On estimation of electrostatic capacity

Abstract

1. Introduction. In this paper we prove, by means of Hadamard's variational formula, several monotonicity theorems for the electrostatic capacity of a solid or a lamina. From these monotonicity theorems follow some well known isoperimetric inequalities for the capacity which are, however, not such strong statements as the monotonicity theorems themselves. A point of interest in the development as we present it here is that, up to this time, it has rarely been found possible to solve extremal problems and obtain isoperimetric inequalities directly using only Hadamard's formula, since the validity of the formula depends on piece-wise smoothness of the boundary of the domain under consideration. 2. Volume and capacity. Let D be a finite 3-dimensional domain including the origin in its interior and bounded by a piece-wise smooth surface S. We denote by G =-t + 0(r) r the Green's function of D for Laplace's equation, with a pole at the origin, where r is the distance from the origin. We shall call 7 the inverse capacity of D with respect to the origin, and we shall denote by V the volume of D. Obviously, 7-1 has the dimension of length. Let a deformation of .S be described by an infinitesimal shift of 5 through 5w units along its inner normal n; we obtain the well known Hadamard variational formulas [2]