Description

• Title:
  Geodesics on ${\bf SO}(n)$ and a class of spherically symmetric maps as solutions to a nonlinear generalised harmonic map problem
• Creator:
  Day, Stuart;
  Taheri, Ali
• Description:
  We address questions on existence, multiplicity as well as qualitative features including rotational symmetry for certain classes of geometrically motivated maps serving as solutions to the nonlinear system $$ \begin{cases} -\text{\rm div}[ F'(|x|,|\nabla u|^2) \nabla u] = F'(|x|,|\nabla u|^2) |\nabla u|^2 u &\text{in } \mathbb{X}^n,\\ |u| = 1 &\text{in } \mathbb{X}^n ,\\ u = \varphi &\text{on } \partial \mathbb{X}^n. \end{cases} $$% Here $\varphi \in \mathscr{C}^\infty(\partial {\mathbb{X}}^n, \mathbb S}^{n-1})$ is a suitable boundary map, $F'$ is the derivative of $F$ with respect to the second argument, $u \in W^{1,p}(\mathbb{X}^n, \mathbb S}^{n-1})$ for a fixed $1< p< \infty$ and ${\mathbb{X}}^n=\{x \in \mathbb R^n : a< |x|< b\}$ is a generalised annulus. Of particular interest are spherical twists and whirls, where following \cite{Taheri2012}, a spherical twist refers to a rotationally symmetric map of the form $u\colon x \mapsto \rom{Q}(|x|)x|x|^{-1}$ with $\rom{Q}$ some suitable path in $\mathscr{C}([a, b], {\rm SO}(n))$ and a whirl has a similar but more complex structure with only $2$-plane symmetries. We establish the existence of an infinite family of such solutions and illustrate an interesting discrepancy between odd and even dimensions.
• Object availability:
  free access
• Rights:
  Prawa autorskie (c) 2018 Topological Methods in Nonlinear Analysis
• Date:
  2018.05.21
• Type:
  info:eu-repo/semantics/article;
  info:eu-repo/semantics/publishedVersion;
  Recenzowany artykuł
• Language:
  English
• Source:
  Topological Methods in Nonlinear Analysis; Vol 51, No 2 (June 2018); 637 - 662;
  1230-3429
• Publisher:
  Nicolaus Copernicus University in Toruń
• Relation: https://apcz.umk.pl/TMNA/article/view/TMNA.2018.020/14814 ;https://apcz.umk.pl/TMNA/article/view/TMNA.2018.020/14815
• Subject:
  Generalised harmonic map problem;
  rotationally symmetric sphere-valued maps;
  spherical twists;
  geodesics on ${\rm SO}(n)$;
  spherical whirls;
  weighted $p$-harmonic maps;
  35J57;
  49J10;
  53C22;
  58D19;
  22C05
• Identifier: https://apcz.umk.pl/TMNA/article/view/TMNA.2018.020
• Data provider:
  Akademicka Platforma Czasopism
• Can I use it?:
  seek permission
• Type:
  other