Harmonic approximation on compact sets
Let \(\Omega\) be a \(C^\infty\) oriented connected non-compact Riemannian manifold, and let \(X\), \(Y\) be compact subsets of \(\Omega\) such that \(X\subset Y\). This paper offers necessary and sufficient conditions on \(X\) and \(Y\), formulated in terms of thinness, so that every harmonic function on \(X\) (or, more generally, every continuous function on \(X\) that is harmonic on \(X^0\)) can be uniformly approximated on \(X\) by functions harmonic on \(Y\). These results generalise, using similar arguments, the known case where \(Y\) is the union of \(X\) and all the bounded components of \(\Omega\setminus X\). The results are then applied to the location of singularities in harmonic Runge-type approximation theorems.
scientific article; zbMATH DE number 2112943