Harmonic approximation and its applications (Q2758408)

This article is a survey originally delivered by the author as a series of four lectures in July 2000. Almost all of the work discussed is due to the author, and an account of results obtained before 1995 is given in his monograph [Harmonic approximation, London Math. Soc. Lecture Note Ser. 221, Cambridge Univ. Press, Cambridge (1995; Zbl 0826.31002)]. The topics treated are briefly indicated below. NEWLINENEWLINENEWLINELecture 1. Let \(\Omega\) be an open subset of \(\mathbb{R}^n\) \((n\geq 2)\) and let \(E\) be a relatively closed subset of \(\Omega\). For a subset \(A\) of \(\mathbb{R}^n\), let \({\mathcal H}(A)\) denote the space of functions that are hamonic on some open set containing \(A\). Under which conditions on \(\Omega\) and \(E\) is it possible to approximate functions in \(C(E)\cup{\mathcal H}(E^0)\) by functions in \({\mathcal H}(\Omega)\), the approximation being either uniform on \(E\) or (if \(E\) is non-compact) tangential in some sense? Precise answers are given; the conditions involve both the Euclidean topology and the fine topology. NEWLINENEWLINENEWLINELecture 2. Harmonic approximation is used to answer an old question about the Dirichlet problem: which open sets \(\Omega\) have the property that for each \(f\in C(\partial \Omega)\) there exists \(h_f\in {\mathcal H}(\Omega)\) such that \(h_f(x)\to f(y)\) as \(x\to y\) for every regular \(y\in\partial \Omega\) and \(h_f\) is bounded near every irregular \(y\in \partial \Omega\)? Here \(\partial \Omega\) is the Euclidean boundary of \(\Omega\), so (if \(\Omega\) is unbounded) functions in \(C(\partial \Omega)\) can be wild near \(\infty\). NEWLINENEWLINENEWLINELecture 3. The author returns to the theory of harmonic approximation. Here the question is: given a particular relatively closed subset \(E\) of \(\Omega\), which functions in \(C(E)\cap{\mathcal H}(E^0)\) can be approximated uniformly on \(E\) by functions in \({\mathcal H}(\Omega)\)? A precise answer is given involving the notions of accessibility, the fine topology, and finely harmonic functions. NEWLINENEWLINENEWLINELecture 4. Two futher applications are presented. (i) A complete answer to the following question is given: for which subsets \(E\) of \(\mathbb{R}^n\) is there a harmonic function \(u\) on \(\mathbb{R}^n\times (0,+\infty)\) such that \(u(x,t)\to +\infty\) as \(t\to 0+\) for each \(x\in E\)? (ii) A result, due to the author, comparing the fine cluster sets and the non-tangential cluster sets of a function on the unit ball of \(\mathbb{R}^n\) is shown to be sharp even for harmonic functions.NEWLINENEWLINEFor the entire collection see [Zbl 0972.00045].