Dirichlet finite harmonic measures on topological balls. (English) Zbl 0959.31005

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.


31B05 Harmonic, subharmonic, superharmonic functions in higher dimensions
31B25 Boundary behavior of harmonic functions in higher dimensions
31B15 Potentials and capacities, extremal length and related notions in higher dimensions
30F20 Classification theory of Riemann surfaces
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