Boundary multifractal behaviour for harmonic functions in the ball

TL;DR

This paper investigates the boundary behaviour of harmonic functions in the ball, focusing on the size and divergence rate of exceptional boundary sets where limits do not exist, revealing a multifractal structure.

Contribution

It establishes the Hausdorff dimension of boundary point sets where harmonic functions exhibit specific divergence rates, demonstrating a boundary multifractal phenomenon.

Findings

Hausdorff dimension of divergence sets is d - β for divergence rate β

Generic harmonic functions exhibit multifractal boundary behaviour

Results extend understanding of boundary limits for harmonic functions

Abstract

It is well known that if $h$ is a nonnegative harmonic function in the ball of $\RR^{d+1}$ or if $h$ is harmonic in the ball with integrable boundary values, then the radial limit of $h$ exists at almost every point of the boundary. In this paper, we are interested in the exceptional set of points of divergence and in the speed of divergence at these points. In particular, we prove that for generic harmonic functions and for any $\beta\in [0,d]$, the Hausdorff dimension of the set of points $\xi$ on the sphere such that $h(r\xi)$ looks like $(1-r)^{-\beta}$ is equal to $d-\beta$.