IJPAM: Volume 77, No. 3 (2012)


A.G. Ramm
Department of Mathematics
Kansas State University
Manhattan, KS 66506, USA

Abstract. Assume that $A$ is a bounded selfadjoint operator in a Hilbert space $H$. Then, the variational principle

\max_{v}\frac{\vert(Au,v)\vert^2}{(Av, v)}
= (Au, u) \eqno{(*)}

holds if and only if $A \geq 0$, that is, if $(Av, v)
\geq 0$ for all $v \in H$. 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_{\sigma \in L^2(S)}\frac{\vert\int_{S}\sigma(t)dt\...
...sigma (t)\sigma
(s)dsdt}{4\pi \vert s - t\vert}},\eqno{(**)}

where $L^2(S)$ is the $L^2$-space of real-valued functions on the connected surface $S$ of a bounded domain $D
\in \mathbb{R}^3$, and $C$ is the electrical capacitance of a perfect conductor $D$.

The classical Gauss' principle for electrical capacitance is an immediate consequence of (*).

Received: April 12, 2012

AMS Subject Classification: 35J05, 47A50

Key Words and Phrases: variational principle, capacitance

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Source: International Journal of Pure and Applied Mathematics
ISSN printed version: 1311-8080
ISSN on-line version: 1314-3395
Year: 2012
Volume: 77
Issue: 3