Inverse mean value property of solutions to the modified Helmholtz equation

TL;DR

This paper proves a theorem characterizing Euclidean balls using solutions to the modified Helmholtz equation, extending previous results based on harmonic functions by utilizing the volume mean value property of these solutions.

Contribution

It introduces a new characterization of Euclidean balls through the inverse mean value property of solutions to the modified Helmholtz equation, generalizing classical harmonic function results.

Findings

Characterization of Euclidean balls via solutions to the modified Helmholtz equation

Establishment of a Kuran type theorem based on volume mean value property

Extension of classical harmonic function results to modified Helmholtz solutions

Abstract

A theorem characterizing analytically balls in the Euclidean space $\RR^m$ is proved. For this purpose positive solutions of the modified Helmholtz equation are used instead of harmonic functions applied in previous results. The obtained Kuran type theorem is based on the volume mean value property of solutions to this equation.