Examples of non-Dini domains with large singular sets

Carlos Kenig; Zihui Zhao

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arXiv · 2212.01541

Examples of non-Dini domains with large singular sets

Carlos Kenig·Zihui Zhao

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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.

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Carlos Kenig, Zihui Zhao. 2022-12-03. Examples of non-Dini domains with large singular sets. https://arxiv.org/abs/2212.01541

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