Examples of non-Dini domains with large singular sets

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.