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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
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