On discontinuous solutions of the integral equations of electrostatics

Abstract

This paper is concerned with the two fundamental integral equations which are satisfied by the surface-charge density on a conductor. One is an integral equation of the first kind, which we call the first integral equation of electrostatics, and the other a Fredholm equation, called the second integral equation of electrostatics, or Robin's integral equation. Existence theory for the latter shows that, for the class of globally twice continuously differentiable surfaces, this equation possesses continuous solutions, and with this knowledge one deduces that both equations have a common solution which is unique in the class of continuous functions. The purpose of this paper is to examine the existence question outside of the class of continuous functions and in a much broader class described only by a general integrability condition. Two theorems are proved, one for each of the respective equations. The first shows that no further solutions of Robin's equation are admitted. The second shows (under conditions not quite so general) that this is nearly so in the case of the first integral equation, in the sense that solutions other than the continuous one are of no physically relevant difference.