A variational principle and its application

Assume that A is a bounded selfadjoint operator in a Hilbert space H. Then, the variational principle max v |(Au, v)| (Av, v) = (Au, u) (*) holds if and only if A ≥ 0, that is, if (Av, v) ≥ 0 for all v ∈ H. We define the left-hand side in (*) to be zero if (Av, v) = 0. As an application of this principle it is proved that C = max v∈L2(S) | ∫ S vdt| ∫ S ∫ S v(t)v(s)dsdt 4π|s−t| , (**) where L(S) is the L-space of real-valued functions on the connected surface S of a bounded domain D ∈ R, and C is the electrical capacitance of a perfect conductor D. The classical Gauss’ principle for electrical capacitance is an immediate consequence of (*).