具有大奇异集的非Dini域的例子
Examples of non-Dini domains with large singular sets
摘要：
设$u$是定义域$D\subset \mathbb{R}^d$中的非平凡调和函数，该函数在边界的开集上为零.在最近的一篇论文中，我们证明了如果$D$是$C^1$ -Dini整环，则在开集内，定义为$\{X\in \overline{D}: u(X) = 0 = |\nabla u(X)|\} $的$u$的奇异集具有有限的$(d-2)$维Hausdorff测度.本文通过构造一类非Dini（但几乎Dini）域，证明了$C^1$ -Dini域的假设是尖锐的，这类域的\textit{singular sets}具有无穷多个$\mathcal{H}^{d-2}$ -测度.
英文摘要：
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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
