Extremal length and rayleigh's reciprocal theorem

Abstract

In a well-known paper R.J. Duffin [J. Math. Anal. Appl., 5 (1962) 200-215] extends the notion of extremal length to electrical networks, by allowing variable specific resistivity along the arcs of the network, and shows extremal length and extremal width are reciprocals for planar networks yielding Rayleigh's Theorem that conjugate* conductors have reciprocal resistances, Similar results are obtained by F. W. Gehring [Mich. Math. J., 9 (1962), 137-150] for continuous conductors with constant specific resistivity in 3-space, This paper unifies the continuous and discrete cases by obtaining the Reciprocity Theorem for p-capacities in n-space provided the specific resistivity satisfies certain realistic smoothness conditions, It follows that Rayleigh's Theorem holds for nonptanar networks confirming a conjecture of Duffin