On the equilibrium measure and the capacity of certain condensers (Q1584499)

Let \(R\) be a domain in \({\mathbf C}_\infty= {\mathbf C}\cup\{\infty\}\) whose complement is the disjoint union of non-empty compact sets \(A\) and \(B\). The corresponding condenser capacity \(\text{cap}(R, A,B)\) is defined to be \(\inf_u \int_R|u|^2\), where the infimum is taken over all continuously differentiable functions \(u\) on \(R\) with boundary values \(0\) on \(A\) and \(1\) on \(B\). This paper studies \(\text{cap}(R, A,B)\), and the equilibrium measure of \((R,A,B)\), for two specific types of condenser. One of the main results is as follows. Let \(0\leq \rho\leq b_1\leq b_1'\leq b_2'\leq b_2< a_1< a_2< \infty\), let \(A= [-a_2, -a_1]\cup [a_1, a_2]\) and \(B= [-b_2', -b_1']\cup [b_1, b_2]\). Also, for each \(\theta\) in \([0,\pi/2]\), let \(R_\theta={\mathbf C}_\infty\setminus (A\cup B_\theta)\), where \(B_\theta\) is the union of \(\{e^{i\theta}x: x\in B\}\) and \(\{z:|z|\leq \rho\}\). Then \(\text{cap}(R_\theta, A,B_\theta)\) is strictly decreasing as a function of \(\theta\) in \([0, \pi/2]\).