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Ferromagnetic thin multi-structures

In this paper, starting from the classical 3D non-convex and nonlocal micromagnetic energy for ferromagnetic materials, we determine, via an asymptotic analysis, the free energy of a multi-structure consisting of a nano-wire in junction with a thin film and of a multi-structure consisting of two joined nano-wires. We assume that the volumes of the two parts composing each multi-structure vanish with same rate. In the first case, we obtain a 1D limit problem on the nano-wire and a 2D limit problem on the thin film, and the two limit problems are uncoupled. In the second case, we obtain two 1D limit problems coupled by a junction condition on the magnetization. In both cases, the limit problem remains non-convex, but now it becomes completely local.