The Dirichlet problem for harmonic functions on compact sets (Q411895)

This article presents an account of the Dirichlet problem with respect to a compact set \(K\) in \(\mathbb{R}^{n}\). The fine topology of classical potential theory plays an essential role, as does the notion of a finely harmonic function on a finely open set. Let \(\partial _{f}K\) denote the fine boundary of \(K\), and \(fH^{c}(K)\) be the collection of finely continuous functions on \(K\) that are finely harmonic on the fine interior of \(K\) and continuous and bounded on \(\partial _{f}K\). Theorem 6.5 says that, for every bounded continuous function \(\phi \) on \(\partial _{f}K\), there is a unique function \(h_{\phi }\in fH^{c}(K)\) that equals \(\phi \) on \(\partial _{f}K\). Further, for each \(x\in K\), the value of \(h_{\phi }(x)\) is given by the integral of \(\phi \) against a certain probability measure on \(\partial _{f}K\).