Singularities of energy minimizing maps from the ball to the sphere: Examples, counterexamples, and bounds
type: Item
instance of: scholarly article
title: Singularities of energy minimizing maps from the ball to the sphere: Examples, counterexamples, and bounds
zbMATH Open document ID: 0673.58013
DOI: 10.2307/1971434
author: Frederick J. jun. Almgren
author: E. H. Lieb
published in: Annals of Mathematics. Second Series
publication date: 1988-01-01T00:00:00Z
review text: The paper studies the singularities of Dirichlet's integral energy upon the mappings from a domain in \(R^ 3\) to \(S^ 2\) with prescribed boundary value functions. The main result in the first part of the paper states that the number of singular points of such a minimizer is linearly dominated by the boundary energy. The second part of the paper is devoted to ``the curious behaviour of singularities''. It is shown that, within an open ball in \(R^ 3\), the minimizers with zero boundary mapping area can have arbitrarily many singularities. It is then constructed a boundary function having a symmetry about the equator in a ball for which every minimizer possesses unsymmetric singularities. Many other interesting results are proved.
Mathematics Subject Classification ID: 58E20
Mathematics Subject Classification ID: 58C99
Mathematics Subject Classification ID: 82D30
Mathematics Subject Classification ID: 28A99
zbMATH DE Number: 4102119
zbMATH Keywords: minimizing map
zbMATH Keywords: harmonic map
zbMATH Keywords: liquid crystals
zbMATH Keywords: Dirichlet's integral energy
zbMATH Keywords: singular points
zbMATH Keywords: unsymmetric singularities
reviewed by: Dumitru Motreanu
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2307/1971434
OpenAlex ID: W2332538224
Wikidata QID: Q124830147
description: scientific article; zbMATH DE number 4102119