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Magnetic clusters and fold energies

Published online by Cambridge University Press:  24 July 2008

Yoshikazu Giga ,
Motohiko Kubo  and
Yoshihiro Tonegawa
Yoshikazu Giga
Affiliation:
Department of Mathematics, Hokkaido University, Sapporo 060-0810, Japan ([email protected])
Motohiko Kubo
Affiliation:
Department of Mathematics, Hokkaido University, Sapporo 060-0810, Japan ([email protected])
Yoshihiro Tonegawa
Affiliation:
Department of Mathematics, Hokkaido University, Sapporo 060-0810, Japan ([email protected])
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Abstract

We are concerned with variational properties of a fold energy for a unit, dilation-invariant gradient field (called a cluster) in the unit disc. We show that boundedness of a fold energy implies $L^{1}$-compactness of clusters. We also show that a fold energy is $L^{1}$-lower semicontinuous. We characterize absolute minimizers. We also give a sequence of stationary states and discuss its stability. Surprisingly, the stability depends upon $q$, the power of modulus of the jump discontinuities, in the definition of the fold energy.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 137 , Issue 1 , February 2007 , pp. 23 - 40
Copyright
2007 Royal Society of Edinburgh

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