Abstract.
We study the Ginzburg-Landau functional
\(I_{\epsilon}(u) := \frac{1}{\ln(1/\epsilon)}\int_U \frac12 |\nabla u|^2 + \frac{1}{4 \epsilon^2} (1-|u|^2)^2 dx ,\)
for \(u\in H^1(U; \R^2)\), where U is a bounded, open subset of \(\mathbb{R}{2}\). We show that if a sequence of functions \(u^\epsilon\) satisfies \(\sup I_{\epsilon}(u^\epsilon)< \infty\), then their Jacobians \(Ju^\epsilon\) are precompact in the dual of \(C^{0,\alpha}_c\) for every \(\alpha \in (0,1]\). Moreover, any limiting measure is a sum of point masses. We also characterize the \(\Gamma\)-limit \(I(\cdot)\) of the functionals \(I_{\epsilon}(\cdot)\), 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 \(u \in B2V(U;S^1)\), and for \( u \in B2V(U;S^1), I(u)\) is equal to the total variation of the Jacobian measure Ju. When the domain U has dimension greater than two, we prove if \(I_\epsilon(u^\epsilon) \le C\) then the Jacobians \(Ju^\epsilon\) are again precompact in \(\left(C^{0,\alpha}_c\right)^*\) for all \(\alpha\in (0,1]\), 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 \(\Gamma\) limit of the Ginzburg-Landau functional.
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Received: 15 December 2000 / Accepted: 23 January 2001 / Published online: 25 June 2001
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Jerrard, R., Soner, H. The Jacobian and the Ginzburg-Landau energy. Calc Var 14, 151–191 (2002). https://doi.org/10.1007/s005260100093
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DOI: https://doi.org/10.1007/s005260100093