A Variational Principle and Its Application to Estimating the Electrical Capacitance of a Perfect Conductor

Assume that A is a bounded self-adjoint operator in a Hilbert space H. Then, the variational principle max v |(Au, v)|2 (Av, v) = (Au, u) (*) holds if and only if A ≥ 0, that is, if (Av, v) ≥ 0 for all v ∈ H. We define the quotient on the left-hand side in (*) to be zero if (Av, v) = 0. As an application of this principle a variational principle for the electrical capacitance of a conductor of an arbitrary shape is derived.