The Dirichlet problem with non-compact boundary
Let \(\Omega\) be an open set in \(\mathbb{R}^ n\) \((n \geq 2)\), let \(\partial \Omega\) denote its Euclidean boundary, let \(\partial \Omega_{\text{reg}}\) be the set of points \(Y\) in \(\partial \Omega\) at which \(R^ n \backslash \Omega\) is non-thin, and let \(\partial \Omega_{\text{irr}}=\partial \Omega \backslash \partial \Omega_{\text{reg} } \). The main result of the paper asserts that the following are equivalent: (a) for each function \(f\) in \(C(\partial \Omega)\) there is a harmonic function \(h_ f\) on \(\Omega\) which satisfies \(h_ f(X) \to f(Y)\) \((X \to Y, Y \in \partial \Omega_{\text{reg}})\) and \(\lim \sup_{X \to Y} | h_ f(X) |<\infty\) \((Y \in \partial \Omega_{\text{irr}})\); (b) for each compact set \(K\) in \(\mathbb{R}^ n\) there is a compact set \(L\) which contains all bounded components of \(\Omega \backslash K\) whose closure intersects \(K\). Uniqueness, or otherwise, of a function \(h_ f\) such as in (a) above is also examined.
scientific article; zbMATH DE number 549786
scientific article