A toroidal tube solution to a problem involving mean curvature and Newtonian potential
type: Item
instance of: scholarly article
title: A toroidal tube solution to a problem involving mean curvature and Newtonian potential
zbMATH Open document ID: 1227.35247
DOI: 10.4171/IFB/251
author: Xiaofeng Ren
author: Juncheng Wei
published in: Interfaces and Free Boundaries
publication date: 2011-06-01T00:00:00Z
review text: This article deals with the following nonlocal geometric problem: find a set \(E\) in \(\mathbb R^2\) and a number \(\lambda\) such that the volume of \(E\) is \(m\) and the equation \(H(\partial E)+\gamma N(E)=\lambda\) is satisfied on the boundary of \(E\). Here \(m\), \(\gamma\) are given positive parameters, \(H(\partial E)\) is the mean curvature of \(\partial E\) and \(N(E)\) is the Newtonian potential operator \(N(E)(x)=\int_E\frac{dy}{4\pi |x-y|}\). The authors prove that for a sufficiently large \(\gamma\) there exists a solution of unit volume, which is closed to a set enclosed by a torus.
Mathematics Subject Classification ID: 35R35
Mathematics Subject Classification ID: 82D60
zbMATH DE Number: 5903244
zbMATH Keywords: torus
zbMATH Keywords: tube
zbMATH Keywords: axisymmetry
zbMATH Keywords: nonlocal geometric problem
zbMATH Keywords: mean curvature
zbMATH Keywords: Newtonian potential
zbMATH Keywords: approximate solution
zbMATH Keywords: Lyapunov-Schmidt reduction
reviewed by: Mykola Krasnoschok
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.4171/ifb/251
OpenAlex ID: W2036836233
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description: scientific article; zbMATH DE number 5903244