Abstract.
We consider the homogenization of a system of second-order equations with a large potential in a periodic medium. Denoting by ε the period, the potential is scaled as ε−2. Under a generic assumption on the spectral properties of the associated cell problem, we prove that the solution can be approximately factorized as the product of a fast oscillating cell eigenfunction and of a slowly varying solution of a scalar second-order equation. This result applies to various types of equations such as parabolic, hyperbolic or eigenvalue problems, as well as fourth-order plate equation. We also prove that, for well-prepared initial data concentrating at the bottom of a Bloch band, the resulting homogenized tensor depends on the chosen Bloch band. Our method is based on a combination of classical homogenization techniques (two-scale convergence and suitable oscillating test functions) and of Bloch waves decomposition.
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Acknowledgments.
This work was partly done when A. Piatnitski and M. Vanninathan were visiting the Centre de Mathématiques Appliquées at Ecole Polytechnique. They gratefully acknowledge the warm hospitality received during their stay. Y. Capdeboscq thanks L.C. Evans for bringing reference [23] to his attention.
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Allaire, G., Capdeboscq, Y., Piatnitski, A. et al. Homogenization of Periodic Systems with Large Potentials. Arch. Rational Mech. Anal. 174, 179–220 (2004). https://doi.org/10.1007/s00205-004-0332-7
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DOI: https://doi.org/10.1007/s00205-004-0332-7