Spectral Stiff Problems in Domains with a Strongly Oscillating Boundary

  • D. Gómez
  • S. A. Nazarov
  • E. Pérez


We consider an asymptotic spectral problem for a second order differential operator, with piecewise constant coefficients, in a two-dimensional domain Ω ε which depends on a small parameter ε. Here, Ω ε is Ω ε =Ω∪ω ε ∪Γ, where Ω is a fixed open bounded domain, ω ε is a curvilinear strip of variable width O(ε), and \(\Gamma={\overline{\Omega}}\cap {\overline{\omega}_{\varepsilon}}\) is the boundary of Ω. The density and stiffness constants are of order O(ε −1) and O(ε t ), respectively, in this strip, while they are of order O(1) in the fixed domain Ω; t is a parameter such that 0≤t<1. Imposing the Neumann condition on the boundary of Ω ε , we study the asymptotic behavior, as ε→0, of the eigenvalues and eigenfunctions. Denoting by (ν,τ) the orthogonal curvilinear coordinates in a neighborhood of Γ, we consider two different types of bands ω ε . First, the case where ω ε ={x : 0<ν<εh(τ)} with h a positive function of the τ variable. Second, the case where ω ε ={x : 0<ν<εh ε (τ)} with h ε (τ)=h(τ/ε) and h a positive and periodic function; namely, a domain with strongly oscillating boundary.


Scalar Product Spectral Problem Neumann Problem Order Differential Operator Thin Band 
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  1. [At84]
    Attouch, H.: Variational Convergence for Functions and Operators, Pitmann (1984). MATHGoogle Scholar
  2. [BuKo87]
    Buttazzo, G., Kohn, R.V.: Reinforcement by a thin layer oscillating thickness. Appl. Math. Optim., 16, 247–261 (1987). MathSciNetCrossRefGoogle Scholar
  3. [Go06a]
    Gómez, D., Lobo, M., Nazarov, S.A., Pérez, E.: Spectral stiff ptoblems in domains surrounded by thin bands: asymptotic and uniform estimates for eigenvalues. J. Math. Pures Appl., 85, 598–632 (2006). MathSciNetMATHCrossRefGoogle Scholar
  4. [Go06b]
    Gómez, D., Lobo, M., Nazarov, S.A., Pérez, E.: Asymptotics for the spectrum of the Wentzell problem with a small parameter and other related stiff problems. J. Math. Pures Appl., 86, 369–402 (2006). MathSciNetMATHCrossRefGoogle Scholar
  5. [GoLo99]
    Gómez, D., Lobo, M., Pérez, E.: On the eigenfunctions associated with the high frequencies in systems with a concentrated mass. J. Math. Pures Appl., 78, 841–865 (1999). MathSciNetMATHCrossRefGoogle Scholar
  6. [GoNa11]
    Gómez, D., Nazarov, S.A., Pérez, E.: Spectral stiff problems in domains surrounded by thin stiff and heavy bands: local effects for eigenfunctions. Netw. Heterog. Media, 6, 1–35 (2011). MathSciNetCrossRefGoogle Scholar
  7. [LoSa79]
    Lobo, M., Sanchez-Palencia, E.: Sur certaines propriétés spectrales des perturbations du domaine dans les problèmes aux limites. Comm. Partial Differential Equations, 4, n. 10, 1085–1098 (1979). MathSciNetMATHCrossRefGoogle Scholar
  8. [Na90]
    Nazarov, S.A.: Binomial asymptotic behavior of solutions of spectral problems with singular perturbations. Mat. Sb., 181, n. 3, 291–320 (1990) (English transl: Math. USSR-Sb., 69, n. 2, 307–340 (1991)). Google Scholar
  9. [Na07]
    Nazarov, S.A.: Eigenoscillations of an elastic body with a rough surface. Prikl. Mekh. Tekhn. Fiz., 48, n. 6, 103–114 (2007) (English transl: J. Appl. Mech. Tech. Phys., 48, n. 6, 861–870 (2007)). MathSciNetGoogle Scholar
  10. [Na08]
    Nazarov, S.A.: Asymptotic behavior of solutions and the modeling of problems in the theory of elasticity in a domain with a rapidly oscillating boundary. Izv. Ross. Akad. Nauk Ser. Mat., 72, n. 3, 103–158 (2008) (English transl: Izv. Math., 72, n. 3, 509–564 (2008)). MathSciNetGoogle Scholar
  11. [OlSh92]
    Oleinik, O.A., Shamaev, A.S., Yosifian, G.A.: Mathematical Problems in Elasticity and Homogenization, North-Holland (1992). Google Scholar
  12. [Sa80]
    Sanchez-Palencia, E.: Nonhomogeneous Media and Vibration Theory, Springer-Verlag (1980). Google Scholar

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© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  1. 1.Universidad de CantabriaSantanderSpain
  2. 2.Institute for Problems in Mechanical EngineeringSt. PetersburgRussia

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