Mean value properties of solutions to the Helmholtz and modified Helmholtz equations

TL;DR

This paper explores mean value properties of solutions to Helmholtz and modified Helmholtz equations, providing elementary derivations and highlighting differences in their implications for bounded domains.

Contribution

It offers a new elementary derivation of mean value properties using the Euler--Poisson--Darboux equation and clarifies how these properties differ between the two equations.

Findings

Elementary derivation of mean value properties for both equations

Distinction in consequences of mean value properties for the two equations

Modified mean value property characterizes solutions to the modified Helmholtz equation

Abstract

Mean value properties of solutions to the $m$-dimensional Helmholtz and modified Helmholtz equations are considered. An elementary derivation of these properties is given; it involves the Euler--Poisson--Darboux equation. Despite the similar form of these properties for both equations, their consequences distinguish essentially. The restricted mean value property for harmonic functions is amended so that a function, satisfying it in a bounded domain of a special class, solves the modified Helmholtz equation in this domain.