    March  2011, 6(1): 1-35. doi: 10.3934/nhm.2011.6.1

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

 1 Departamento de Matemáticas, Estadística y Computación, Universidad de Cantabria, Avenida de los Castros s/n., Santander, 39005, Spain 2 Institute of Mechanical Engineering Problems, RAN V.O.Bol'shoi pr., 61, StPetersburg, 199178, Russian Federation 3 Departamento de Matemática Aplicada y Ciencias de la Computación, Universided de Cantabria, Avenida de los Castros s/n, 39005 Santander

Received  June 2010 Revised  December 2010 Published  March 2011

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.
Citation: Delfina Gómez, Sergey A. Nazarov, Eugenia Pérez. Spectral stiff problems in domains surrounded by thin stiff and heavy bands: Local effects for eigenfunctions. Networks & Heterogeneous Media, 2011, 6 (1) : 1-35. doi: 10.3934/nhm.2011.6.1
##### References:
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##### References:
  H. Attouch, "Variational Convergence for Functions and Operators,", Pitmann, (1984). Google Scholar  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.  doi: 10.1016/S0021-7824(97)89944-8.  Google Scholar  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.  doi: 10.1137/090755680.  Google Scholar  C. Castro and E. Zuazua, Une remarque sur l'analyse asymptotique spectrale en homogénéisation,, C. R. Acad. Sci. Paris S\'er. I, 322 (1996), 1043. Google Scholar  E. A. Coddington and N. Levinson, "Theory of Ordinary Differential Equations,", McGraw-Hill, (1955). Google Scholar  L. Friedlander and M. Solomyak, On the spectrum of the Dirichlet Laplacian in a narrow strip,, Israel J. Math., 170 (2009), 337.  doi: 10.1007/s11856-009-0032-y.  Google Scholar  V. Mazýa, S. Nazarov and B. Plamenevskij, "Asymptotic Theory of Elliptic Boundary Value Problems in Singularly Perturbed Domains,", Birkhäuser, (2000). Google Scholar  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.  doi: 10.1142/S0218202504003520.  Google Scholar  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.  doi: 10.1016/S0021-7824(99)00009-4.  Google Scholar  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.  doi: 10.1016/j.matpur.2005.10.013.  Google Scholar  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.  doi: 10.1016/j.matpur.2006.08.003.  Google Scholar  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.  doi: 10.1007/BF02672180.  Google Scholar  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.  doi: 10.1093/imamat/hxh039.  Google Scholar  M. Lobo and E. Pérez, Local problems in vibrating systems with concentrated masses: A review,, C. R. Mecanique, 331 (2003), 303.  doi: 10.1016/S1631-0721(03)00058-5. Google Scholar  M. Lobo and E. Pérez, High frequency vibrations in a stiff problem,, Math. Models Methods Appl. Sci., 7 (1997), 291.  doi: 10.1142/S0218202597000177.  Google Scholar  S. A. Nazarov and M. Specovius-Neugebauer, Approximation of exterior problems. Optimal conditions for the Laplacian,, Analysis, 16 (1996), 305. Google Scholar  S. A. Nazarov, Localization effects for eigenfunctions near to the edge of a thin domain,, Math. Bohem, 127 (2002), 283. Google Scholar  S. A. Nazarov, "Asymptotic Theory of Thin Plates and Rods. Vol.1. Dimension Reduction and Integral Estimates,", Nauchnaya Kniga, (2002).   Google Scholar  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.  doi: 10.1023/A:1022364812273.  Google Scholar  O. A. Oleinik, A. S. Shamaev and G. A. Yosifian, "Mathematical Problems in Elasticity and Homogenization,", North-Holland, (1992). Google Scholar  E. Pérez, Long time approximations for solutions of wave equations via standing waves from quasimodes,, J. Math. Pures Appl., 90 (2008), 387.  doi: 10.1016/j.matpur.2008.06.003.  Google Scholar  J. Sanchez-Hubert and E. Sanchez-Palencia, "Vibration and Coupling of Continuous Systems. Asymptotic Methods,", Springer, (1988). Google Scholar
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