Proof of a conjecture on the conductivity of checkerboards
Graeme W. Milton
Source abstract
In 1985 Mortola and Steffé conjectured a formula for the effective conductivity tensor of a checkerboard structure where the unit cell of periodicity is square and subdivided into four equal squares each having a different conductivity. In this article their conjecture is proven. The key idea is to superimpose suitably reflected potentials to obtain the solution to the dual problem. This is then related back to the original problem using a well known theorem of Keller, thereby proving the conjecture. The analysis also yields formulas relating the potentials in the four squares. Independently, Craster and Obnosov have obtained a completely different proof of the conjecture.
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