Harmonic functions expressible as Dirichlet solutions (Q1300300)

Let \(R\) be a bounded domain in \(\mathbb{R}^d\), where \(d\geq 2\). Further, let \(H_{qb}(R)\) denote the collection of quasi-bounded harmonic functions on \(R\), and let \(H_{ds}(R)\) denote the collection of harmonic functions on \(R\) which are expressible as the Perron-Wiener-Brelot solution to the Dirichlet problem for some resolutive function on the Euclidean boundary. It is elementary to observe that \(H_{ds}(R) \subseteq H_{qb}(R)\) for any choice of \(R\), but the inclusion may be strict. (If, instead, the Martin boundary were used, then the two collections would always be equal.) This paper shows that \(H_{ds}(R)= H_{qb}(R)\) for a large class of domains \(R\) called the ``continuous domains''. The precise definition of this concept is rather technical, but it covers star-shaped domains, Lipschitz domains and certain Hölder domains.