We prove various estimates that relate the Ginzburg-Landau energy E ε (u) = ∫Ω | ∇u| 2/2 + (|u|2 - 1)2/(4ε2)dx of a function u ε H1(Ωℝ2), Ω ∪ ℝ2, to the distance in the W-1,1 norm between the Jacobian J(u) = det ∇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.
- Gamma convergence
- Ginzburg-Landau functional