Tangential harmonic approximation on relatively closed sets (Q1342701)

Let \(\Omega\) be a connected open set in \(\mathbb{R}^ n\) \((n \geq 2)\) and \(E\) be a relatively closed proper subset of \(\Omega\). If \(A \subseteq \mathbb{R}^ n\), then let \({\mathcal H} (A)\) (resp. \({\mathcal S}^ + (A))\) denote the collection of functions which are harmonic (resp. positive and superharmonic) on some open set containing \(A\). In an earlier paper [Ann. Inst. Fourier 44, No. 1, 65-91 (1994; Zbl 0795.31004)] the author characterized the pairs \((\Omega,E)\) such that any function \(h\) in \({\mathcal H} (E)\) (or in \(C(E) \cap{\mathcal H}(E^ 0))\) can be uniformly approximated on \(E\) by functions \(H\) in \({\mathcal H} (\Omega)\). The present paper shows that any pair \((\Omega, E)\) which permits such uniform approximation also permits approximation by functions \(H\) in \({\mathcal H} (\Omega)\) such that \(0 < H - h < s\) for any predetermined function \(s\) in \({\mathcal S}^ + (\widehat E)\). (Here \(\widehat E\) denotes the union of \(E\) with the components of \(\Omega \backslash E\) which are relatively compact in \(\Omega\).) Thus, for example, if \(\Omega = \mathbb{R}^ n\) and \(E\) is contained in a strip, then the error bound can be chosen to decay exponentially near infinity. The proofs rely on recent work of \textit{D. H. Armitage} and \textit{M. Goldstein} [Proc. Lond. Math. Soc., III. Ser. 68, No. 1, 112-126 (1994; Zbl 0795.31002)]. These results are then used to study approximation problems in which the error bound can decay arbitrarily quickly. As a sample we quote the following result with solves a problem posed by Boivin and Gauthier. Theorem. The following are equivalent: (a) For each \(h\) in \(C(E) \cap {\mathcal H} (E^ 0)\) and each continuous function \(\varepsilon : E \to (0,1]\) there exists \(H\) in \({\mathcal H}(E)\) such that \(| H - h | < \varepsilon\) on \(E\). (b) The pair \((\Omega,E)\) satisfies: (i) \(\Omega \backslash E\) and \(\Omega \backslash E^ 0\) are thin at the same points of \(E\), and (ii) for each compact subset \(K\) of \(\Omega\) there is a compact subset \(L\) of \(\Omega\) which contains every component of \(E^ 0\) that intersects \(K\).