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



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.


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

Full Text:



F. Alouges and J. Ghidaglia, Minimizing Oseen–Frank energy for nematic liquid crystals: algorithms and numerical results, Annales de l’IHP Physique Théorique 66 (1977), 411–447.

F. Bethuel, The approximation problem for Sobolev maps between two manifolds, Acta Math. 167 (1991), 153–206.

J.C. Bourgoin, The minimality of the map x/||x|| for weighted energy, Calc. Var. Partial Differential Equations 25 (2006), 469–489.

H. Brezis and Y. Li, Topology and Sobolev spaces, J. Funct. Anal. 183 (2001), 321–369.

L. Cesari, Optimization Theory and Applications: Problems with Ordinary Differential Equations, Applications of Mathematics, Vol. 17, Springer, 1983.

S. Day and A. Taheri, Stability of spherical harmonic twists, second variations and conjugate points on SO(n), (2018). (submitted)

S. Day and A. Taheri, A class of extremising sphere-valued maps with inherent maximal tori symmetries in SO(n), Bound. Value Probl. (2017).

J. Eells and L. Lemaire, Two reports on harmonic maps, Bull. Lond. Math. Soc. 10 (1978), 1–68; 20 (1988), 385–524.

J.L. Ericksen, Inequalities in liquid crystal theory, Physics of Fluids 9 (1966), 1205–1207.

M. Giaquinta, G. Modica and J. Soucek, Cartesian Currents in the Calculus of Variations, Vol. I, II, Springer, Berlin, 1998.

R. Gulliver and J.M. Coron, Minimizing p-harmonic maps into spheres, J. Reine Angew. Math. 401 (1989), 82–100.

F.B. Hang and F.H. Lin, Topology of Sobolev mappings, Math. Res. Lett. 8 (2001), 321–330.

R. Hardt and F.H. Lin, Mappings minimizing the Lp norm of the gradient, Comm. Pure Appl. Math. 40 (1987), 555–588.

F. Hélein and J.C. Wood, Harmonic maps, Handbook of Global Analysis, Elsevier, Amsterdam, 2008, 417–491.

M.C. Hong, On the minimality of the p-harmonic map x/|x| : Bn → Sn−1 , Calc. Var. Partial Differential Equations 13 (2001), 459–468.

F.H. Lin, A remark on the map x/|x|, C.R. Acad. Sci. 305 (1987), 529–531.

S. Luckhaus, Partial Hölder continuity for minima of certain energies among maps into a Riemannian manifold, Indiana Univ. Math. J. 37 (1988), 349–367.

C. Morris and A. Taheri, Annular rearrangements, incompressible axi-symmetric whirls and L1 -local minimisers of the distortion energy, NoDEA 25 (2018), Art. No. 2.

C. Morris and A. Taheri, Whirl mappings on generalised annuli and incompressible symmetric equilibria of the Dirichlet energy, J. Elasticity (2018). (to appear)

M.S. Shahrokhi-Dehkordi and A. Taheri, Generalised twists, SO(n) and the p-energy over a space of measure preserving maps, Ann. Inst. H. Poincaré Anal. Non Linéaire (C) 26 (2009), 1897–1924.

M.S. Shahrokhi-Dehkordi and A. Taheri, Generalised twists, stationary loops, and the Dirichlet energy over a space of measure preserving maps, Calc. Var. Partial Differential Equations 35 (2009), 191–213.

L. Simon, Theorems on Regularity and Singularity of Energy Minimizing Maps, Lectures in Mathematics, ETH Zürich, Birkhäuser, Basel, 1996.

A. Taheri, Local minimizers and quasiconvexity – the impact of topology, Arch. Rational Mech. Anal. 176 (2005), 363–414.

A. Taheri, Homotopy classes of self-maps of annuli, generalised twists and spin degree, Arch. Rational Mech. Anal. 197 (2010), 239–270.

A. Taheri, Spherical twists, SO(n) and the lifting of their twist paths to Spin(n) in low dimensions, Q. J. Math. 63 (2012), 723–751.

A. Taheri, Spherical twists, stationary paths and harmonic maps from generalised annuli into spheres, NoDEA 19 (2012), 79–95.

A. Taheri, Function Spaces and Partial Differential Equations, Vol. I, II, Oxford Lecture Series in Mathematics and its Applications, OUP, Oxford, 2015.

E.G. Virga, Variational Theories for Liquid Crystals, Applied Mathematics and Mathematical Computation, Vol. 8, Chapman & Hall, London, 1994.


  • There are currently no refbacks.

Partnerzy platformy czasopism