Minimization of conformally invariant energies in homotopy classes
type: Item
instance of: scholarly article
title: Minimization of conformally invariant energies in homotopy classes
zbMATH Open document ID: 0909.49008
author: Frank Duzaar
author: Ernst Kuwert
published in: Calculus of Variations and Partial Differential Equations
publication date: 1999-03-22T00:00:00Z
full work available at URL: http://cds.cern.ch/record/330053
review text: Let \(X\subset \mathbb{R}^N\) be compact and \(\alpha\in \Pi_n(X)\) a nontrivial homotopy class of maps \(u: S^n\to X\). The authors analyze the behavior of a minimizing sequence \(u_k\in\alpha\) for a conformally invariant functional such as the \(n\)-energy \[ E(u)= \int_{S^n}| du(\xi)|^nd\xi. \] A collection \((u^i)_{i\in I}\) of nonconstant maps is a \textit{weak limit set} of a sequence \(u_k\) if there exist mutually unbounded conformal automorphisms \(g^i_k\) of the sphere such that \(u_k\circ g^i_k\) converges to \(u^i\) weakly for any \(i\in I\). If in addition the sum of the energies of the \(u^i\) equals the limit of energy along the sequence, the convergence is said to be \textit{in energy}. The main results follow from compactness and continuity results formulated in terms of weak limit set and convergence in energy and the reasoning follows the direct method of calculus of variations.
Mathematics Subject Classification ID: 49J45
Mathematics Subject Classification ID: 58E20
zbMATH DE Number: 1160791
zbMATH Keywords: conformally invariant energies
zbMATH Keywords: homotopy classes
zbMATH Keywords: weak limit set
zbMATH Keywords: convergence in energy
reviewed by: Ubiratan D'Ambrosio
MaRDI profile type: Publication
OpenAlex ID: W2010988046
Wikidata QID: Q125274587
DOI: 10.1007/S005260050092
description: scientific article; zbMATH DE number 1160791