通过双尺度均匀化研究高对比度周期性介质中缺陷引起的局域模
Localised modes due to defects in high contrast periodic media via two-scale homogenization
摘要：
研究了受紧缺陷扰动的无限大周期介质的谱问题。对于高对比度小$\ve$尺寸周期性和有限尺寸缺陷，我们考虑对比度的临界$\ve^2$缩放。我们采用（高对比度）两个尺度的均匀化渐近显式极限方程的本地化模式（指数衰减的本征函数）和相关的本征值。这些都表示的特征值和特征函数的扰动版本的两个规模的极限运营商介绍了由V. V.日科夫与紧急显式非线性依赖的谱参数的谱问题在宏观尺度上。利用高对比度边界层分析补充的渐近展开法，我们建立了极限算子的本征值附近的实际本征值的存在性，与%紧“$ \ve$平方根“误差界。给出了具有均匀各向同性性质的周期介质中圆形或球形缺陷的例子，并显示了显式的极限本征值和本征函数。将我们的结果与M. Cherdantsev（{ \it Mathematika. 2009;55：29--57}）基于强双尺度预解式收敛技术和相关联的双尺度紧致性。
英文摘要：
The spectral problem for an infinite periodic medium perturbed by a compact defect is considered. For a high contrast small $\ve$-size periodicity and a finite size defect we consider the critical $\ve^2$-scaling for the contrast. We employ (high contrast) two-scale homogenization for deriving asymptotically explicit limit equations for the localised modes (exponentially decaying eigenfunctions) and associated eigenvalues. Those are expressed in terms of the eigenvalues and eigenfunctions of a perturbed version of a two-scale limit operator introduced by V.V. Zhikov with an emergent explicit nonlinear dependence on the spectral parameter for the spectral problem at the macroscale. Using the method of asymptotic expansions supplemented by a high contrast boundary layer analysis we establish the existence of the actual eigenvalues near the eigenvalues of the limit operator, with %tight "$ \ve$ square root" error bounds. An example for circular or spherical defects in a periodic medium with isotropic homogenized properties is given and displays explicit limit eigenvalues and eigenfunctions. Further results on improved error bounds for the eigenfunctions are discussed, by combining our results with those of M. Cherdantsev ({\it Mathematika. 2009;55:29--57}) based on the technique of strong two-scale resolvent convergence and associated two-scale compactness properties.
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