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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)].

MSC:

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
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