Homogenization of a class of linear partial differential equations
Homogenization of a class of linear partial differential equations
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一类线性偏微分方程的齐次化
DOI：
10.3233/asy-2012-1145
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发表时间：
2013
期刊：
影响因子：
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通讯作者：
M. Waurick
中科院分区：
文献类型：
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作者：
M. Waurick
给出了一类数学物理发展方程均匀化的统一Hilbert空间观点。我们制定一个纯粹的运营商理论设置均匀化。利用R. Picard（Mathematical Methods in the Applied Sciences 32（2009），1768-1803）中，我们讨论了作为有界、解析和算子值函数M：E → L（H）的哈代空间H ∞（E; L（H））的某些元素的本构关系，其中E C开，H Hilbert空间。其核心思想是在本构关系集上引入一定的拓扑。给出一个收敛序列的本构关系，相应的问题的解决方案的行为进行了讨论。我们将结果应用于声学、热力学、弹性力学或耦合系统（如热弹性）的方程。相应的方程还可以包含记忆或延迟项和分数导数。特别是，本构关系通过微分方程也可以处理。
We present a unified Hilbert space perspective to homogenization of a class of evolutionary equations of mathematical physics. We formulate homogenization in a purely operator-theoretic setting. Using "A structural observation for linear material laws in classical mathematical physics" by R. Picard (Mathematical Methods in the Applied Sciences 32 (2009), 1768-1803), we discuss constitutive relations as certain elements of the Hardy space H ∞ (E; L(H)) of bounded, analytic and operator- valued functions M : E → L(H), where E C open, H Hilbert space. The core idea is to introduce a certain topology on the set of constitutive relations. Given a convergent sequence of constitutive relations, the behavior of solutions to the respective problems is discussed. We apply the results to the equations of acoustics, thermodynamics, elasticity or coupled systems such as thermo-elasticity. The respective equations may also incorporate memory or delay terms and fractional derivatives. In particular, constitutive relations via differential equations can also be treated.