The Jacobian and the Ginzburg-Landau energy

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.