Proof of a conjecture on the conductivity of checkerboards

In 1985 Mortola and Steffe conjectured a formula for the effective conducti vity tensor of a checkerboard structure where the unit cell of periodicity is square and subdivided into four equal squares each having a different co nductivity. In this article their conjecture is proven. The key idea is to superimpose suitably reflected potentials to obtain the solution to the dua l problem. This is then related back to the original problem using a well k nown theorem of Keller, thereby proving the conjecture. The analysis also y ields formulas relating the potentials in the four squares. Independently, Craster and Obnosov have obtained a completely different proof of the conje cture. (C) 2001 American Institute of Physics.