Bloch wave homogenization and spectral asymptotic analysis
Bloch wave homogenization and spectral asymptotic analysis
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DOI：
10.1016/s0021-7824(98)80068-8
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发表时间：
1998-02-01
影响因子：
2.3
通讯作者：
Conca, C
中科院分区：
文献类型：
--
作者：
Allaire, G;Conca, C
考虑有界周期非均质介质中的二阶椭圆方程，研究其谱在结构周期趋于零时的渐近行为。我们使用了一种新的布洛赫波均匀化方法，它与经典均匀化方法不同，它表征了谱的重整化极限，即中周期平方阶的特征值序列。我们证明了这样一个重整化的极限谱由两部分组成：所谓的Bloch谱，它被明确地定义为一类极限问题的谱；以及所谓的边界层谱，它由集中在定义域边界上的特征向量序列所对应的极限特征值组成。这种分析还依赖于布洛赫测量的概念，在半经典分析的背景下，它可以被看作是临时的维格纳测量。最后，对于由整个周期性单元构成的矩形畴，布洛赫波均匀化方法的一种变体也给出了边界层谱的明确表征。(C)爱思唯尔，巴黎。
We consider a second-order elliptic equation in a bounded periodic heterogeneous medium and study the asymptotic behavior of its spectrum, as the structure period goes to zero. We use a new method of Bloch wave homogenization which, unlike the classical homogenization method, characterizes a renormalized limit of the spectrum, namely sequences of eigenvalues of the order of the square of the medium period. We prove that such a renormalized limit spectrum is made of two parts: the so-called Bloch spectrum, which is explicitly defined as the spectrum of a family of limit problems, and the so-called boundary layer spectrum, which is made of limit eigenvalues corresponding to sequences of eigenvectors concentrating on the boundary of the domain. This analysis relies also on a notion of Bloch measures which can be seen as ad hoc Wigner measures in the context of semi-classical analysis. Finally, for rectangular domains made of entire periodicity cells, a variant of the Bloch wave homogenization method gives an explicit characterization of the boundary layer spectrum too. (C) Elsevier, Paris.