Discontinuity of best harmonic approximants (Q1283897)

Let \(D\) be a (Jordan) domain in \(\mathbb{R}^2\) and \(f\) some real-valued and bounded continuous function on \(\overline D\). Then \(f\) has some best harmonic approximant \(h:D\in\mathbb{R}\) in the sense of uniform approximation on \(D\), but it is known that \(h\) must not have a continuous extension on \(\overline D\). The authors present very concrete examples as well as classes of real analytic \(f\) showing this phenomenon.