On relations between harmonic functions and solutions of the modified Helmholtz equation

TL;DR

This paper explores the connections between solutions of the modified Helmholtz equation and harmonic functions in bounded domains, utilizing properties of the Newtonian potential to establish new relations and characteristics.

Contribution

It introduces two novel relations linking solutions of the modified Helmholtz equation with harmonic functions, expanding understanding of their properties.

Findings

Established two key relations between modified Helmholtz solutions and harmonic functions.

Derived new characteristics of solutions using Newtonian potential properties.

Enhanced theoretical understanding of the modified Helmholtz equation in bounded domains.

Abstract

We consider the $m$-dimensional modified Helmholtz equation and establish two relations between its solutions in a bounded domain and harmonic functions. Both relations essentially rely on properties of the Newtonian potential. Some other characteristics of these solutions are also obtained.