On the Optimal Conductivity of Packed Two-Dimensional Dispersed Composites.

Optimal packing is applied in physics for investigation of macroscopic properties of composites and porous media. The present paper demonstrates that the optimal packing of perfectly conducting disks on the plane corresponds to the minimal effective conductivity for macroscopically isotropic composites. Such an optimal location is studied and compared to the pure geometric problem for N \leq 8 disks packing on the flat torus. The hexagonal (triangular) array of disks solves the problem for the triangle lattice numbers N = 1,3, 4, 7, .... The rest of the cases N = 2,5,6,8 are investigated separately by minimization of structural sums constructed by means of the Eisenstein functions.