The Jacobian and the Ginzburg-Landau energy |
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Authors: | Robert L Jerrard Halil Mete Soner |
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Institution: | Department of Mathematics, University of Illinois, Urbana, IL 61801, USA (e-mail: rjrrard@math.uiuc.edu), US Department of Mathematics, Ko? University, Istanbul, Turkey (e-mail: msoner@ku.edu.tr), TR
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Abstract: | We study the Ginzburg-Landau functional
for , where U is a bounded, open subset of . We show that if a sequence of functions satisfies , then their Jacobians are precompact in the dual of for every . Moreover, any limiting measure is a sum of point masses. We also characterize the -limit of the functionals , in terms of the function space B2V introduced by the authors in 16,17]: we show that I(u) is finite if and only if , and for is equal to the total variation of the Jacobian measure Ju. When the domain U has dimension greater than two, we prove if then the Jacobians are again precompact in for all , and moreover we show that any limiting measure must be integer multiplicity rectifiable. We also show that the total variation
of the Jacobian measure is a lower bound for the limit of the Ginzburg-Landau functional.
Received: 15 December 2000 / Accepted: 23 January 2001 / Published online: 25 June 2001 |
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Keywords: | Mathematics Subject Classification (2000): 35J50 35Q80 |
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