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 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.