The Martin kernel for unbounded domains
Let \((x_{1},\widetilde{x})\) denote a typical point of \(\mathbb{R}\times \mathbb{R}^{d-1}\), where \(d\geq 3\), and let \(a:[0,\infty )\rightarrow (0,\infty )\) be Lipschitz. This paper is concerned with unbounded domains \(D\) of the form \(\{(x_{1},\widetilde{x}):x_{1}>0,\;\left| \widetilde{x}\right| <a(x_{1})\}\). Under certain hypotheses on \(a(\cdot )\) the author establishes: (i)~that the Martin boundary of \(D\) is homeomorphic to \(\partial D\cup \{\infty \}\) and all boundary points are minimal; (ii)~the asymptotic behaviour of the Martin kernel with pole at \(\infty \) in terms of the function \(a(\cdot )\).
scientific article; zbMATH DE number 5693237
scientific article