Potentials for a three-dimensional elliptic equation with one singular coefficient and their application

Abstract

A potential theory for a three-dimensional elliptic equation with one singular coefficient is considered. Double- and simple-layer potentials with unknown density are introduced, which are expressed in terms of the fundamental solution of the mentioned elliptic equation. When studying these potentials, the properties of the Gaussian hypergeometric function are used. Theorems are proved on the limiting values of the introduced potentials and their conormal derivatives, which make it possible to equivalently reduce boundary value problems for singular elliptic equations to an integral equation of the second kind, to which the Fredholm theory is applicable. The Holmgren problem is solved for a three-dimensional elliptic equation with one singular coefficient in the domain bounded \(x=0\) by the coordinate plane and the Lyapunov surface for \(x0\) as an application of the stated theory. The uniqueness of the solution to the stated problem is proved by the well-known \(abc\) method, and existence is proved by the method of the Green's function, the regular part of which is sought in the form of the double-layer potential with an unknown density. The solution to the Holmgren problem is found in a form convenient for further research.

Keywords

• three-dimensional elliptic equation with one singular coefficient
• fundamental solution
• potential theory
• green's function
• holmgren problem