A problem of minimization with relaxed energy
type: Item
instance of: scholarly article
title: A problem of minimization with relaxed energy
zbMATH Open document ID: 0861.58013
author: Rejeb Hadiji
author: Feng Zhou
published in: Annales de la Faculté des Sciences de Toulouse. Mathématiques. Série VI
publication date: 1996-12-12T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/73359
review text: Let \(\Omega\) denote the unit ball in \(\mathbb{R}^3\) and \(S^2\) the unit sphere and let \[ E(u)=\int_\Omega |\nabla u|^2dxdydz. \] The authors consider the minimization problem \[ \alpha =\inf\{E(u-f),\quad u\in C^1(\overline{\Omega},S^2)\} \] where \(f\) is a \(H^1\) map from \(\Omega\) to \(\mathbb{R}^3\) which has values almost everywhere in \(S^2\) and which is regular except at a finite number of points. They generalize the result of \textit{F. Bethuel} and \textit{H. Brezis} [C.R. Acad. Sci., Paris, Sér. I 310, No. 13, 859-864 (1990; Zbl 0717.49005)] and show that the infimum is not attained and that every minimizing sequence converges strongly in \(H^1\) to a regular mapping of degree one from \(S^2\) to \(S^2\).
Mathematics Subject Classification ID: 58E30
Mathematics Subject Classification ID: 49J10
zbMATH DE Number: 942769
zbMATH Keywords: relaxed energy
zbMATH Keywords: minimization
zbMATH Keywords: minimizing sequence
reviewed by: Chandra Shekhar Sharma
MaRDI profile type: Publication
OpenAlex ID: W2017499492
cites work: A characterization of maps in \(H^ 1(B^ 3,S^ 2)\) which can be approximated by smooth maps
cites work: Q5748480
cites work: Q3972645
cites work: Density of smooth functions between two manifolds in Sobolev spaces
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cites work: Harmonic maps with defects
cites work: The problem of Plateau on a Riemannian manifold
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DOI: 10.5802/AFST.804
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description: scientific article; zbMATH DE number 942769