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Domain branching in uniaxial ferromagnets: A scaling law for the minimum energy. (English) Zbl 1023.82011

Summary: We address the branching of magnetic domains in a uniaxial ferromagnet. Our thesis is that branching is required by energy minimization. To show this, we consider the nonlocal, nonconvex variational problem of micromagnetics. We identify the scaling law of the minimum energy by proving a rigorous lower bound which matches the already known upper bound. We further show that any domain pattern achieving this scaling law must have average width of order \(L^{2/3}\), where \(L\) is the length of the magnet in the easy direction. Finally, we argue that branching is required, by considering the constrained variational problem in which branching is prohibited and the domain structure is invariant in the easy direction. Its scaling law is different.

MSC:

82D40 Statistical mechanics of magnetic materials
49N90 Applications of optimal control and differential games
58E50 Applications of variational problems in infinite-dimensional spaces to the sciences
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