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Asymptotics for eigenvalues of the Laplacian in higher dimensional periodically perforated domains

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Abstract

This paper considers the periodic spectral problem associated with the Laplace operator written in \({\mathbb{R}^N}\) (N = 3, 4, 5) periodically perforated by balls, and with homogeneous Dirichlet condition on the boundary of holes. We give an asymptotic expansion for all simple eigenvalues as the size of holes goes to zero. As an application of this result, we use Bloch waves to find the classical strange term in homogenization theory, as the size of holes goes to zero faster than the microstructure period.

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Authors and Affiliations

  1. Departamento de Ingeniería Matemática, Facultad de Ciencias Físicas y Matemáticas, Universidad de Chile, Casilla 170/3, Correo 3, Santiago, Chile

    Jorge San Martín

  2. Centro de Modelamiento Matemático, UMR 2071, CNRS-UChile, Casilla 170/3, Correo 3, Santiago, Chile

    Jorge San Martín

  3. Department of Mathematics, Faculty of Mathematics and Computer Science, University of Piteşti, Str. Târgu din Vale nr. 1, 110040, Piteşti, Romania

    Loredana Smaranda

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  1. Jorge San Martín
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  2. Loredana Smaranda
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Correspondence to Loredana Smaranda.

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San Martín, J., Smaranda, L. Asymptotics for eigenvalues of the Laplacian in higher dimensional periodically perforated domains. Z. Angew. Math. Phys. 61, 401–424 (2010). https://doi.org/10.1007/s00033-009-0036-9

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  • DOI: https://doi.org/10.1007/s00033-009-0036-9

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