Geodesics on ${\bf SO}(n)$ and a class of spherically symmetric maps as solutions to a nonlinear generalised harmonic map problem

Stuart Day, Ali Taheri

DOI: http://dx.doi.org/10.12775/TMNA.2018.020

Abstract


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.

Keywords


Generalised harmonic map problem; rotationally symmetric sphere-valued maps; spherical twists; geodesics on ${\rm SO}(n)$; spherical whirls; weighted $p$-harmonic maps

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