An Extremal Problem Involving Current Flow Through Distributed Resistance

We treat essentially the following problem in the context of electrostatics: Given a compact, convex set $Q \subset \mathbb{R}^{2} $ (of positive area), which is perfectly conducting and held at potential 1, how should a total amount $A > 0$ of resistance be distributed in ${{\mathbb{R}^2 }\setminus Q}$ (subject to an upper bound on resistivity) in order that the flow of current from Q to t (assumed to have potential 0) be minimized?