A variational principle and its application (Q2913912)

This paper gives a way to estimate \((Au,u)\) for a positive and selfadjoint operator \(A\) on a Hilbert space \(H\). The main result in the paper is that, for a linear bounded selfadjoint operator \(A\), \((Au,u)=\max_{v \in H} \frac{{|(Av,u)|}^2}{(Av,v)}\) holds if and only if \(A\) is non-negative. As an immediate consequence, it is shown that the electrical capacitance \(C\) of a perfect conductor \(D\) is evaluated by NEWLINE\[NEWLINEC=\max_{\sigma \in L^2(S)} \frac{|\int_S \sigma(t) \,dt|^2}{\int_S\int_S \frac{\sigma(t)\sigma(s)\, ds\,dt}{4\pi|s-t|}},NEWLINE\]NEWLINEwhere \(L^2(S)\) is the \(L^2\)-space of real-valued functions on the connected surface \(S\) of a bounded domain \(D \subset \mathbb{R}^3\).