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Article Dans Une Revue Archive for Rational Mechanics and Analysis Année : 2004

Homogenization of periodic systems with large potentials

Résumé

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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Dates et versions

hal-02083813 , version 1 (04-08-2020)

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Grégoire Allaire, Yves Capdeboscq, Andrey Piatnitski, Vincent Siess, Muthusamy Vanninathan. Homogenization of periodic systems with large potentials. Archive for Rational Mechanics and Analysis, 2004, 174 (2), pp.179--220. ⟨10.1007/s00205-004-0332-7⟩. ⟨hal-02083813⟩
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