Concavity of Condenser Energy Under Boundary Variations

Let D, D₁ be two bounded domains in Rⁿ, n≥ 2 , such that D¯ ⊂ D₁ and ∂D and ∂D₁ are closed surfaces. Consider a variation of D to D₁ via a family of smooth domains D_t, t∈ (0 , 1) , whose boundaries ∂D_t are level sets of a C² function V on D₁\ D. Let K be an arbitrary compact subset of D 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 ∂D_t are level sets of the equilibrium potential of the condenser (D₁, K). This is a generalization of a result obtained by R. Laugesen [14] when the domains D_t are concentric balls.