Hostname: page-component-78c5997874-dh8gc Total loading time: 0 Render date: 2024-11-03T02:17:10.686Z Has data issue: false hasContentIssue false
  • English
  • Français

Stability and constant boundary-value problems of harmonic maps with potential

Part of: Variational problems in infinite-dimensional spaces Global differential geometry

Published online by Cambridge University Press:  09 April 2009

Qun Chen
Qun Chen
Affiliation:
Mathematics Department Central China Normal University Wushan 430079 China School of Mathematical Science Wuhan University Wuhan 430072 China e-mail: [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Let M, N be Riemannian manifolds, f: M → N a harmonic map with potential H, namely, a smooth critical point of the functional EH(f) = ∫M[e(f)−H(f)], where e(f) is the energy density of f. Some results concerning the stability of these maps between spheres and any Riemannian manifold are given. For a general class of M, this paper also gives a result on the constant boundary-value problem which generalizes the result of Karcher-Wood even in the case of the usual harmonic maps. It can also be applied to the static Landau-Lifshitz equations.

Keywords

Harmonic mapspotentialstabilityboundary-value.

MSC classification

Secondary: 58E20: Harmonic maps , etc. 53C21: Methods of Riemannian geometry, including PDE methods; curvature restrictions
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 68 , Issue 2 , April 2000 , pp. 145 - 154
Copyright
Copyright © Australian Mathematical Society 2000

References

[BE]Baird, P. and Eells, J., A conservation law for harmonic maps, Lecture Notes in Math. 894 (Springer, Berlin, 1981) pp. 1–15.Google Scholar
[C1]Chen, Q., ‘Liouville theorem for harmonic maps with potential’, Manuscripta Math. 95 (1998), 507–517.CrossRefGoogle Scholar
[C2]Chen, Q., ‘On harmonic maps with potential from complete manifolds’, Chinese Sci. Bull. 43 (1988), 1780–1786.CrossRefGoogle Scholar
[C3]Chen, Q., ‘Maximum principles, uniqueness and existence for harmonic maps with potential and Landau-Lifshitz equations’, Cal. Var. Partial Differential Equations 8 (1999), 91–107.CrossRefGoogle Scholar
[Di]Ding, Q., ‘The Dirichlet problem at infinity for manifolds of nonpositive curvature’, in: Differential geometry(Shanghai, 1991) (World Sci. Publ., River Edge N.J. 1991) pp. 49–58.Google Scholar
[EL]Eells, J. and Lemaire, L., ‘Selected topics in harmonic maps’, CBMS Regional Conference Series in Math. 50, Amer. Math. Soc., Providence, 1983.CrossRefGoogle Scholar
[FR]Fardoun, A. and Ratto, A., ‘Harmonic maps with potential’, Calc. Var. Partial Differential Equations 5 (1997), 183–197.CrossRefGoogle Scholar
[Ho]Hong, M. C., ‘The Landau-Lifshitz equation with the external field—a new extension for harmonic maps with values in S 2’, Math. Z. 220 (1995), 171–188.CrossRefGoogle Scholar
[HL]Hong, M. C. and Lemaire, L., ‘Multiple solutions of the static Landau-Lifshitz equation from B 2 into S 2’, Math. Z. 220 (1995), 295–306.CrossRefGoogle Scholar
[KW]Karcher, J. and Wood, J. C., ‘Non existence results and growth properties for harmonic maps and forms’, J. Reine. Angew. Math. 353 (1984), 165–180.Google Scholar
[Lg]Leung, P. F., On the stability of harmonic maps, Lecture Notes in Math. 949 (Springer, Berlin, 1982), pp. 122–129.CrossRefGoogle Scholar
[Li]Li, P., Lecture notes on geometric analysis, Lecture Notes Series No. 6 (Research Institute of Mathematics and Global Analysis Research Center, Seoul National University, Seoul, 1993).Google Scholar
[Xin1]Xin, Y. L., ‘Some results on stable harmonic maps’, Duke Math. J. 47 (1980), 609–613.CrossRefGoogle Scholar
[Xin2]Xin, Y. L., ‘Differential forms, conservation law and monotonicity formula’, Sci. Sinica (Ser. A) 29 (1986), 40–50.Google Scholar
[Xin3]Xin, Y. L., Liouville type theorems and regularity of harmonic maps, Lecture Notes in Math. 1255 (Springer, Berlin, 1987) pp. 198–208.Google Scholar
[Xin4]Xin, Y. L., ‘Harmonic maps of bounded symmetric domains’, Math. Ann. 303 (1995), 417–433.CrossRefGoogle Scholar