Potentials for a multidimensional elliptic equation with one line of degeneration and their applications to boundary value problems

TL;DR

This paper develops potential theory for a multidimensional elliptic equation with a line of degeneration, deriving integral equations and limit theorems to advance boundary value problem solutions.

Contribution

It introduces new potential functions for a multidimensional elliptic equation with a degeneration line and establishes related integral equations and limit theorems.

Findings

Derived integral equations with potential densities

Proved limit theorems for potentials

Extended potential theory to multidimensional equations with degeneracy

Abstract

Potentials play an important role in solving boundary value problems for elliptic equations. In the middle of the last century, a potential theory was constructed for a two-dimensional elliptic equation with one singular coefficient. In the study of potentials, the properties of the fundamental solutions of the given equation are essentially used. At the present time, fundamental solutions of a multidimensional elliptic equation with one degeneration line are already known. In this paper, we investigate the potentials of the double- and simple-layers for this equation, with the help of which limit theorems are proved and integral equations containing in the kernel the density of the above potentials are derived.