We study the asymptotic behavior of the spectrum of the Dirichlet problem for a formally selfadjoint elliptic system of differential equations with rapidly oscillating coefficients and changing sign density ρ. Since the factor ρ at the spectral parameter changes sign, the problem possesses two – positive and negative – infinitely large sequences of eigenvalues. Their asymptotic structure essentially depends on whether the mean \( \overline \rho \) over the periodicity cell vanishes. In particular, in the case \( \overline \rho = 0 \), the homogenized problem becomes a quadratic pencil. Bibliography: 20 titles.
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Translated from Problems in Mathematical Analysis 48, July 2010, pp. 75–107
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Nazarov, S.A., Pyatnitskii, A.L. Homogenization of the spectral Dirichlet problem for a system of differential equations with rapidly oscillating coefficients and changing sign density. J Math Sci 169, 212–248 (2010). https://doi.org/10.1007/s10958-010-0047-2
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DOI: https://doi.org/10.1007/s10958-010-0047-2