Refined jacobian estimates for Ginzburg-Landau functional

Robert Jerrard, Daniel Spirn

Research output: Contribution to journalArticlepeer-review

17 Scopus citations

Abstract

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.

Original languageEnglish (US)
Pages (from-to)135-186
Number of pages52
JournalIndiana University Mathematics Journal
Volume56
Issue number1
DOIs
StatePublished - 2007

Keywords

  • Gamma convergence
  • Ginzburg-Landau functional
  • Jacobian

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