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:
[1]

H. Attouch, "Variational Convergence for Functions and Operators,", Pitmann, (1984).   Google Scholar

[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.  doi: 10.1016/S0021-7824(97)89944-8.  Google Scholar

[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.  doi: 10.1137/090755680.  Google Scholar

[4]

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

[5]

E. A. Coddington and N. Levinson, "Theory of Ordinary Differential Equations,", McGraw-Hill, (1955).   Google Scholar

[6]

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

[7]

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

[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.  doi: 10.1142/S0218202504003520.  Google Scholar

[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.  doi: 10.1016/S0021-7824(99)00009-4.  Google Scholar

[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.  doi: 10.1016/j.matpur.2005.10.013.  Google Scholar

[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.  doi: 10.1016/j.matpur.2006.08.003.  Google Scholar

[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.  doi: 10.1007/BF02672180.  Google Scholar

[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.  doi: 10.1093/imamat/hxh039.  Google Scholar

[14]

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

[15]

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

[16]

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

[17]

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

[18]

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

[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.  doi: 10.1023/A:1022364812273.  Google Scholar

[20]

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

[21]

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

[22]

J. Sanchez-Hubert and E. Sanchez-Palencia, "Vibration and Coupling of Continuous Systems. Asymptotic Methods,", Springer, (1988).   Google Scholar

show all references

References:
[1]

H. Attouch, "Variational Convergence for Functions and Operators,", Pitmann, (1984).   Google Scholar

[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.  doi: 10.1016/S0021-7824(97)89944-8.  Google Scholar

[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.  doi: 10.1137/090755680.  Google Scholar

[4]

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

[5]

E. A. Coddington and N. Levinson, "Theory of Ordinary Differential Equations,", McGraw-Hill, (1955).   Google Scholar

[6]

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

[7]

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

[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.  doi: 10.1142/S0218202504003520.  Google Scholar

[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.  doi: 10.1016/S0021-7824(99)00009-4.  Google Scholar

[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.  doi: 10.1016/j.matpur.2005.10.013.  Google Scholar

[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.  doi: 10.1016/j.matpur.2006.08.003.  Google Scholar

[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.  doi: 10.1007/BF02672180.  Google Scholar

[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.  doi: 10.1093/imamat/hxh039.  Google Scholar

[14]

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

[15]

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

[16]

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

[17]

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

[18]

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

[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.  doi: 10.1023/A:1022364812273.  Google Scholar

[20]

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

[21]

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

[22]

J. Sanchez-Hubert and E. Sanchez-Palencia, "Vibration and Coupling of Continuous Systems. Asymptotic Methods,", Springer, (1988).   Google Scholar

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