论文标题
具有较大奇异集的非迪尼域的示例
Examples of non-Dini domains with large singular sets
论文作者
论文摘要
令$ u $为域中的非平凡谐波函数$ d \ subset \ mathbb {r}^d $，它消失在开放的边界集中。在最近的一篇论文中，我们表明，如果$ d $是一个$ c^1 $ -Dini域，则在公开内集合$ u $的单数集，定义为$ \ {x \ in \ in \ overline {d}：u（x）= 0 = | \ nabla u（x）在本文中，我们表明，$ c^1 $ -DINI域的假设是锋利的，它通过构建了一大批的非迪尼（但几乎是dini）域，其\ textIt {singular sets}具有无限$ \ Mathcal $ \ Mathcal {H}^{D-2} $ - 测量。
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.
