Two-scale homogenization for a general class of high contrast PDE systems with periodic coefficients
Two-scale homogenization for a general class of high contrast PDE systems with periodic coefficients
复制标题
具有周期系数的一般类高对比度偏微分方程系统的两尺度均质化
DOI：
10.1080/00036811.2018.1441994
复制
发表时间：
2013
影响因子：
1.1
通讯作者：
V. Smyshlyaev
中科院分区：
文献类型：
--
作者：
I. Kamotski;V. Smyshlyaev
摘要对于一类具有临界尺度的小周期高对比度周期系数的渐近退化强椭圆对称偏微分方程系统的双尺度均匀化问题，我们在对刚性部分的一般分解假设下得到了一个双尺度极限预解问题。我们表明，这一关键假设确实适用于大量具有高对比度的示例，无论是之前研究的还是最近研究的一些示例，包括线性弹性和电磁学中的示例。根据V.V.Zhikov的思想，在对域的正则性的非常温和的限制下，我们证明了极限预解式问题是适定的，并且证明了它是一个定义良好的非负自伴双尺度极限算子的伪预解式问题。这里的一个关键的新的技术成分是证明产品测试功能的线性跨度在相应的功能空间的退化是密集的相关的两个尺度之间的一般耦合的能量空间。其结果是，我们建立（弱和强）两个规模的预解收敛，以及它的一些进一步的影响的谱收敛和收敛的抛物和双曲半群和相关的时间依赖的初始边值问题。这项工作的一些结果在Kamotski IV，Smyshlyaev VP中宣布。一类部分退化PDE系统的双尺度均匀化。arXiv：1309.4579v1。2013年（https：//arxiv.org/abs/1309.4579v1）。
ABSTRACT For two-scale homogenization of a general class of asymptotically degenerating strongly elliptic symmetric PDE systems with a critically scaled high contrast periodic coefficients of a small period , we derive a two-scale limit resolvent problem under a single generic decomposition assumption for the ‘stiff’ part. We show that this key assumption does hold for a large number of examples with a high contrast, both studied before and some recent ones, including those in linear elasticity and electromagnetism. Following ideas of V. V. Zhikov, under very mild restrictions on the regularity of the domain we prove that the limit resolvent problem is well-posed and turns out to be a pseudo-resolvent problem for a well-defined non-negative self-adjoint two-scale limit operator. A key novel technical ingredient here is a proof that the linear span of product test functions in the functional spaces corresponding to the degeneracies is dense in associated two-scale energy space for a general coupling between the scales. As a result, we establish (both weak and strong) two-scale resolvent convergence, as well as some of its further implications for the spectral convergence and for convergence of parabolic and hyperbolic semigroups and of associated time-dependent initial boundary value problems. Some of the results of this work were announced in Kamotski IV, Smyshlyaev VP. Two-scale homogenization for a class of partially degenerating PDE systems. arXiv:1309.4579v1. 2013 (https://arxiv.org/abs/1309.4579v1).