## Compactness, kinetic formulation, and entropies for a problem related to micromagnetics.(English)Zbl 1094.35125

Summary: We carry on the study of a former paper in [Commun. Pure Appl. Math. 54, No. 3, 294–338 (2001; Zbl 1031.35142)] on the asymptotics of a family of energy-functionals related to micromagnetics. We prove compactness for families of uniformly bounded energies releasing the LBP condition we had previously set. Such families converge to unit-valued divergence-free vector-fields that are tangent to the boundary of the domain, and we found in the paper cited above that the energy-functionals $$\Gamma$$-converge to a limiting jump-energy of such configurations. We examine the behavior of certain truncated fields which serve to construct “entropies,” and to provide an improved lower bound. We give a kinetic formulation of the problem, and show that the limiting divergence-free problem is supplemented, in the case of minimizers, with a sign condition which can in turn, using the kinetic formulation, be interpreted as an entropy condition that plays a role in uniqueness questions.

### MSC:

 35Q60 PDEs in connection with optics and electromagnetic theory 35A15 Variational methods applied to PDEs 35B27 Homogenization in context of PDEs; PDEs in media with periodic structure 35J35 Variational methods for higher-order elliptic equations 82D40 Statistical mechanics of magnetic materials

Zbl 1031.35142
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