Averaging of dilute random media: a rigorous proof of the Clausius-Mossotti formula (Q1944730)

An \(H^1\)-error estimate between the physical fields \(\phi_\varepsilon\) and an average one is established via the derivations of estimates for solutions of the one- and two-particle problems. For each \(\varepsilon>0\), which represents the radius of a finite number of spherical particles of conductivity \(\sigma\) immersed into a medium of conductivity 1, \(\phi_\varepsilon\) is the unique weak solution to a Dirichlet problem: \(\nabla\cdot(a_\varepsilon\nabla\phi)=0\) in a bounded domain \(\Omega\subset\mathbb{R}^3\) of class \(C^{2,\alpha}\), for some \(\alpha>0\), with the input data belonging to \(C^{2,\alpha}(\partial\Omega)\). Here, the coefficient \(a_\varepsilon\) is given by the Clausius-Mossotti approximation. The proof is based on the potential theory.