Criteria for approximation by harmonic functions

Abstract

In [1], P. R. Ahern gives “geometric” conditions which ensure that every continuous function on $K$, harmonic in the interior of $\mathrm{K}$, can be approximated uniformly on $K$ by functions harmonic in a neighborhood of $K$. Here we observe that Ahern's conditions can be sharpened to yield necessary and sufficient conditions for such approximation to obtain. The proof depends on a simple characterization of stable boundary points, which facilitates the evaluation of certain logarithmic potentials.