Isoperimetric inequalities for an electrostatic problem

We study the problem of the (p-)capacity cp of a multiconnected configuration Ω = (G E) (∪ Hi) when ∂G and ∂E have given potentials. Here Ω represents a nonhomogeneous medium and the Hi, which separate the different connected components of Ω, represent perfect conductors. By comparison with a similar configuration with spherical symmetry, we give isoperimetric inequalities for cp and the unknown potentials on Hi