Examples of non-Dini domains with large singular sets

Abstract

Let uu be a nontrivial harmonic function in a domain D⊂RdD\subset {{\mathbb{R}}}^{d}, which vanishes on an open set of the boundary. In a recent article, we showed that if DD is a C1{C}^{1}-Dini domain, then, within the open set, the singular set of uu, defined as {X∈D¯:u(X)=0=∣∇u(X)∣}\left\{X\in \overline{D}:u\left(X)=0=| \nabla u\left(X)| \right\}, has finite (d−2)\left(d-2)-dimensional Hausdorff measure. In this article, we show that the assumption of C1{C}^{1}-Dini domains is sharp, by constructing a large class of non-Dini (but almost Dini) domains whose singular sets have infinite ℋd−2{{\mathcal{ {\mathcal H} }}}^{d-2}-measures.

Keywords

• boundary unique continuation
• harmonic measures
• singular sets of harmonic functions
• conformal mapping
• 35j25
• 42b37
• 31b35