# IJPAM: Volume 77, No. 3 (2012)

A VARIATIONAL PRINCIPLE AND ITS APPLICATION

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

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

holds if and only if , that is, if for all . We define the left-hand side in (*) to be zero if . As an application of this principle it is proved that

where is the -space of real-valued functions on the connected surface of a bounded domain , and is the electrical capacitance of a perfect conductor .

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