Malmheden's theorem revisited

TL;DR

This paper revisits Malmheden's theorem, providing an alternative proof that extends to polyharmonic functions and explores applications to harmonic measures in Euclidean balls.

Contribution

It offers a new proof of Malmheden's theorem, generalizes it to polyharmonic functions, and discusses its implications for harmonic measure properties.

Findings

Alternative proof of Malmheden's theorem

Extension to polyharmonic functions

Applications to harmonic measures

Abstract

In 1934 H. Malmheden discovered an elegant geometric algorithm for solving the Dirichlet problem in a ball. Although his result was rediscovered independently by Duffin 23 years later, it still does not seem to be widely known. In this paper we return to Malmheden's theorem, give an alternative proof of the result that allows generalization to polyharmonic functions and, also, discuss applications of his theorem to geometric properties of harmonic measures in balls in Euclidean spaces.