Examples of non-Dini domains with large singular sets

TL;DR

This paper constructs examples of non-Dini domains where the singular set of harmonic functions has infinite Hausdorff measure, demonstrating the sharpness of previous regularity results.

Contribution

It provides explicit examples of non-Dini domains with large singular sets, showing the boundary regularity condition is optimal.

Findings

Constructed non-Dini domains with infinite singular set measure

Demonstrated sharpness of regularity assumptions in harmonic analysis

Extended understanding of boundary behavior of harmonic functions

Abstract

Let $u$ be a non-trivial harmonic function in a domain $D\subset \mathbb{R}^d$ which vanishes on an open set of the boundary. In a recent paper, we showed that if $D$ is a $C^1$-Dini domain, then within the open set the singular set of $u$, defined as $\{X\in \overline{D}: u(X) = 0 = |\nabla u(X)|\} $, has finite $(d-2)$-dimensional Hausdorff measure. In this paper, we show that the assumption of $C^1$-Dini domains is sharp, by constructing a large class of non-Dini (but almost Dini) domains whose \textit{singular sets} have infinite $\mathcal{H}^{d-2}$-measures.