlin, 1984, pp. 346–368. MR 755735, DOI https://doi.org/10.1007/3-540-12916-2_66
- E. Sánchez-Palencia and H. Tchatat, Vibration de systèmes élastiques avec des masses concentrées, Rend. Sem. Mat. Univ. Politec. Torino 42 (1984), no. 3, 43–63 (French). MR 834781
H. Tchatat, Thèse de 3ème cycle, University Pierre et Marie Curie, 1984
- O. A. Oleĭnik, Homogenization problems in elasticity. Spectra of singularly perturbed operators, Nonclassical continuum mechanics (Durham, 1986) London Math. Soc. Lecture Note Ser., vol. 122, Cambridge Univ. Press, Cambridge, 1987, pp. 53–95. MR 926498, DOI https://doi.org/10.1017/CBO9780511662911.005
- J. Sanchez Hubert and E. Sánchez-Palencia, Vibration and coupling of continuous systems, Springer-Verlag, Berlin, 1989. Asymptotic methods. MR 996423
- J. Kevorkian and Julian D. Cole, Perturbation methods in applied mathematics, Applied Mathematical Sciences, vol. 34, Springer-Verlag, New York-Berlin, 1981. MR 608029
G. Goursat, Cours d’ Analyse mathématique, Gauthier-Villars, Paris, Vol. 3 (1921)
R. Dautray and J. L. Lions, Analyse mathématique et calcul numérique pour les sciences et les techniques, Tome 1, Masson, Paris
O. A. Ladyzenskaia, Mathematical theory of viscous fluid flows, Gordon and Breach, New York, 1963
E. Sanchez-Palencia, Perturbation of eigenvalues in thermoelasticity and vibration of systems with concentrated masses, In Trends in applications of pure mathematics to mechanics, P. Ciarlet and M. Roseau, editors, Springer, 1984
E. Sanchez-Palencia and H. Tchatat, Vibration de systèmes élastiques avec masses concentrées, Rend. Sem. Mat. Univ., Politecn. Torino, 42, 43–63 (1984)
H. Tchatat, Thèse de 3ème cycle, University Pierre et Marie Curie, 1984
O. A. Oleinik, Homogenization problems in elasticity. Spectra of singularly perturbed operators in Nonclassical continuum mechanics, Ed. R. J. Knops and A. A. Lacey, Cambridge Univ. Press, 80–95 (1987)
J. Sanchez-Hubert and E. Sanchez-Palencia, Vibration and coupling of continuous systems. Asymptotic methods, Springer, Heidelberg (To be published 1988–89)
J. D. Cole and J. Kevorkian, Perturbation methods in applied mathematics, Springer
G. Goursat, Cours d’ Analyse mathématique, Gauthier-Villars, Paris, Vol. 3 (1921)
R. Dautray and J. L. Lions, Analyse mathématique et calcul numérique pour les sciences et les techniques, Tome 1, Masson, Paris
O. A. Ladyzenskaia, Mathematical theory of viscous fluid flows, Gordon and Breach, New York, 1963
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Article copyright:
© Copyright 1989
American Mathematical Society
