IUMJ

Title: Refined Jacobian estimates for Ginzburg-Landau functionals

Authors: Robert Jerrard and Daniel Spirn

Issue: Volume 56 (2007), Issue 1, 135-186

Abstract:

We prove various estimates that relate the Ginzburg-Landau energy $E_{\epsilon}(u) = \int_{\Omega} |\nabla u|^2 /2 + (|u|^2 - 1)^2 /(4\epsilon^2) dx$ of a function $u \in H^1(\Omega; \mathbb{R}^2)$, $\Omega \subset \mathbb{R}^2$, to the distance in the $W^{-1,1}$ norm between the Jacobian $J(u) = \det\nabla u$ and a sum of point masses. These are interpreted as quantifying the precision with which "vortices" in a function $u$ can be located via measure-theoretic tools such as the Jacobian, and the extent to which variations in the Ginzburg-Landau energy due to translation of vortices can be detected using the Jacobian. We give examples to show that some of our estimates are close to optimal.