On the anisotropic Landau-Lifshitz equations. (English) Zbl 1002.35117
Author’s summary: We discuss the anisotropic static Landau-Lifshitz equations. We obtain maximum principles and existence results for the small solutions to the Dirichlet problems and the initial-boundary value problems of the corresponding evolution equations. We also construct multiple large smooth solutions in the two dimensions. These problems are dealt with from the viewpoint of harmonic maps with potential.
Reviewer: Marco Biroli (Monza)
MSC:
| 35Q72 | Other PDE from mechanics (MSC2000) |
| 58E20 | Harmonic maps, etc. |
| 82D40 | Statistical mechanics of magnetic materials |
| 31C12 | Potential theory on Riemannian manifolds and other spaces |
Keywords:
maximum principles; existence; Dirichlet problems; initial-boundary value problems; harmonic mapsReferences:
| [1] | Brezis, H.; Coron, J.-M., Large solutions for harmonic maps in two dimensions, Comm. Math. Phys., 92, 203-215 (1983) · Zbl 0532.58006 |
| [2] | P. Courilleau, S. Dumont, and, R. Hadiji, Regularity of minimizing maps with values in \(S^2\); P. Courilleau, S. Dumont, and, R. Hadiji, Regularity of minimizing maps with values in \(S^2\) · Zbl 0979.49029 |
| [3] | Chen, Q., Maximum principles, uniqueness and existence for harmonic maps with potential and Landau-Lifshitz equations, Calc. Var., 8, 91-107 (1999) · Zbl 0933.58016 |
| [4] | Chen, Q., Liouville theorem for harmonic maps with potential, Manuscripta Math., 95, 507-517 (1998) · Zbl 0911.58008 |
| [5] | Chen, Q., Stability and constant boundary value problems for harmonic maps with potential, J. Austral. Math. Soc. Ser. A, 68, 145-154 (2000) · Zbl 0956.58010 |
| [6] | Fardoun, A.; Ratto, A., Harmonic maps with potential, Calc. Var., 5, 183-197 (1997) · Zbl 0874.58010 |
| [7] | Fardoun, A.; Ratto, A.; Regbaoui, R., Sur l’équation de la chaleur pour lés applications harmoniques avec potential, C.R. Acad. Sci. Paris, 327, 569-574 (1998) · Zbl 1004.58510 |
| [8] | Hamilton, R., Harmonic Maps of Manifolds with Boundary (1975), Springer-Verlag: Springer-Verlag Berlin/Heidelberg/New York · Zbl 0308.35003 |
| [9] | Hong, M. C.; Lemaire, L., Multiple solutions of the static Landau-Lifshitz equation from \(B^2\) into \(S^2\), Math. Z., 220, 295-306 (1995) · Zbl 0843.35029 |
| [10] | 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, 171-188 (1995) · Zbl 0843.35028 |
| [11] | Hadiji, R.; Zhou, F., Regularity of ∫\(_Ω\)|∇\(u|^2\)+λ∫\(_Ω|u\)−\(f|^2\) and some gap phenomenon, Potential Anal., 1, 385-400 (1992) · Zbl 0794.35036 |
| [12] | Jost, J., The Dirichlet problem for harmonic maps from a surface with boundary onto a 2-sphere with nonconstant boundary values, J. Differential Geom., 19, 393-401 (1984) · Zbl 0551.58012 |
| [13] | Jost, J., Ein Existenzbeweis für harmonische Abbildungen, die ein Dirichlet-problem lösen, mittels der Methode des Wärmeflusses, Manuscripta Math., 34, 17-25 (1981) · Zbl 0459.58013 |
| [14] | Jost, J., Harmonic Mappings between Riemannian Manifolds. Harmonic Mappings between Riemannian Manifolds, Proc. Centre Math. Analysis (1983), Aust. Nat. Univ. Press: Aust. Nat. Univ. Press Canberra · Zbl 0542.58001 |
| [15] | Shen, Y. T.; Yan, S. S., Existence and nonexistence of nontrivial axially symmetric solution for quasilinear elliptic systems with inhomogeneous term, Comm. Partial Differential Equations, 1477-1491 (1993) · Zbl 0792.35060 |
| [16] | Shen, Y. T.; Yan, S. S., Boundary value problems for the equilibrium systems of ferromagnetic chain, Acta Math. Sci., 15, 144-151 (1995) · Zbl 0836.35136 |
| [17] | Peng, X. W.; Wang, G. F., Harmonic maps with a prescribed potential, C.R. Acad. Sci. Paris, 327, 271-276 (1998) · Zbl 0930.58009 |
| [18] | Zhai, J., Non-constant stable solutions to Landau-Lifshitz equation, Calc. Var., 7, 159-171 (1998) · Zbl 0909.35043 |
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