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

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 $$ \begin{cases} -{\rm div}[ F'(|x|,|\nabla u|^2) \nabla u] = F'(|x|,|\nabla u|^2) |\nabla u|^2 u & \text{in } \mathbb{X}^n,\\ |u| = 1 &\text{in } \mathbb{X}^n ,\\ u = \varphi &\text{on } \partial \mathbb{X}^n. \end{cases} $$ 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 [26], a spherical twist refers to a rotationally symmetric map of the form $u\colon x \mapsto {\rm Q}(|x|)x|x|^{-1}$ with ${\rm Q}$ 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.