Skip to main content
Log in

The Landau-Lifshitz equation with the external field—a new extension for harmonic maps with values inS 2

  • Published:
Mathematische Zeitschrift Aims and scope Submit manuscript

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

References

  • [AS] Alouges, F., Soyeur, A.: On global weak solutions for Landau-Lifshitz equations: Existence and non-uniqueness, To appear in Nonlinear Analysis, Theory, Methods and Applications

  • [BC] Benci, V., Coron, J.M.: The Dirichlet problem for harmonic maps from the disk into the Euclidean n-sphere, Ann. Inst. H. Poincaré Anal. Non Lin.2, 119–141 (1985)

    MATH  MathSciNet  Google Scholar 

  • [BB] Bethuel, F., Brezis, H.: Minimisation de\(\int | \nabla (u - \tfrac{x}{{\left| x \right|}})|^2 \) et divers phénomènes de gap. C.R. Acad. Sci. Paris310, 859–864 (1990)

    MATH  MathSciNet  Google Scholar 

  • [BBC] Bethuel, F., Brezis, H., Coron, J.M.: Relaxed energies for harmonic maps, in Variational Problems, H. Berestycki, J.M. Coron and I. Ekeland Eds, (1990)

  • [Br] Brezis, H.:S k-valued maps with singularities, Lecture Notes in Mathematics1365, 1–30 (1989)

    Article  MathSciNet  Google Scholar 

  • [CD] Chang, K.-C., Ding, W.-Y.: A result on the global existence for heat flows of harmonic maps fromD 2 intoS 2,Kluwer Academic Publisher, J.M. Coron et al (eds) 37–47 (1991)

  • [Ch] Chen, Y.: Weak solutions to the evolution problem of harmonic maps, Math. Z.201, 69–74 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  • [CHH] Chen, Y., Hong, M.-C., Hungerbühler, N.: Heat flow of p-harmonic maps with values into spheres, Math. Z.215, 25–35 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  • [CL] Chen, Y., Lin, F.-H.: Evolution of harmonic maps with Dirichlet boundary conditions, To appear in Comm. of Geometric Analysis

  • [CS] Chen, Y., Struwe, M.: Existence and partial regularity results for the heat flow for harmonic maps, Math. Z.201, 83–103 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  • [Co] Coron, J.M.: Harmonic maps with values into spheres, Proc. ICM, Kyoto (1990)

  • [EL] Eells, J., Lemaire, L.: Another report on Harmonic maps, Bull. London Math. Soc.20, 385–524 (1988)

    Article  MATH  MathSciNet  Google Scholar 

  • [ES] Eells, J., Sampson, J.H.: Harmonic mappings of Riemannian manifolds, Amer. J. Math.86, 109–160 (1964)

    Article  MATH  MathSciNet  Google Scholar 

  • [Fo] Fogedby, H.C.: Theoretical aspects of mainly low dimensional magnetic systems, Lecture Notes in Phy. (131) Springer-Verlag, Berlin, Heidelberg, 1980

    Google Scholar 

  • [GMS] Giaquinta, M., Modica, G., Soucek, J. The gap phenomenon for variational integrals in Sobolev space, Proc. of the Royal Socia. of Edinburgh120 A, 93–98 (1992)

    MathSciNet  Google Scholar 

  • [GH] Guo, B., Hong, M.-C.: The Landau-Lifshitz equation of the ferromagnetic spin chain and harmonic maps, Calc. Var.1, 311–334 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  • [HKL] Hardt, R., Kinderlehrer, D., Luskin, M. Remarks about the mathematical theory of liquid crystals, Lect. Note in Math.1340, 121–138 (1986)

    Google Scholar 

  • [HL] Hardt, R., Lin, F.H.: A remark onH 1 mappings, Manuscripta Math.56, 1–10 (1986)

    Article  MATH  MathSciNet  Google Scholar 

  • [HZ] Hadiji, R., Zhou, F.: Regularity of\(\smallint _\Omega \left| {\nabla u} \right|^2 + \lambda \left| {u - f} \right|^2 \) and some gap phenomenon, to appear

  • [Hle] Hong, M.-C., Lemaire, L.: Multiple solutions of the static Landau-Lifshitz equation fromB 2intoS 2,Math. Z.220 295–306 (1995)

    MATH  MathSciNet  Google Scholar 

  • [LL] Landau, L.D., Lifshitz, E.M.: On the theory of the dispersion of magnetic permeability in ferromagnetic bodies, Phys. Z. Sowjetunion,8, (1935), 153-D. ter Haar (eds.), Reproduced in, Collected Papers of L.D. Landau, Pergamon, New York, 1965, pp. 101–114

    MATH  Google Scholar 

  • [LN] Lakshmanan, M., Nakamura, K.: Landau-Lifshitz equation of ferromagnetism: Exact Treatment of the Gilbert Damping, Phy. Rew. Lett.53(26), 2497–2499 (1984)

    Article  Google Scholar 

  • [LSU] Ladyzenskaja, O.A., Solonnikov, V.A., Ural'ceva, N.N.: Linear and quasilinear equations of parabolic type, Transl. Math. Monogr. 23 (1968)

  • [Le] Lemaire, L.: Applications harmoniques de surfaces Riemanniennes, J. Diff. Geom.13, 51–78 (1978)

    MATH  MathSciNet  Google Scholar 

  • [Sc] Schoen, R.: Analytic aspects of the harmonic map problem, Seminar on Nonlinear Partial Differential Equation (Chern, ed.) Springer, Berlin, 1984

    Google Scholar 

  • [SU] Schoen, R., Uhlenbeck, K.: Boundary regularity and the Dirichlet problem for harmonic maps. J. Diff. Geom.18, 253–268 (1983)

    MATH  MathSciNet  Google Scholar 

  • [St1] Struwe, M.: On the evolution of Harmonic maps of Riemannian surfaces, Commun. Math. Helv.60, 558–581 (1985)

    Article  MATH  MathSciNet  Google Scholar 

  • [St2] Struwe, M.: On the evolution of harmonic maps in higher dimensions, J. Diff. Geom.28, 485–502 (1988)

    MATH  MathSciNet  Google Scholar 

  • [St3] Struwe, M.: The evolution of harmonic maps, Proc. ICM, Kyoto 1197–1203 (1990)

  • [Wo] Wood, J.C.: Non existence of solutions to certain Dirichlet problems for harmonic maps, preprint Leeds Univ. (1981)

Download references

Author information

Authors and Affiliations

Authors

Rights and permissions

Reprints and permissions

About this article

Cite this article

Hong, MC. The Landau-Lifshitz equation with the external field—a new extension for harmonic maps with values inS 2 . Math Z 220, 171–188 (1995). https://doi.org/10.1007/BF02572608

Download citation

  • Received:

  • Accepted:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02572608

Keywords

Navigation