Abstract
Let \({{\bf X} \subset {\mathbb R}^n}\) be a generalised annulus and consider the Dirichlet energy functional
on the space of admissible maps
Here \({\varphi \in {\bf C}(\partial {\bf X}, {\mathbb S}^{n-1})}\) is fixed and \({{\mathcal A}_\varphi({\bf X})}\) is non-empty. In this paper we introduce a class of maps referred to as spherical twists and examine them in connection with the Euler–Lagrange equation associated with \({{\mathbb E}[\cdot, {\bf X}]}\) on \({{\mathcal A}_\varphi({\bf X})}\) [the so-called harmonic map equation on X]. The main result here is an interesting discrepancy between even and odd dimensions. Indeed for even n subject to a compatibility condition on φ the latter system admits infinitely many smooth solutions modulo isometries whereas for odd n this number reduces to one or none. We discuss qualitative features of the solutions in view of their novel and explicit representation through the exponential map of the compact Lie group SO(n).
Article PDF
Similar content being viewed by others
References
Ball J.M.: Convexity conditions and existence theorems in nonlinear elasticity. Arch. Ration. Mech. Anal. 63, 337–403 (1977)
Birman J.S.: Braids, Links and Mapping Class Groups. Annals of Mathematics Studies, Study, vol. 82. Princeton University Press, NJ (1975)
Chang K.C.: Infinite Dimensional Morse Theory and Multiple Solution Problems. PNLDE, vol. 6. Birkhäuser, Basel (1993)
Dunford N., Schwartz J.T.: Linear Operators, vol. I. Wiley Interscience, London (1988)
Eells, J., Lemaire, L.: Two reports on harmonic maps. Bull. Lond. Math. Soc. 10 & 20, 1–68 & 385–524 (1978 & 1988)
Evans L.C.: Partial regularity for stationary harmonic maps into spheres. Arch. Ration. Mech. Anal. 116, 101–113 (1991)
Gilbarg D., Trudinger N.S.: Elliptic Partial Differential Equations of Second Order. Springer, Berlin (1998)
Helein F.: Harmonic Maps, Conservation Laws and Moving Frames. CUP, Cambridge (2002)
Lemaire L.: Applications harmoniques des surfaces Riemanniennes. J. Differ. Geom. 13, 51–87 (1978)
Riviere T.: Everywhere discontinuous harmonic maps into sphere. Acta Math. 175, 197–226 (1995)
Simon L.: Theorems on Regularity and Singularity of Energy Minimizing Maps. Birkhäuser, Basel (1996)
Taheri A.: Homotopy classes of self-maps of annuli, generalised twists and spin degree. Arch. Ration. Mech. Anal. 197, 239–270 (2010)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Taheri, A. Spherical twists, stationary paths and harmonic maps from generalised annuli into spheres. Nonlinear Differ. Equ. Appl. 19, 79–95 (2012). https://doi.org/10.1007/s00030-011-0119-0
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00030-011-0119-0