Homogenization of Differential Operators and Integral Functionals
Homogenization of Differential Operators and Integral Functionals
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DOI：
10.1007/978-3-642-84659-5
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发表时间：
1994
期刊：
影响因子：
--
通讯作者：
V. Zhikov;S. Kozlov;O. Oleinik
中科院分区：
文献类型：
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作者：
V. Zhikov;S. Kozlov;O. Oleinik
偏微分方程的均匀化或平均化理论主要是在过去二十年中形成的一门独特的数学学科。这一理论在复合材料和多孔材料力学、过滤、分散介质以及物理学、力学和现代技术的许多分支中有许多重要的应用。关于这个问题有大量的文献。长期平均通常与方法的非线性力学和常微分方程的工作庞加莱，货车德尔波尔，克雷洛夫，Bogoliubov等长期以来，工程后的麦克斯韦和瑞利，齐次化问题的偏微分方程大多被认为是专家在物理和力学，并停留在范围之外的数学家。大量的注意力被给予了所谓的分散介质，在最简单的情况下，分散介质是由包含小的外来颗粒（晶粒、夹杂物）的主要均质材料形成的两相介质。这类两相体的尺寸远大于单个包裹体的尺寸，具有稳定的物理性质（如传热、导电等）。其不同于组成相的那些。出于这个原因，均匀化或有效这个词与这些特征有关。在诸如电磁波在小颗粒上的散射、两相介质中的有效传热等问题上，已经得到了大量的结果、近似公式和估计。
It was mainly during the last two decades that the theory of homogenization or averaging of partial differential equations took shape as a distinct mathe matical discipline. This theory has a lot of important applications in mechanics of composite and perforated materials, filtration, disperse media, and in many other branches of physics, mechanics and modern technology. There is a vast literature on the subject. The term averaging has been usually associated with the methods of non linear mechanics and ordinary differential equations developed in the works of Poincare, Van Der Pol, Krylov, Bogoliubov, etc. For a long time, after the works of Maxwell and Rayleigh, homogeniza tion problems for· partial differential equations were being mostly considered by specialists in physics and mechanics, and were staying beyond the scope of mathematicians. A great deal of attention was given to the so called disperse media, which, in the simplest case, are two-phase media formed by the main homogeneous material containing small foreign particles (grains, inclusions). Such two-phase bodies, whose size is considerably larger than that of each sep arate inclusion, have been discovered to possess stable physical properties (such as heat transfer, electric conductivity, etc.) which differ from those of the con stituent phases. For this reason, the word homogenized, or effective, is used in relation to these characteristics. An enormous number of results, approximation formulas, and estimates have been obtained in connection with such problems as electromagnetic wave scattering on small particles, effective heat transfer in two-phase media, etc.