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First-order corrections to the homogenized eigenvalues of a periodic composite medium. (English) Zbl 0808.35085
SIAM J. Appl. Math. 53, No. 6, 1636-1668 (1993); erratum ibid. 55, No. 3, 864 (1995).
The authors study the eigenvalues of the Dirichlet boundary value problem associated to the operator $$\nabla\cdot a(x/\varepsilon)\nabla$$ and to a convex Lipschitz open set $$\Omega$$ of $$\mathbb{R}^ n$$. The function $$a$$ is assumed to be smooth and periodic, and it is shown that if an eigenvalue $$\lambda^ \varepsilon_ .$$ has some limit $$\lambda^ 0$$ as $$\varepsilon$$ tends to $$0_ +$$, then the behavior of the ratio $$r(\varepsilon)= (\lambda^ \varepsilon- \lambda^ 0)/ \varepsilon$$ is very sensitive to the way $$\varepsilon$$ approaches 0. More precisely, a set of values is explicited, for which there exists a sequence $$(\varepsilon_ k)$$ tending to zero such that the value is the limit of $$r(\varepsilon_ k)$$. Further precisions are given in the case of polygons, as well as numerical computations.

##### MSC:
 35P15 Estimates of eigenvalues in context of PDEs 35B27 Homogenization in context of PDEs; PDEs in media with periodic structure 74E05 Inhomogeneity in solid mechanics
##### Keywords:
homogenization; Dirichlet conditions; periodic medium
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