Concavity of Condenser Energy Under Boundary Variations

Abstract

Let $D_{0}$, $D_{1}$ be two bounded domains in ${\mathbb {R}}^{n}$, $n\ge 2$, such that $\overline{D_{0}}\subset D_{1}$ and $\partial D_{0}$ and $\partial D_{1}$ are closed surfaces. Consider a variation of $D_{0}$ to $D_{1}$ via a family of smooth domains $D_{t}$, $t\in (0,1)$, whose boundaries $\partial D_{t}$ are level sets of a $C^{2}$ function V on $D_{1}\setminus D_{0}$. Let K be an arbitrary compact subset of $D_{0}$ and let $I(D_{t},K)$ be the equilibrium energy of the condenser $(D_{t},K)$. We show that the function $f(t):=I(D_{t},K)$ is continuously differentiable. In addition, we show that, if V is subharmonic, then f is a concave function. We characterize the cases where f is affine by showing that this occurs if and only if $\partial D_{t}$ are level sets of the equilibrium potential of the condenser $(D_{1},K)$. This is a generalization of a result obtained by R. Laugesen [14] when the domains $D_{t}$ are concentric balls.