On an isoperimetric problem with power-law potentials and external attraction
type: Item
instance of: scholarly article
title: On an isoperimetric problem with power-law potentials and external attraction
zbMATH Open document ID: 1430.49046
author: Guoqing Zhang
author: Xiaoqian Geng
published in: Journal of Mathematical Analysis and Applications
publication date: 2019-11-28T00:00:00Z
review text: The authors consider the following energy (or shape functional:) \[ E(\Omega):=P(\Omega)+\int_\Omega\!\!\int_\Omega K(x-y) u(x) u(y) dxdy -\int_\Omega V(x) dx \] where \(\Omega\subset \mathbb{R}^N\), \(N\geq 3\) is a set of finite perimeter, \(P(\Omega)\) is the classical perimeter (in the sense of De Giorgi), \(K(x)\) is a power law potential: \[ K(x)=\frac{1}{p} |x|^{-p} - \frac{1}{q} |x|^{-q}\,, \] and \(V\) is a function in \(L^1_{\mathrm{loc}}(\mathbb{R}^N)\), \(V\geq 0\) and \(V(x)\) tends to 0 when \(|x|\to \infty\). They are interested in the minimization of the functional \(E\), in the class of sets of finite perimeter, with the volume constraint: \(|\Omega|=m\). The authors prove that, when \(0<q<p<N\), there exists a minimizing domain for any volume constraint (for any \(m\)). The proof relies on the use of the concentration-compactness Lemma for sets of finite perimeter due to \textit{R. L. Frank} and \textit{E. H. Lieb} [SIAM J. Math. Anal. 47, No. 6, 4436--4450 (2015; Zbl 1332.49042)]. Then, the authors look at the minimization problem in the case \(q=-2, 0<p<N\) and \(V\equiv 0\). They prove that, for \(m\) sufficiently large, the unique minimizer is the ball. They use a relaxation technique together with the optimality condition to derive this property.
Mathematics Subject Classification ID: 49Q10
Mathematics Subject Classification ID: 49J45
zbMATH DE Number: 7138379
zbMATH Keywords: isoperimetric problem
zbMATH Keywords: power-law potentials
zbMATH Keywords: minimizers
reviewed by: Antoine Henrot
MaRDI profile type: Publication
full work available at URL: j.jmaa.2019.123521
OpenAlex ID: W2976038408
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Wikidata QID: Q127198164
DOI: 10.1016/J.JMAA.2019.123521
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description: scientific article; zbMATH DE number 7138379