A note on the Almansi property

TL;DR

This paper investigates the Almansi property across different geometric settings, establishing its uniqueness in Euclidean space, exploring its relation to biharmonic functions, and extending the analysis to semi-Euclidean spaces.

Contribution

It proves that the Almansi property holds only in Euclidean space among models, studies its relation to biharmonic functions, and extends the property to semi-Euclidean spaces.

Findings

The Almansi property is unique to Euclidean space R^m.

Connections between Almansi's property and biharmonic functions are analyzed.

Extension of the Almansi property to semi-Euclidean space R^{p,q} is provided.

Abstract

The first goal of this note is to study the Almansi property on an m-dimensional model in the sense of Greene and Wu and, more generally, in a Riemannian geometric setting. In particular, we shall prove that the only model on which the Almansi property is verified is the Euclidean space R^m. In the second part of the paper we shall study Almansi's property and biharmonicity for functions which depend on the distance from a given submanifold. Finally, in the last section we provide an extension to the semi-Euclidean case R^{p,q} which includes the proof of the classical Almansi property in R^m as a special instance.