On characterization of balls via solutions to the Helmholtz equation

TL;DR

This paper introduces a novel analytical method to characterize Euclidean balls using solutions to the Helmholtz equation, expanding the tools available for geometric analysis despite domain size restrictions.

Contribution

It provides a new characterization of Euclidean balls based on Helmholtz equation solutions, differing from prior harmonic or modified Helmholtz approaches.

Findings

Characterizes Euclidean balls via Helmholtz solutions

Introduces domain size restrictions in the characterization

Extends inverse mean value property techniques

Abstract

A new analytical characterization of balls in the Euclidean space $\RR^m$ is obtained. Unlike previous results of this kind, using either harmonic functions or solutions to the modified Helmholtz equation, the present one is based on solutions to the Helmholtz equation. This is achieved at the expense of a restriction imposed on the size of a domain -- a feature absent in the inverse mean value properties known before.