A Free Boundary Optimization Problem. II

Given a compact set $Q \subset R^2 $, a function $a(p) > 0$ continuous on $R^2$, and a sufficiently large constant $A > 0$, we determine (under suitable assumptions) the doubly-connected region $\Omega \subset R^2 $ encircling (but not intersecting) Q which has the least capacitance subject to the constraint that $|\Omega |: = \int \int _\Omega a^2 (p)dxdy \leqq A$.