On a conjecture of Gustafsson and Lin concerning Laplacian growth
This paper proves a conjecture of \textit{B. Gustafsson} and \textit{Y.-L. Lin} [Laplacian growth on branched Riemann surfaces. Cham: Springer (2021; Zbl 07325956)] as follows. Let \(g\) be a holomorphic function on a connected neighbourhood \(\omega\) of of \(\mathbb{D}\), where \(\mathbb{D}\) denotes the unit disc in the plane, and let \(\lambda\) denote the planar Lebesgue measure. For each \(t>0\) define \[ u_t=\inf\{w\in C^2(\mathbb{R}^2\setminus\{0\}: w\geq 0, \Delta w\leq \lambda\mid_{\mathbb{R}^2\setminus\mathbb{D}}-t\delta_0\} \] and \(\Omega(t)=\{u_t>0\}\). The conjecture of Gustafsson and Lin asserts that the domains \(\Omega(t)\) are simply connected for all sufficiently small \(t > 0\). The main result of the paper establishes that there exists \(\delta>0\) such that for all \(0<t<\delta\) the domains \(\Omega(t)\) are all starshaped about \(0\), and so in particular are simply connected. The approach relies on the use of partial balayage techniques.
scientific article; zbMATH DE number 7474295