A Duality Method for Micromagnetics

We present a new method for micromagnetics based on replacing the nonlocal total energy of magnetizations by a new local energy for divergence-free fields and then studying the dual Legendre functional of this new energy restricted on gradient fields. We establish a Fenchel-type duality principle relevant to the minimization for these problems. The dual functional may be written as a convex integral functional of gradients, and its minimization problem will be solved by standard minimization procedures in the calculus of variations. Special emphasis is placed on the analysis of existence/nonexistence, depending on the applied field and the physical domain. In particular, we describe a precise procedure to check the existence of magnetization of minimal energy for ellipsoid domains.

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