On domains in which harmonic functions satisfy generalized mean value properties (Q1368571)

A bounded domain \(\Omega\) in \(\mathbb{R}^N\) \((N\geq 2)\) is called (in the paper under review) a quadrature domain \((QD)\) if there exists a signed measure \(\mu\) with compact support in \(\Omega\) such that \(\int_\Omega hdx= \int hd \mu\) for every integrable harmonic function \(h\) on \(\Omega\). Similarly, \(\Omega\) is called a positive quadrature domain \((PQD)\) if the same is true for some positive measure \(\mu\) with compact support in \(\mu\). Is every \(QD\) a \(PQD\)? The paper provides partial answers in \(\mathbb{R}^N\) and a complete answer in \(\mathbb{R}^2\). In fact it is proved that if \(\Omega\) is a \(QD\), then \(\Omega\) is a \(PQD\), provided that every positive harmonic function on \(\Omega\) is the pointwise limit of a sequence of integrable positive harmonic functions. (Indeed, it is shown that a somewhat weaker condition suffices.) This result is used in conjunction with a geometric description of all plane \(QD\)s, due to \textit{M. Sakai} [Acta Math. 166, 263-297 (1991; Zbl 0728.30007) and SIAM J. Math. Anal. 24, 341-364 (1993; Zbl 0771.30041)], to show that in \(\mathbb{R}^2\) every \(QD\) is a \(PQD\).