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Spectral stiff problems in domains surrounded by thin stiff and heavy bands: Local effects for eigenfunctions

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  • We consider the Neumann spectral problem for a second order differential operator, with piecewise constants coefficients, in a domain $\Omega_\varepsilon$ of $R^2$. Here $\Omega_\varepsilon$ is $\Omega \cup \omega_\varepsilon \cup \Gamma$, where $\Omega$ is a fixed bounded domain with boundary $\Gamma$, $\omega_\varepsilon$ is a curvilinear band of variable width $O(\varepsilon)$, and $\Gamma=\overline{\Omega}\cap \overline {\omega_\varepsilon}$. The density and stiffness constants are of order $O(\varepsilon^{-m-1})$ and $O(\varepsilon^{-1})$ respectively in this band, while they are of order $O(1)$ in $\Omega$; $m$ is a positive parameter and $\varepsilon \in (0,1)$, $\varepsilon\to 0$. Considering the range of the low, middle and high frequencies, we provide asymptotics for the eigenvalues and the corresponding eigenfunctions. For $m>2$, we highlight the middle frequencies for which the corresponding eigenfunctions may be localized asymptotically in small neighborhoods of certain points of the boundary.
    Mathematics Subject Classification: Primary: 35P05, 35P20; Secondary: 35B25, 73D30, 47A55, 47A75, 49R05.


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  • [1]

    H. Attouch, "Variational Convergence for Functions and Operators," Pitmann, London, 1984.


    A. Campbell and S. A. Nazarov, Une justification de la méthode de raccordement des développements asymptotiques appliquée a un probléme de plaque en flexion. Estimation de la matrice d'impedance, J. Math. Pures Appl., 76 (1997), 15-54.doi: 10.1016/S0021-7824(97)89944-8.


    G. Cardone, T. Durante and S. A. Nazarov, The localization effect for eigenfunctions of the mixed boundary value problem in a thin cylinder with distorted ends, SIAM J. Math. Anal., 42 (2010), 2581-2609.doi: 10.1137/090755680.


    C. Castro and E. Zuazua, Une remarque sur l'analyse asymptotique spectrale en homogénéisation, C. R. Acad. Sci. Paris Sér. I, 322 (1996), 1043-1047.


    E. A. Coddington and N. Levinson, "Theory of Ordinary Differential Equations," McGraw-Hill, London, 1955.


    L. Friedlander and M. Solomyak, On the spectrum of the Dirichlet Laplacian in a narrow strip, Israel J. Math., 170 (2009), 337-354.doi: 10.1007/s11856-009-0032-y.


    V. Mazýa, S. Nazarov and B. Plamenevskij, "Asymptotic Theory of Elliptic Boundary Value Problems in Singularly Perturbed Domains," Birkhäuser, Basel, 2000.


    Yu. D. Golovaty, D. Gómez, M. Lobo and E. Pérez, On vibrating membranes with very heavy thin inclusions, Math. Models Methods Appl. Sci., 14 (2004), 987-1034.doi: 10.1142/S0218202504003520.


    D. Gómez, M. Lobo and E. Pérez, On the eigenfunctions associated with the high frequencies in systems with a concentrated mass, J. Math. Pures Appl., 78 (1999), 841-865.doi: 10.1016/S0021-7824(99)00009-4.


    D. Gómez, M. Lobo, S. A. Nazarov and E. Pérez, Spectral stiff problems in domains surrounded by thin bands: asymptotic and uniform estimates for eigenvalues, J. Math. Pures Appl., 85 (2006), 598-632.doi: 10.1016/j.matpur.2005.10.013.


    D. Gómez, M. Lobo, S. A. Nazarov and E. Pérez, Asymptotics for the spectrum of the Wentzell problem with a small parameter and other related stiff problems, J. Math. Pures Appl., 86 (2006), 369-402.doi: 10.1016/j.matpur.2006.08.003.


    I. V. Kamotskii and S. A. Nazarov, On eigenfunctions localized in a neighborhood of the lateral surface of a thin domain, Probl. Mat. Analiz., 19 (1999), 105-148;doi: 10.1007/BF02672180.


    M. Lobo, S. A. Nazarov and E. Pérez, Eigenoscillations of contrasting non-homogeneous elastic bodies. Asymptotic and uniform estimates for eigenvalues, IMA J. Appl. Math., 70 (2005), 419-458.doi: 10.1093/imamat/hxh039.


    M. Lobo and E. Pérez, Local problems in vibrating systems with concentrated masses: A review, C. R. Mecanique, 331 (2003), 303-317.doi: 10.1016/S1631-0721(03)00058-5.


    M. Lobo and E. Pérez, High frequency vibrations in a stiff problem, Math. Models Methods Appl. Sci., 7 (1997), 291-311.doi: 10.1142/S0218202597000177.


    S. A. Nazarov and M. Specovius-Neugebauer, Approximation of exterior problems. Optimal conditions for the Laplacian, Analysis, 16 (1996), 305-324.


    S. A. Nazarov, Localization effects for eigenfunctions near to the edge of a thin domain, Math. Bohem, 127 (2002), 283-292.


    S. A. Nazarov, "Asymptotic Theory of Thin Plates and Rods. Vol.1. Dimension Reduction and Integral Estimates," Nauchnaya Kniga, Novosibirsk, 2002 (Russian).


    S. A. Nazarov, Uniform estimates of remainders in asymptotic expansions of solutions to the problem on eigen-oscillations of a piezoelectric plate, Probl. Mat. Analiz., 25 (2003), 99-188;doi: 10.1023/A:1022364812273.


    O. A. Oleinik, A. S. Shamaev and G. A. Yosifian, "Mathematical Problems in Elasticity and Homogenization," North-Holland, London, 1992.


    E. Pérez, Long time approximations for solutions of wave equations via standing waves from quasimodes, J. Math. Pures Appl., 90 (2008), 387-411.doi: 10.1016/j.matpur.2008.06.003.


    J. Sanchez-Hubert and E. Sanchez-Palencia, "Vibration and Coupling of Continuous Systems. Asymptotic Methods," Springer, Heidelberg, 1988.

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