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On two-dimensional ferromagnetism

Published online by Cambridge University Press:  26 May 2009

Pablo Pedregal
Affiliation:
ETSI Industriales, Universidad de Castilla-La Mancha, 13071 Ciudad Real, Spain ([email protected])
Baisheng Yan
Affiliation:
Department of Mathematics, Michigan State University, East Lansing, MI 48824, USA ([email protected])

Abstract

We present a new method for solving the minimization problem in ferromagnetism. Our method is based on replacing the non-local non-convex total energy of magnetization by a new local non-convex energy of divergence-free fields. Such a general method works in all dimensions. However, for the two-dimensional case, since the divergence-free fields are equivalent to the rotated gradients, this new energy can be written as an integral functional of gradients and hence the minimization problem can be solved by some recent non-convex minimization procedures in the calculus of variations. We focus on the two-dimensional case in this paper and leave the three-dimensional situation to future work. Special emphasis is placed on the analysis of the existence/non-existence depending on the applied field and the physical domain.

Type
Research Article
Copyright
Copyright © Royal Society of Edinburgh 2009

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