2-d stability of the Néel wall

  • Antonio DeSimone
  • Hans Knüpfer
  • Felix Otto
Article

Abstract

We are interested in thin-film samples in micromagnetism, where the magnetization m is a 2-d unit-length vector field. More precisely we are interested in transition layers which connect two opposite magnetizations, so called Néel walls.

We prove stability of the 1-d transition layer under 2-d perturbations. This amounts to the investigation of the following singularly perturbed energy functional:
$$ E_{2d}(m)= \epsilon \int |\nabla m|^2 \,{\rm d}x + \frac{1}{2} \int |\nabla^{-1/2}\nabla \cdot m|^2\,{\rm d}x. $$
The topological structure of this two-dimensional problem allows us to use a duality argument to infer the optimal lower bound. The lower bound relies on an ε-perturbation of the following logarithmically failing interpolation inequality
$$ \int |\nabla^{1/2}\phi|^2 \, {\rm d}x \, \not\lesssim \, {\rm sup} |\phi| \, \int |\nabla \phi| \, {\rm d}x. $$

Keywords

Micromagnetics Thin films Néel wall 

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References

  1. 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)MATHGoogle Scholar
  2. 2.
    Alberti, G., Bouchitté, G., Seppecher, P.: Phase transition with the line-tension effect. Arch. Rational Mech. Anal. 144(1), 1–46 (1998)MATHCrossRefMathSciNetGoogle Scholar
  3. 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. LionsMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Bronstein, Semendjajew, Musiol, Mühlig, Taschenbuch der Mathematik. Verlag Harri DeutschGoogle Scholar
  5. 5.
    Cabré, Solà-Morales, Layer solutions in a halfspace for bondary reactions. to appearGoogle Scholar
  6. 6.
    Cervera, G.: Magnetic domains and magnetic domain walls. PhD thesis, New York University (1999)Google Scholar
  7. 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)MATHCrossRefMathSciNetGoogle Scholar
  8. 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)MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Garroni, A., Müller, S.: γ-limit of a phase-field model of dislocations. SIAM J. Math. Anal. 36(6), 1943–1964 (2005)Google Scholar
  10. 10.
    Holz, A., Hubert, A.: Phys. Stat. Sol. (B) 46, 377–384 (1971)CrossRefGoogle Scholar
  11. 11.
    Hubert, A., Schäfer, R.: Magnetic domains: The analysis of magnetic microstructures. Springer-Verlag (1998)Google Scholar
  12. 12.
    Jin, W., Kohn, R.V.: Singular perturbation and the energy of folds. J. Non-linear Sci. 10(3), 355–390 (2000)MATHMathSciNetGoogle Scholar
  13. 13.
    Kohn, Slastikov: Another thin-film limit of micromagnetics. submitted to Arch. Rat. Mech. Anal.Google Scholar
  14. 14.
    Koslowski, M., Cuiti~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)MATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Kurzke, M.: A nonlocal singular perturbation problem with periodic well potential. ESAIM Control. Optim. Calc. Var. 12(1), 52–63 (2006)Google Scholar
  16. 16.
    Melcher, C.: Logarithmic lower bounds for Néel walls. Calc. Var. Partial Differential Equations 21(2), 209–219 (2004)Google Scholar
  17. 17.
    Melcher, C.: Micromagnetic treatment of Néel walls. Arch. Rat. Mech. 168, 83–113 (2003)MATHMathSciNetCrossRefGoogle Scholar
  18. 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)Google Scholar
  19. 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)CrossRefGoogle Scholar
  20. 20.
    Verhulst, F.: Nonlinear Differential Equations and Dynamical Systems. Springer-Verlag (1989)Google Scholar

Copyright information

© Springer-Verlag 2006

Authors and Affiliations

  • Antonio DeSimone
    • 1
  • Hans Knüpfer
    • 2
  • Felix Otto
    • 2
  1. 1.SISSA, International School for Advanced MathematicsTriesteItaly
  2. 2.Institute of Applied MathematicsUniversity of BonnBonnGermany

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