Hostname: page-component-7bb8b95d7b-w7rtg Total loading time: 0 Render date: 2024-09-18T23:59:18.824Z Has data issue: false hasContentIssue false

Magnetic clusters and fold energies

Published online by Cambridge University Press:  24 July 2008

Yoshikazu Giga
Affiliation:
Department of Mathematics, Hokkaido University, Sapporo 060-0810, Japan (tonegawa@math.sci.hokudai.ac.jp)
Motohiko Kubo
Affiliation:
Department of Mathematics, Hokkaido University, Sapporo 060-0810, Japan (tonegawa@math.sci.hokudai.ac.jp)
Yoshihiro Tonegawa
Affiliation:
Department of Mathematics, Hokkaido University, Sapporo 060-0810, Japan (tonegawa@math.sci.hokudai.ac.jp)

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
Copyright
2007 Royal Society of Edinburgh

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)