Bounds on the effective behaviour of a square conducting lattice

A collection of resistors with two possible resistivities is considered. This paper investigates the overall or macroscopic behaviour of a square two-dimensional lattice of such resistors when both types coexist in fixed proportions in the lattice. The macroscopic behaviour is that of an anisotropic conductor at the continuum level and the goal of the paper is to describe the set of all possible such conductors. This is thus a problem of bounds, following in the footsteps of an abundant literature on the topic in the continuum case. The originality of the paper is that the investigation focuses on the interplay between homogenization and the passage from a discrete network to a continuum. A set of bounds is proposed and its optimality is shown when the proportion of each resistor on the discrete lattice is 1/2. We conjecture that the derived bounds are optimal for all proportions.