Dirichlet finite harmonic measures on topological balls (Q1585900)

Let \(\Omega\) be a bounded domain in \(\mathbb{R}^d\) and let \(\omega^\Omega_E\) denote the harmonic measure of a boundary set \(E\). We say that \(\Omega\in O_{HmD}\) if \(\int|\nabla \omega^\Omega_E|^2= +\infty\) whenever \(E\subseteq \partial\Omega\) and \(\omega^\Omega_E\) is nonconstant on \(\Omega\). It is known that the unit ball is in \(O_{HmD}\). When \(d= 2\), conformal mapping arguments show that any Jordan domain belongs to \(O_{HmD}\). When \(d= 3\), physical intuition suggests that any topological ball \(\Omega\) should belong to \(O_{HmD}\). Previous work of the author showed that this is the case under the additional assumption that \(\Omega\) is Lipschitz. Contrary to what one might expect the author now constructs, for each \(d\geq 3\), a topological ball in \(\mathbb{R}^d\) that does not belong to \(O_{HmD}\). The paper is clearly written.