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

  • Received: 01 June 2010 Revised: 01 December 2010
  • Primary: 35P05, 35P20; Secondary: 35B25, 73D30, 47A55, 47A75, 49R05.

  • 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 surroundedby thin stiff and heavy bands: Local effects for eigenfunctions[J]. Networks and Heterogeneous Media, 2011, 6(1): 1-35. doi: 10.3934/nhm.2011.6.1

    Related Papers:

    [1] Delfina Gómez, Sergey A. Nazarov, Eugenia Pérez . Spectral stiff problems in domains surrounded by thin stiff and heavy bands: Local effects for eigenfunctions. Networks and Heterogeneous Media, 2011, 6(1): 1-35. doi: 10.3934/nhm.2011.6.1
    [2] Natalia O. Babych, Ilia V. Kamotski, Valery P. Smyshlyaev . Homogenization of spectral problems in bounded domains with doubly high contrasts. Networks and Heterogeneous Media, 2008, 3(3): 413-436. doi: 10.3934/nhm.2008.3.413
    [3] Robert Carlson . Spectral theory for nonconservative transmission line networks. Networks and Heterogeneous Media, 2011, 6(2): 257-277. doi: 10.3934/nhm.2011.6.257
    [4] Robert Carlson . Myopic models of population dynamics on infinite networks. Networks and Heterogeneous Media, 2014, 9(3): 477-499. doi: 10.3934/nhm.2014.9.477
    [5] Alessandro Gondolo, Fernando Guevara Vasquez . Characterization and synthesis of Rayleigh damped elastodynamic networks. Networks and Heterogeneous Media, 2014, 9(2): 299-314. doi: 10.3934/nhm.2014.9.299
    [6] Olivier Delestre, Arthur R. Ghigo, José-Maria Fullana, Pierre-Yves Lagrée . A shallow water with variable pressure model for blood flow simulation. Networks and Heterogeneous Media, 2016, 11(1): 69-87. doi: 10.3934/nhm.2016.11.69
    [7] Michael T. Redle, Michael Herty . An asymptotic-preserving scheme for isentropic flow in pipe networks. Networks and Heterogeneous Media, 2025, 20(1): 254-285. doi: 10.3934/nhm.2025013
    [8] Dilip Sarkar, Shridhar Kumar, Pratibhamoy Das, Higinio Ramos . Higher-order convergence analysis for interior and boundary layers in a semi-linear reaction-diffusion system networked by a $ k $-star graph with non-smooth source terms. Networks and Heterogeneous Media, 2024, 19(3): 1085-1115. doi: 10.3934/nhm.2024048
    [9] David Gérard-Varet, Alexandre Girodroux-Lavigne . Homogenization of stiff inclusions through network approximation. Networks and Heterogeneous Media, 2022, 17(2): 163-202. doi: 10.3934/nhm.2022002
    [10] Eric Chung, Yalchin Efendiev, Ke Shi, Shuai Ye . A multiscale model reduction method for nonlinear monotone elliptic equations in heterogeneous media. Networks and Heterogeneous Media, 2017, 12(4): 619-642. doi: 10.3934/nhm.2017025
  • 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.


    [1] H. Attouch, "Variational Convergence for Functions and Operators," Pitmann, London, 1984.
    [2] 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
    [3] 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
    [4] 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.
    [5] E. A. Coddington and N. Levinson, "Theory of Ordinary Differential Equations," McGraw-Hill, London, 1955.
    [6] 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
    [7] V. Mazýa, S. Nazarov and B. Plamenevskij, "Asymptotic Theory of Elliptic Boundary Value Problems in Singularly Perturbed Domains," Birkhäuser, Basel, 2000.
    [8] 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
    [9] 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
    [10] 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
    [11] 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
    [12] 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
    [13] 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
    [14] 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
    [15] 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
    [16] S. A. Nazarov and M. Specovius-Neugebauer, Approximation of exterior problems. Optimal conditions for the Laplacian, Analysis, 16 (1996), 305-324.
    [17] S. A. Nazarov, Localization effects for eigenfunctions near to the edge of a thin domain, Math. Bohem, 127 (2002), 283-292.
    [18] S. A. Nazarov, "Asymptotic Theory of Thin Plates and Rods. Vol.1. Dimension Reduction and Integral Estimates," Nauchnaya Kniga, Novosibirsk, 2002 (Russian).
    [19] 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
    [20] O. A. Oleinik, A. S. Shamaev and G. A. Yosifian, "Mathematical Problems in Elasticity and Homogenization," North-Holland, London, 1992.
    [21] 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
    [22] J. Sanchez-Hubert and E. Sanchez-Palencia, "Vibration and Coupling of Continuous Systems. Asymptotic Methods," Springer, Heidelberg, 1988.
  • This article has been cited by:

    1. Delfina Gómez, Sergei A. Nazarov, Maria-Eugenia Pérez-Martínez, Localization effects for Dirichlet problems in domains surrounded by thin stiff and heavy bands, 2021, 270, 00220396, 1160, 10.1016/j.jde.2020.09.011
    2. Huicong Li, Jingyu Li, Lifespan of effective boundary conditions for the heat equation, 2022, 63, 0022-2488, 091509, 10.1063/5.0083001
    3. V. Chiadò Piat, L. D’Elia, S.A. Nazarov, The stiff Neumann problem: Asymptotic specialty and “kissing” domains, 2022, 128, 18758576, 113, 10.3233/ASY-211701
    4. Huicong Li, Jingyu Li, Xuefeng Wang, Error estimates and lifespan of effective boundary conditions for 2-dimensional optimally aligned coatings, 2021, 303, 00220396, 1, 10.1016/j.jde.2021.09.015
    5. D. Gómez, S. A. Nazarov, E. Pérez, 2011, Chapter 15, 978-0-8176-8237-8, 159, 10.1007/978-0-8176-8238-5_15
    6. Sergei A. Nazarov, M. Eugenia Pérez, On multi-scale asymptotic structure of eigenfunctions in a boundary value problem with concentrated masses near the boundary, 2018, 31, 1139-1138, 1, 10.1007/s13163-017-0243-4
    7. Antonio Gaudiello, Delfina Gómez, Maria-Eugenia Pérez-Martínez, Asymptotic analysis of the high frequencies for the Laplace operator in a thin T-like shaped structure, 2020, 134, 00217824, 299, 10.1016/j.matpur.2019.06.005
    8. Lixin Meng, Zhitong Zhou, Error estimates of effective boundary conditions for the heat equation with optimally aligned coatings, 2025, 543, 0022247X, 128972, 10.1016/j.jmaa.2024.128972
    9. Yuriy Golovaty, Delfina Gómez, Maria-Eugenia Pérez-Martínez, On eigenvibrations of branched structures with heterogeneous mass density, 2025, 0022247X, 129586, 10.1016/j.jmaa.2025.129586
  • Reader Comments
  • © 2011 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
通讯作者: 陈斌, bchen63@163.com
  • 1. 

    沈阳化工大学材料科学与工程学院 沈阳 110142

  1. 本站搜索
  2. 百度学术搜索
  3. 万方数据库搜索
  4. CNKI搜索

Metrics

Article views(4065) PDF downloads(56) Cited by(9)

Article outline

Other Articles By Authors

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog