Selected topics on quadrature domains (Q2460552)

Given finitely many points \(a_1,\dots,a_n \in \mathbb{C}\) and \(c_j>0, j=1,\dots,n\), there is a unique (up to null sets) open set \(\Omega \subset \mathbb{C}\) such that, for all subharmonic and integrable functions \(\varphi\) on \(\Omega\), \(\int \varphi\, d\mu \leq \int_{\Omega} \varphi \,dA\), where \(\mu= c_1 \delta_{a_1} +\cdots+ c_n \delta_{a_n}\) and \(d A\) is the Lebesgue measure. In this case, \(\Omega\) is called the quadrature domain for subharmonic functions with respect to \(\mu\). This holds for example for \(n=1, c_1= \pi r^2\) and \(\Omega\) the open ball of center \(a_1\) and radius \(r\). It was proved by Gustafsson and Sakai that, denoting by \(K\) the convex hull of the support of \(\mu\), the boundary \(\partial \Omega\) of \(\Omega\) may have singular points, all located inside \(K\) and that \(\partial \Omega\) is smooth algebraic outside \(K\). The present paper is an informative survey covering a variety of phenomena related to quadrature domains. The investigation of these domains was initiated by Shapiro. Richardson discovered the connections to the Hele-Shaw flow moving boundary problem, and Putinar showed that quadrature domains are relevant within the theory of hyponormal operators; in fact the exponential transforms revolutionized the theory of quadrature domains. They also appear in modern physics in connection with Laplacian growth problems. Among other topics, the following ones are discussed: the differences between quadrature domains for subharmonic, harmonic and complex analytic functions, geometric properties of the boundary of \(\Omega\), instability of the reverse Hele-Shaw flow, a matrix model and a reconstruction algorithm. Several examples are explained, giving the computations in detail.