Isotropic conductivity of two-dimensional three- and four-phase symmetric composites: duality and universal bounds

We consider the problem of isotropic effective conductivity $\sigma_e(\sigma_1,\ldots,\sigma_n)$ in two-dimensional three- and four-phase symmetric composites with a partial isotropic conductivity $\sigma_j$ of the $j$-th phase. The upper $\Omega(\sigma_1,\ldots,\sigma_n)$ and lower $\omega(\sigma_1,\ldots,\sigma_n)$, $n=3,4$, bounds for effective conductivity, found by the algebraic approach, are universal (independent of the composite micro-structure) and possess all algebraic properties of $\sigma_e(\sigma_1,\ldots,\sigma_n)$ that follow from physics: first-order homogeneity, full permutation invariance, Keller's self-duality, positivity, and monotony. The bounds are compatible with the trivial solution $\sigma_e(\sigma,\ldots,\sigma)=\sigma$ and satisfy Dykhne's ansatz. Their comparison with previously known numerical calculations, asymptotic analysis, and exact results for isotropic effective conductivity $\sigma_e(\sigma_1,\ldots,\sigma_n)$ of two-dimensional three- and four-phase composites showed complete agreement. The bounds $\Omega(\sigma_1,\ldots,\sigma_n)$ and $\omega(\sigma_1,\ldots,\sigma_n)$ in both cases $n=3,4$ are stronger than the currently known variational bounds.