Balayage properties related to rational interpolation (Q1862750)

Let \(D\) be a bounded simply connected domain in the complex plane \(\mathbb{C}\) and let \(\Omega\) denote the complement of \(\overline D\) with respect to the extended plane \(\mathbb{C}_\infty\). Moreover, let \(\alpha\) and \(\beta\) be unit measures with \(\alpha(D)=1\) and \(\beta (\Omega)=1\). Denoting by \(\alpha' \) the balayage of \(\alpha\) onto \(\mathbb{C}_\infty\setminus D\) and by \(\beta'\) the baylayage of \(\beta\) onto \(\overline D\), the authors describe under which conditions on \(D\) the following properties hold: P1: There is a measure \(\alpha\) such that \(\alpha'\) is the equilibrium measure for \(\overline D\). P2: There exist measures \(\alpha\), and \(\beta\) such that \(\alpha'=\beta'\). P3: For every \(\alpha\) there exists \(\beta\) such that \(\alpha'=\beta'\). They show that P1 (respectively, P3) holds if \(D\) is a Jordan domain which satisfies the so-called exterior (respectively, interior) Dini-condition and that these one-sided Dini-conditions turn out to be crucial for P1 and P3. Moreover, they prove that for Jordan domains with rectifiable boundary P2 holds. This implies, in particular, that P2 is strictly weaker than P1 and P3.