Inverse mean value properties (a survey)

TL;DR

This survey reviews inverse mean value properties for harmonic and panharmonic functions, highlighting how these properties characterize geometric shapes like balls, annuli, and strips analytically.

Contribution

It compiles and discusses inverse mean value properties for harmonic and panharmonic functions, emphasizing their role in geometric characterization.

Findings

Inverse mean value properties characterize specific geometric domains.

Harmonic and panharmonic functions are used to analytically identify shapes.

The survey consolidates known inverse properties and their applications.

Abstract

Several mean value identities for harmonic and panharmonic functions are reviewed along with the corresponding inverse properties. The latter characterize balls, annuli and strips analytically via these functions.