On a conjecture of Gustafsson and Lin concerning Laplacian growth
type: Item
instance of: scholarly article
title: On a conjecture of Gustafsson and Lin concerning Laplacian growth
zbMATH Open document ID: 1486.31006
author: Stephen J. Gardiner
author: Tomas Sjödin
published in: Analysis and Mathematical Physics
publication date: 2022-02-16T00:00:00Z
review text: 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.
reviewed by: Marius Ghergu
Mathematics Subject Classification ID: 31A15
zbMATH DE Number: 7474295
zbMATH Keywords: Laplacian growth
zbMATH Keywords: partial balayage
zbMATH Keywords: potential
zbMATH Keywords: starshaped domains
MaRDI profile type: Publication
full work available at URL: s13324-022-00647-z
OpenAlex ID: W4207022791
Wikidata QID: Q113899300
cites work: Q4513495
cites work: Q3654930
cites work: Laplacian Growth on Branched Riemann Surfaces
cites work: Properties of some balayage operators, with applications to quadrature domains and moving boundary problems
cites work: Q5703494
cites work: Q3032559
DOI: 10.1007/S13324-022-00647-Z
description: scientific article; zbMATH DE number 7474295