Asymptotic mean value properties of meta- and panharmonic functions

TL;DR

This paper investigates the asymptotic mean value properties of solutions to the Helmholtz equation and its modified form, revealing unique properties not found in harmonic functions.

Contribution

It introduces new asymptotic mean value properties for metaharmonic and panharmonic functions, expanding understanding beyond classical harmonic functions.

Findings

Some properties have no analogues for harmonic functions

Asymptotic mean value properties are established for metaharmonic functions

Results include converse and related properties for these functions

Abstract

Asymptotic mean value properties, their converse and some other related results are considered for solutions to the $m$-dimensional Helmholtz equation (metaharmonic functions) and solutions to its modified counterpart (panharmonic functions). Some of these properties have no analogues for harmonic functions.