A viscosity property of minimizing micromagnetic configurations. (English) Zbl 1121.35309

Summary: We study the limit as \(\varepsilon\downarrow 0\) of the minimizers of a singularly perturbed problem arising in micromagnetics. Using a sign condition and a kinetic interpretation of the limit problem we show that limiting vector fields are, after a rotation, gradients of viscosity solutions of the eikonal equation. This leads to a characterization of limiting configurations, once boundary conditions are imposed. This solves a problem left open in previous papers by S. Serfaty and T.Rivière [Commun. Pure Appl. Math. 54, No. 3, 294–338 (2001; Zbl 1031.35142), Commun. Partial Differ. Equations 28, No. 1–2, 249–269 (2003; Zbl 1094.35125)].


35J20 Variational methods for second-order elliptic equations
35B25 Singular perturbations in context of PDEs
35Q30 Navier-Stokes equations
58E50 Applications of variational problems in infinite-dimensional spaces to the sciences
82D40 Statistical mechanics of magnetic materials
Full Text: DOI


[1] Alouges, Néel walls and cross-tie walls for micromagnetic materials having a strong planar anisotropy, ESAIM Control Optim Calc Var 8 pp 31– (2002) · Zbl 1092.82047 · doi:10.1051/cocv:2002017
[2] Ambrosio, Calculus of variations and partial differential equations (2000) · doi:10.1007/978-3-642-57186-2
[3] Ambrosio, Line energies for gradient vector fields in the plane, Calc Var Partial Differential Equations 9 (4) pp 327– (1999) · Zbl 0960.49013 · doi:10.1007/s005260050144
[4] Aviles , P. Giga , Y. A mathematical problem related to the physical theory of liquid crystal configurations Miniconference on geometry and partial differential equations, 2 (Canberra, 1986) 1987
[5] Aviles, On lower semicontinuity of a defect energy obtained by a singular limit of the Ginzburg-Landau type energy for gradient fields, Proc Roy Soc Edinburgh Sect A 129 (1) pp 1– (1999) · Zbl 0923.49008 · doi:10.1017/S0308210500027438
[6] Crandall, Some properties of viscosity solutions of Hamilton-Jacobi equations, Trans Amer Math Soc 282 (2) pp 487– (1984) · Zbl 0543.35011 · doi:10.1090/S0002-9947-1984-0732102-X
[7] DeSimone , A. Müller , S. Kohn , R. V. Otto , F. Magnetic microstructures-a paradigm of multiscale problems 175 190 Oxford University Oxford 2000 · Zbl 0991.82038
[8] DeSimone, A compactness result in the gradient theory of phase transitions, Proc Roy Soc Edinburgh Sect A 131 (4) pp 833– (2001) · Zbl 0986.49009 · doi:10.1017/S030821050000113X
[9] Jabin, Compactness in Ginzburg-Landau energy by kinetic averaging, Comm Pure Appl Math 54 (9) pp 1096– (2001) · Zbl 1124.35312 · doi:10.1002/cpa.3005
[10] Jin, Singular perturbation and the energy of folds, J Nonlinear Sci 10 (3) pp 355– (2000) · Zbl 0973.49009 · doi:10.1007/s003329910014
[11] Lecumberry , M. The structure of micromagnetic defects · Zbl 1172.35501
[12] Lecumberry, Regularity for micromagnetic configurations having zero jump energy, Calc Var Partial Differential Equations (2002) · Zbl 1021.35023 · doi:10.1007/s005260100132
[13] Rivière, Limiting domain wall energy for a problem related to micromagnetics, Comm Pure Appl Math 54 (3) pp 294– (2001) · Zbl 1031.35142 · doi:10.1002/1097-0312(200103)54:3<294::AID-CPA2>3.0.CO;2-S
[14] Rivière, Compactness, kinetic formulation and entropies for a problem related to micromagnetics, Comm Partial Differential Equations (2002)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.