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2-D stability of the Néel wall. (English) Zbl 1158.78300

The authors study transition layers which connect two opposite magnetizations. The mathematical framework is described by a singular energy functional involving a small parameter \(\varepsilon\). The main result of the paper asserts that if \(\varepsilon <<1\) then the minimum of this energy behaves like \(\pi/2\ln | \varepsilon| \). The proof combines elliptic estimates, variational arguments, and asymptotic analysis techniques.

MSC:

78A30 Electro- and magnetostatics
49S05 Variational principles of physics
78M30 Variational methods applied to problems in optics and electromagnetic theory
82D40 Statistical mechanics of magnetic materials
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[1] Alberti, G., Bouchitté, G., Seppecher, P.: Un résultat de perturbations sin- gulières avec la norme H1/2. C. R. Acad. Sci. Paris Sér. I Math. 319(4), 333–338 (1994) · Zbl 0845.49008
[2] Alberti, G., Bouchitté, G., Seppecher, P.: Phase transition with the line-tension effect. Arch. Rational Mech. Anal. 144(1), 1–46 (1998) · Zbl 0915.76093 · doi:10.1007/s002050050111
[3] Alouges, F., Rivière, T., Serfaty, S.: Néel and cross-tie wall energies for planar micromagnetic configurations. ESAIM Control Optim. Calc. Var. 8, 31–68 (2002) (electronic). A tribute to J. L. Lions · Zbl 1092.82047 · doi:10.1051/cocv:2002017
[4] Bronstein, Semendjajew, Musiol, Mühlig, Taschenbuch der Mathematik. Verlag Harri Deutsch
[5] Cabré, Solà-Morales, Layer solutions in a halfspace for bondary reactions. to appear
[6] Cervera, G.: Magnetic domains and magnetic domain walls. PhD thesis, New York University (1999)
[7] Desimone, A., Kohn, R.V., Müller, S., Otto, F.: A reduced theory for thin-film micromagnetics. Comm. Pure Appl. Math. 55(11), 1408–1460 (2002) · Zbl 1027.82042 · doi:10.1002/cpa.3028
[8] Desimone, A., Kohn, R.V., Müller, S., Otto, F.: Repulsive interaction of Néel walls, and the internal length scale of the cross-tie wall. Multiscale Model. Simul. 1(1), 57–104 (2003) (electronic) · Zbl 1059.82046 · doi:10.1137/S1540345902402734
[9] Garroni, A., Müller, S.: {\(\gamma\)}-limit of a phase-field model of dislocations. SIAM J. Math. Anal. 36(6), 1943–1964 (2005) · Zbl 1094.82008
[10] Holz, A., Hubert, A.: Phys. Stat. Sol. (B) 46, 377–384 (1971) · doi:10.1002/pssb.2220460136
[11] Hubert, A., Schäfer, R.: Magnetic domains: The analysis of magnetic microstructures. Springer-Verlag (1998)
[12] Jin, W., Kohn, R.V.: Singular perturbation and the energy of folds. J. Non-linear Sci. 10(3), 355–390 (2000) · Zbl 0973.49009
[13] Kohn, Slastikov: Another thin-film limit of micromagnetics. submitted to Arch. Rat. Mech. Anal.
[14] Koslowski, M., Cuiti\(\sim\)no, A.M., Ortiz, M.: A phase-field theory of dislocation dynamics, strain hardening and hysteresis in ductile single crystals. J. Mech. Phys. Solids 50(12), 2597–2635 (2002) · Zbl 1094.74563 · doi:10.1016/S0022-5096(02)00037-6
[15] Kurzke, M.: A nonlocal singular perturbation problem with periodic well potential. ESAIM Control. Optim. Calc. Var. 12(1), 52–63 (2006) · Zbl 1107.49016
[16] Melcher, C.: Logarithmic lower bounds for Néel walls. Calc. Var. Partial Differential Equations 21(2), 209–219 (2004) · Zbl 1054.78011
[17] Melcher, C.: Micromagnetic treatment of Néel walls. Arch. Rat. Mech. 168, 83–113 (2003) · Zbl 1151.82437 · doi:10.1007/s00205-003-0248-7
[18] Otto, F.: Cross-over in scaling laws: a simple example from micromagnetics. In: Proceedings of the International Congress of Mathematicians, vol. III (Beijing, 2002), pp. 829–838, Beijing, Higher Ed. Press (2002) · Zbl 1067.74020
[19] van den Berg, H.A.M.: Self-consistent domain theory in soft ferromagnetic media. ii. basic domain structures in thin film objects. J. Appl. Phys. 60, 1104–1113 (1986) · doi:10.1063/1.337352
[20] Verhulst, F.: Nonlinear Differential Equations and Dynamical Systems. Springer-Verlag (1989) · Zbl 0685.34002
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