## Abstract

Let D, D_{1} be two bounded domains in R^{n}, n≥ 2 , such that D¯ ⊂ D_{1} and ∂D and ∂D_{1} are closed surfaces. Consider a variation of D to D_{1} via a family of smooth domains D_{t}, t∈ (0 , 1) , whose boundaries ∂D_{t} are level sets of a C^{2} function V on D_{1}\ 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 (D_{1}, K). This is a generalization of a result obtained by R. Laugesen [14] when the domains D_{t} are concentric balls.

Original language | English |
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Pages (from-to) | 7726-7740 |

Number of pages | 15 |

Journal | Journal of Geometric Analysis |

Volume | 31 |

Issue number | 8 |

DOIs | |

State | Published - Aug 2021 |

## Keywords

- Capacity constant
- Condenser energy
- Harmonic radius
- Parametric deformation