A compactness result for Landau state in thin-film micromagnetics

Ignat, Radu; Otto, Felix

Annales de l'I.H.P. Analyse non linéaire, Tome 28 (2011) no. 2, pp. 247-282

We deal with a nonconvex and nonlocal variational problem coming from thin-film micromagnetics. It consists in a free-energy functional depending on two small parameters ε and η and defined over vector fields $m:\Omega \subset {ℝ}^2\to S^2$ that are tangent at the boundary ∂Ω. We are interested in the behavior of minimizers as $ϵ,\eta \to 0$. They tend to be in-plane away from a region of length scale ε (generically, an interior vortex ball or two boundary vortex balls) and of vanishing divergence, so that $S^1$-transition layers of length scale η (Néel walls) are enforced by the boundary condition. We first prove an upper bound for the minimal energy that corresponds to the cost of a vortex and the configuration of Néel walls associated to the viscosity solution, so-called Landau state. Our main result concerns the compactness of vector fields ${\left\{m_{ϵ,\eta }\right\}}_{ϵ,\eta \downarrow 0}$ of energies close to the Landau state in the regime where a vortex is energetically more expensive than a Néel wall. Our method uses techniques developed for the Ginzburg–Landau type problems for the concentration of energy on vortex balls, together with an approximation argument of $S^2$-vector fields by $S^1$-vector fields away from the vortex balls.

Classification : 49S05, 82D40, 35A15, 35B25
Keywords: Compactness, Singular perturbation, Vortex, Néel wall, Micromagnetics, Ginzburg–Landau energy

[1] Fabrice Béthuel, Haïm Brezis, Frédéric Hélein, Ginzburg–Landau Vortices, Progr. Nonlinear Differential Equations Appl. vol. 13, Birkhäuser Boston Inc., Boston, MA (1994) | MR | Zbl

[2] H. Brezis, L. Nirenberg, Degree theory and BMO. I. Compact manifolds without boundaries, Selecta Math. (N.S.) 1 no. 2 (1995), 197-263 | MR | Zbl

[3] Robert Dautray, Jacques-Louis Lions, Analyse mathématique et calcul numérique pour les sciences et les techniques, INSTN: Collection Enseignement vol. 6, Masson, Paris (1988) | MR | Zbl

[4] Antonio Desimone, Hysteresis and imperfection sensitivity in small ferromagnetic particles, Microstructure and Phase Transitions in Solids Udine, 1994 Meccanica 30 no. 5 (1995), 591-603 | MR | Zbl

[5] Antonio Desimone, Hans Knüpfer, Felix Otto, 2-D stability of the Néel wall, Calc. Var. Partial Differential Equations 27 (2006), 233-253 | MR | Zbl

[6] Antonio Desimone, Robert V. Kohn, Stefan Müller, Felix Otto, A reduced theory for thin-film micromagnetics, Comm. Pure Appl. Math. 55 (2002), 1408-1460 | MR | Zbl

[7] Antonio Desimone, Robert V. Kohn, Stefan Müller, Felix Otto, Recent analytical developments in micromagnetics, Giorgio Bertotti, Isaak Mayergoyz (ed.), The Science of Hysteresis, vol. 2, Elsevier, Academic Press (2005), 269-381 | MR | Zbl

[8] Radu Ignat, A Γ-convergence result for Néel walls in micromagnetics, Calc. Var. Partial Differential Equations 36 no. 2 (2009), 285-316 | MR | Zbl

[9] Radu Ignat, A survey of some new results in ferromagnetic thin films, Séminaire: Équations aux Dérivées Partielles, 2007–2008, Sémin. Équ. Dériv. Partielles, École Polytech., Palaiseau (2009) | MR | EuDML | Zbl | Numdam

[10] Radu Ignat, Hans Knüpfer, Vortex energy and 360°-Néel walls in thin-film micromagnetics, Comm. Pure Appl. Math. 63 (2010), 1677-1724 | MR | Zbl

[11] Radu Ignat, Felix Otto, A compactness result in thin-film micromagnetics and the optimality of the Néel wall, J. Eur. Math. Soc. (JEMS) 10 no. 4 (2008), 909-956 | MR | EuDML | Zbl

[12] Robert L. Jerrard, Lower bounds for generalized Ginzburg–Landau functionals, SIAM J. Math. Anal. 30 no. 4 (1999), 721-746 | MR | Zbl

[13] Robert V. Kohn, Valeriy V. Slastikov, Another thin-film limit of micromagnetics, Arch. Ration. Mech. Anal. 178 no. 2 (2005), 227-245 | MR | Zbl

[14] Matthias Kurzke, Boundary vortices in thin magnetic films, Calc. Var. Partial Differential Equations 26 no. 1 (2006), 1-28 | MR | Zbl

[15] Fang Hua Lin, Vortex dynamics for the nonlinear wave equation, Comm. Pure Appl. Math. 52 (1999), 737-761 | MR

[16] Roger Moser, Ginzburg–Landau vortices for thin ferromagnetic films, AMRX Appl. Math. Res. Express 1 (2003), 1-32 | MR | Zbl

[17] Felix Otto, Cross-over in scaling laws: a simple example from micromagnetics, Proceedings of the International Congress of Mathematicians, vol. III, Beijing, 2002, Higher Ed. Press, Beijing (2002), 829-838 | MR | Zbl

[18] Etienne Sandier, Lower bounds for the energy of unit vector fields and applications, J. Funct. Anal. 152 (1998), 379-403 | MR | Zbl

[19] Etienne Sandier, Sylvia Serfaty, Vortices in the Magnetic Ginzburg–Landau Model, Progr. Nonlinear Differential Equations Appl. vol. 70, Birkhäuser Boston Inc., Boston, MA (2007) | MR | Zbl