The Clausius--Mossotti Formula for Dilute Random Media of Perfectly Conducting Inclusions
Yaniv Almog · SIAM Journal on Mathematical Analysis · 2017
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.