理想导电夹杂稀随机介质的Clausius-Mossotti公式
The Clausius-Mossotti formula for dilute random media of perfectly conducting inclusions
摘要：
我们考虑了大量的随机分散的球形，相同的，完美的导电夹杂物（无限的电导率）在一个有界域。主介质的电导率是有限的，并且可以是不均匀的。在稀释极限下，在大量（与整体体积分数的-1/2次方成比例）边际概率密度的有界性假设下，我们证明了稳态热方程解的期望在H^1范数下收敛于有效介质问题的解，其中电导率由Clausius-Mossotti公式给出。还提供了误差估计。
英文摘要：
We consider a large number of randomly dispersed spherical, identical, perfectly conducting inclusions (of infinite conductivity) in a bounded domain. The host medium's conductivity is finite and can be inhomogeneous. In the dilute limit, with some boundedness assumption on a large number (proportional to the global volume fraction raised to the power of -1/2) of marginal probability densities, we prove convergence in H^1 norm of the expectation of the solution of the steady state heat equation, to the solution of an effective medium problem, where the conductivity is given by the Clausius-Mossotti formula. Error estimates are provided as well.
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