We study a two-dimensional model for micromagnetics, which consists in an energy functional over ${S}^{2}$-valued vector fields. Bounded-energy configurations tend to be planar, except in small regions which can be described as vortices (Bloch lines in physics). As the characteristic “exchange-length” tends to 0, they converge to planar divergence-free unit norm vector fields which jump along line singularities. We derive lower bounds for the energy, which are explicit functions of the jumps of the limit. These lower bounds are proved to be optimal and are achieved by one-dimensional profiles, corresponding to Néel walls, if the jump is small enough (less than $\pi /2$ in angle), and by two-dimensional profiles, corresponding to cross-tie walls, if the jump is bigger. Thus, it provides an example of a vector-valued phase-transition type problem with an explicit non-one-dimensional energy-minimizing transition layer. We also establish other lower bounds and compactness properties on different quantities which provide a good notion of convergence and cost of vortices.

Keywords: micromagnetics, thin films, cross-tie walls, gamma-convergence

@article{COCV_2002__8__31_0, author = {Alouges, Fran\c{c}ois and Rivi\`ere, Tristan and Serfaty, Sylvia}, title = {N\'eel and {Cross-Tie} wall energies for planar micromagnetic configurations}, journal = {ESAIM: Control, Optimisation and Calculus of Variations}, pages = {31--68}, publisher = {EDP-Sciences}, volume = {8}, year = {2002}, doi = {10.1051/cocv:2002017}, mrnumber = {1932944}, zbl = {1092.82047}, language = {en}, url = {http://archive.numdam.org/articles/10.1051/cocv:2002017/} }

TY - JOUR AU - Alouges, François AU - Rivière, Tristan AU - Serfaty, Sylvia TI - Néel and Cross-Tie wall energies for planar micromagnetic configurations JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2002 SP - 31 EP - 68 VL - 8 PB - EDP-Sciences UR - http://archive.numdam.org/articles/10.1051/cocv:2002017/ DO - 10.1051/cocv:2002017 LA - en ID - COCV_2002__8__31_0 ER -

%0 Journal Article %A Alouges, François %A Rivière, Tristan %A Serfaty, Sylvia %T Néel and Cross-Tie wall energies for planar micromagnetic configurations %J ESAIM: Control, Optimisation and Calculus of Variations %D 2002 %P 31-68 %V 8 %I EDP-Sciences %U http://archive.numdam.org/articles/10.1051/cocv:2002017/ %R 10.1051/cocv:2002017 %G en %F COCV_2002__8__31_0

Alouges, François; Rivière, Tristan; Serfaty, Sylvia. Néel and Cross-Tie wall energies for planar micromagnetic configurations. ESAIM: Control, Optimisation and Calculus of Variations, Volume 8 (2002), pp. 31-68. doi : 10.1051/cocv:2002017. http://archive.numdam.org/articles/10.1051/cocv:2002017/

[1] Asymptotic behavior of the Landau-Lifschitz model of ferromagnetism. Appl. Math. Optim. 23 (1991) 171-193. | MR | Zbl

, and ,[2] Line energies for gradient vector fields in the plane. Calc. Var. Partial Differential Equation 9 (1999) 327-355. | MR | Zbl

, and ,[3] A mathematical problem related to the physical theory of liquid crystals configurations. Proc. Centre Math. Anal. Austral. Nat. Univ. 12 (1987) 1-16. | MR

and ,[4] On lower semicontinuity of a defect obtained by a singular limit of the Ginzburg-Landau type energy for gradient fields. Proc. Royal Soc. Edinburgh Sect. A 129 (1999) 1-17. | MR | Zbl

and ,[5] A Viscosity Property of Minimizing Micromagnetic Configurations. Preprint. | MR | Zbl

, and ,[6] On nematics stabilized by a large external field. Rev. Math. Phys. 11 (1999) 653-710. | MR | Zbl

and ,[7] Ginzburg-Landau vortices. Birkhauser (1994). | MR | Zbl

, and ,[8] Density of smooth functions between two manifolds in Sobolev spaces. J. Func. Anal. 80 (1988) 60-75. | MR | Zbl

and ,[9] Micromagnetics. Wiley, New York (1963).

,[10] Regularity for critical points of a non local energy. Calc. Var. Partial Differential Equation 5 (1997) 409-433. | MR | Zbl

,[11] A $\Gamma $-convergence result for the two-gradient theory of phase transitions. Preprint. | MR | Zbl

, and ,[12] Energy minimizers for large ferromagnetic bodies. Arch. Rational Mech. Anal. 125 (1993) 99-143. | MR | Zbl

,[13] Minima absolus pour des énergies ferromagnétiques. C. R. Acad. Sci. Paris 331 (2000) 497-500. | MR | Zbl

and ,[14] A compactness result in the gradient theory of phase transitions. Proc. Roy. Soc. Edinburgh 131 (2001) 833-844. | MR | Zbl

, , and ,[15] Magnetic microstructures, a paradigm of multiscale problems. Proceedings of ICIAM. | Zbl

, , and ,[16] A reduced theory for thin-film micromagnetics. Preprint (2001). | MR | Zbl

, , and ,[17]

, , and (in preparation).[18] Measure theory and fine properties of functions. CRC Press, Boca Raton, FL, Stud. Adv. Math. (1992). | MR | Zbl

and ,[19] Some regularity results in ferromagnetism. Comm. Partial Differential Equation 25 (2000) 1235-1258. | MR | Zbl

and ,[20]

, Ph.D. Thesis. Courant Institute (2001).[21] Magnetic Domains. Springer (1998).

and ,[22] Singular Perturbation and the Energy of Folds. J. Nonlinear Sci. 10 (2000) 355-390. | MR | Zbl

and ,[23] Frustration in ferromagnetic materials, Continuum Mech. Thermodynamics 2 (1990) 215-239. | MR

and ,[24] Ginzburg-Landau line energies: The zero-energy case (to appear).

, , and ,[25] Compactness in Ginzburg-Landau energy by kinetic averaging. Comm. Pure Appl. Math. 54 (2001) 1096-1109. | Zbl

and ,[26] Regularity for micromagnetic configurations having zero jump energy. Calc. Var. Partial Differential Equations (to appear). | MR | Zbl

and ,[27] $\Gamma $-convergenza di una famiglia di funzionali ellittici. Boll. Un. Mat. Ital. A (5) 14 (1977) 526-529. | MR | Zbl

and Mortola, Il limite nella[28] Direct solution of the Landau-Lifshitz-Gilbert equation for micromagnetics. Japanese J. Appl. Phys. 28 (1989) 2485-2507.

, and ,[29] The Magnetic Ground State of a Thin-Film Element. IEEE Trans. Mag. 36 (2000) 3886-3899.

and ,[30] Limiting Domain Wall Energy for a Problem Related to Micromagnetics. Comm. Pure Appl. Math. 54 (2001) 294-338. | MR | Zbl

and ,[31] Compactness, kinetic formulation and entropies for a problem related to micromagnetics. Comm. in Partial Differential Equations (to appear). | MR | Zbl

and ,[32] On Landau-Lifschitz equations for ferromagnetism, Japanese J. Appl. Math. 2 (1985) 69-84. | Zbl

,[33]

, preprint (1999) and habilitation thesis. University of Tours (2000).[34] The effect of a singular perturbation on nonconvex variational problems. Arch. Rational Mech. Anal. 101 (1988) 209-260. | MR | Zbl

,[35] Self-consistent domain theory in soft micromagnetic media, II, Basic domain structures in thin film objects. J. Appl. Phys. 60 (1986) 1104-1113.

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