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The Bloch Approximation in Periodically Perforated Media

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Abstract

We consider a periodically heterogeneous and perforated medium filling an open domain Ω of ℝN. Assuming that the size of the periodicity of the structure and of the holes is O(ε), we study the asymptotic behavior, as ε → 0, of the solution of an elliptic boundary value problem with strongly oscillating coefficients posed in Ωε (Ωε being Ω minus the holes) with a Neumann condition on the boundary of the holes. We use Bloch wave decomposition to introduce an approximation of the solution in the energy norm which can be computed from the homogenized solution and the first Bloch eigenfunction. We first consider the case where Ωis ℝN and then localize the problem for a bounded domain Ω, considering a homogeneous Dirichlet condition on the boundary of Ω.

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Authors and Affiliations

  1. Departamento de Ingenieria Matem\’atica and CMM, UMI 2807, CNRS-UChile, Facultad de Ciencias Fisicas y Matematicas, Universidad de Chile, Casilla 170/3 - Correo 3, Santiago, Chile

    C. Conca

  2. Departamento de Matematicas, Estadistica y Computacion, Universidad de Cantabria, Av. de los Castros s/n, 39071 Santander, Spain

    D. Gómez & M. Lobo

  3. Departamento de Matematica Aplicada y Ciencias de la Computacion, Universidad de Cantabria, Av. de los Castros s/n, 39071 Santander, Spain

    E. Pérez

Authors
  1. C. Conca
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  2. D. Gómez
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  3. M. Lobo
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  4. E. Pérez
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Corresponding authors

Correspondence to C. Conca, D. Gómez, M. Lobo or E. Pérez.

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Conca, C., Gómez, D., Lobo, M. et al. The Bloch Approximation in Periodically Perforated Media. Appl Math Optim 52, 93–127 (2005). https://doi.org/10.1007/s00245-005-0822-5

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  • DOI: https://doi.org/10.1007/s00245-005-0822-5

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