Martin boundary points of a John domain and unions of convex sets (Q819566)

Let \(D\) be a bounded domain in \(\mathbb{R}^{n}\) \((n\geq 2)\) and let \(K_{0}\) be a compact subset of \(D\). Then \(D\) is called a general John domain with centre \(K_{0}\) and constant \(c_{J}>0\) if each \(x\in D\) can be joined to \( K_{0}\) by a rectifiable curve \(\gamma \) such that \(\mathrm{dist}(y,\partial D)\geq c_{J}\ell (\gamma (x,y))\) for all \(y\in \gamma \), where \(\ell (\gamma (x,y))\) is the length of the subarc of \(\gamma \) from \(x\) to \(y\). The authors show that, for such a domain, the number of minimal Martin boundary points at any Euclidean boundary point is bounded by a constant depending only on \(c_{J}\). Further, if \(c_{J}>\sqrt{3}/2\), then there are at most two minimal Martin boundary points at any Euclidean boundary point. The constant \(\sqrt{3}/2\) here is sharp. For a certain class of John domains expressible as a union of convex domains a sufficient condition is given for the Martin boundary and Euclidean boundary to coincide. This well written paper generalizes work of several authors.